A Fuzzy-Random Extension of the Lee–Carter Mortality Prediction Model

Jorge de Andrés-Sánchez; Laura González-Vila Puchades

doi:10.2991/ijcis.d.190626.001

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Volume 12, Issue 2, 2019, Pages 775 - 794

A Fuzzy-Random Extension of the Lee–Carter Mortality Prediction Model

Authors

Jorge de Andrés-Sánchez¹^{, *}, Laura González-Vila Puchades²

¹Social and Business Research Laboratory, Business Management Department, Rovira i Virgili University, Av. Universitat 1, 43204 Reus, Spain

²Department of Mathematics for Economics, Finance and Actuarial Science, University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain

^*Corresponding author. Email: jorge.deandres@urv.cat

Corresponding Author

Jorge de Andrés-Sánchez

Received 12 September 2018, Accepted 17 June 2019, Available Online 18 July 2019.

DOI: 10.2991/ijcis.d.190626.001 How to use a DOI?
Keywords: Lee–Carter model; Fuzzy numbers; Fuzzy regression; Fuzzy-random modeling
Abstract: The Lee–Carter model is a useful dynamic stochastic model to represent the evolution of central mortality rates throughout time. This model only considers the uncertainty about the coefficient related to the mortality trend over time but not to the age-dependent coefficients. This paper proposes a fuzzy-random extension of the Lee–Carter model that allows quantifying the uncertainty of both kinds of parameters. As it is commonplace in actuarial literature, the variability of the time-dependent index is modeled as an ARIMA time series. Likewise, the uncertainty of the age-dependent coefficients is also quantified, but by using triangular fuzzy numbers. The consideration of this last hypothesis requires developing and solving a fuzzy regression model. Once the fuzzy-random extension has been introduced, it is also shown how to obtain some variables linked with central mortality rates such as death probabilities or life expectancies by using fuzzy numbers arithmetic. It is simultaneously shown the applicability of our developments with data of Spanish male population in the period 1970–2012. Finally we make a comparative assessment of our method with alternative Lee–Carter model estimates on 16 Western Europe populations.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Classical actuarial methods graduate mortality by only taking into account the age of persons without calendar year considerations. Due to the progressive increase of life expectancy in all developed countries, this kind of methods systematically overestimate the mortality rates and, as a consequence, may increase the longevity risk when pricing life annuities.

In the last decades of the 20th century, several papers developed dynamic stochastic approaches for the evolution of mortality rates throughout calendar time and, so, projecting mortality to the future with these models became more accurate. In this way, the method in [1], that we will name Lee–Carter (LC), is one of the most extended methodologies. The LC model proposed adjusting a linear function to the logarithm of central mortality rates of each year and age, mx,t. The coefficients of the linear function depend on the age x whereas the independent variable is a nonobserved intensity index kt associated to the time calendar t. Once the parameters of the model have been adjusted, to make predictions on mortality dynamics it is necessary projecting kt to the future. It is commonly made with an ARIMA model.

There are two main reasons why the LC model boasts great acceptance. On the one hand, it has been applied in many countries with good results [2–10]. Likewise, the LC method is relatively easy to compute in its seminal version, either by using the singular value decomposition (SVD) method or with the approximation to the SVD solution suggested in [1].

Several papers proposed technical extensions to the original LC model as [3,5–7,11–13]. All these extensions have two common features. Firstly, more completeness and sophistication of the model suppose more computational effort. Secondly, all of them consider that the age-specific historical influences not captured by the model are due to a stochastic error-term, as the LC model does. However, stochastic variability may not be the unique source of uncertainty since it can also come from fuzziness (e.g., due to incomplete or imprecise information) and, consequently, it can be modeled with Fuzzy Sets Theory tools. In this way, [14] developed two alternative fuzzy formulations of the LC model. The first model considers that all the parameters are fuzzy numbers (FNs) and arithmetical operations are carried out by means of the weakest t-norm. This first approach was object of several refinements in [15,16]. In the second approach, the centers and spreads of the FNs that estimate the parameters of the LC model are supposed to be random variables (RVs) and are estimated with Bayesian methods. The comparison between the fuzzy and the fuzzy-stochastic models seems to show very similar results, but the second model requires much more computational effort.

Mixing fuzziness and randomness in actuarial modeling is not new. [17] described fuzzy RVs with actuarial applications in view and [18] developed a non-life individual risk model where the number of claims follows a Poisson process and their amount is estimated with a triangular fuzzy number (TFN). In a life insurance context, [19,20] used fuzzy RVs for the valuation of life contingencies.

This paper also blends fuzziness and randomness and proposes a fuzzy-random approach of the LC model which is conceptually different to those developed in [14]. We consider that the behavior of the independent variable kt follows an ARIMA stochastic process. Likewise, we assume that the variability of the age-specific coefficients is due to fuzziness and it is captured by means of FNs. Under these hypotheses, for a given outcome of the RV kt, we will have a concrete result of the central mortality rate, mx,t, which will be given by an FN.

In order to adjust the fuzzy coefficients of the logarithm of mx,t, we use the model of fuzzy regression (FR) developed by [21], that mixes ordinary least squares (OLSs) regression, and the FR method by [22], but also allows a nonsymmetrical shape for the coefficients.

The rest of the paper is organized as follows: We firstly make a brief review of the LC model. Then, we describe some concepts of FNs and FR that will be necessary to develop our work. Section 4 features our fuzzy-random extension to the LC model, exposes how to project future mortality from this formulation, and shows an empirical application for Spanish male population, evaluating both the capability of the model to adjust central mortality rates to sample data and to predict out-of-sample data. In the fifth section, we show how some mortality tables variables can be obtained from the results of previous sections. Subsequently, focused on life expectancies of Spanish male population, we test the capability of our model to predict future values. Section 6 includes a complete comparison of the predictive performance of the proposed method with both the basic LC method in [1] and the fuzzy extension by Koissi and Shapiro in [14]. This comparison shows the advantages of our fuzzy-random extension of the LC model. We finish the work by pointing out the main conclusions and suggesting possible extensions.

2. OVERVIEW OF THE LEE–CARTER MODEL

Lee and Carter in [1] proposed modeling the logarithm of the central death rate for each specific age and each year with a linear function. In such a way, if mx,t is the central death rate of a person aged x in the calendar year t, the Lee–Carter (LC) model considers

(1)lnmx,t=ax+bxkt+εx,t

or, equivalently

(2)mx,t=expax+bxkt+εx,t

where

expax is the specific value of the central mortality rate at age x regardless of the time calendar t,
bx quantifies the sensitivity of the central death logarithm rate for age x in year t respect to changes in kt d ln(mx,t)dt=bxdktdt,
kt is a specific mortality index for each year t that represents the trend of the mortality across time,
εx,t is a random error term, with mean 0 and standard deviation σε, which reflects particular age-specific historical influences not captured by the model.

Notice that whereas the parameters ax and bx are age-dependent, the parameter kt is time-dependent. To estimate the model for a given matrix of rates mx,t, the authors seek the least squares solution to the Equation (1). This model is undetermined since, given a solution ax,bx,kt, any transformation of the type ax,bx/c,ckt or ax+cbx,bx,kt−c, ∀c∈ℜ, is also a solution. In order to avoid this issue [1] introduced the constraints ∑xbx=1 and ∑tkt=0. So, the estimations of ax are simply the averages over time of lnmx,t, that is,

(3a)ax*=∑t=0Tlnmx,tT+1

being the considered calendar years t=0,1,…,T and x=0,1,…,ω the different ages with ω the maximum attainable age.

The model Equation (1) cannot be fitted by ordinary regression techniques because on its right side there are two parameters to be estimated but kt is unobservable. However, [1] showed that a least squares solution can be obtained by applying SVD to the matrix Zx,t=lnmx,t−ax*. Alternatively, in the Appendix A of [1], it is also proposed an approximation to the SVD solution. As the parameters bx are assumed to follow the constraint ∑xbx=1, the parameter kt, for each t=0,1,…,T, can be fitted as

(3b)kt*=∑xlnmx,t−∑xax*

And finally, each bx can be found by regressing through OLSs the linear model Zx,t=bxkt+εx,t. So

(3c)bx*=∑tZx,tkt*∑tkt*2=∑tlnmx,t−ax*kt*∑tkt*2

Once the parameters of the model have been fitted, for t=0,1,…,T and x=0,1,…,ω, the trend of the mortality across time, k_t, for t=T+1,… can be forecasted with an ARIMA model and related variables, as confidence intervals for k_t, survival and mortality probabilities or life expectancies, can be obtained.

3. FNS AND FR

3.1. FNs and Their Arithmetic

This paper quantifies uncertain quantities as a common type of FN, TFNs, that will be symbolized as A˜=A,lA,rA being A the core of the TFN μA˜A=1 and lA and rA its left and right spreads, respectively. The α-cuts of this kind of FNs are closed and bounded intervals ∀α∈0,1

(4a)Aα=A_α,A¯α=A−lA1−α,A+rA1−α

The expected interval of a FN A˜, eIA˜, is a crisp interval that in the case of TFNs is

(4b)eIA˜=∫01A_αdα,∫01A¯αdα=A−lA2,A+rA2

Let f be a continuous real-valued function of n-real variables xj, j=1,2,…,n,. If xj are not crisp numbers, but the FNs A˜j, j = 1, 2, …, n, f induces the FN B˜ in such a way that B˜=fA˜1,A˜2,…,A˜n. In order to obtain the α-cuts of B˜, Bα, the results of [23] can be used. If the function f is increasing respect to the first m variables, m≤n, and decreasing respect to the last n−m variables

(5)Bα=[B_(α),B¯(α)] =[f(A_1(α),A_2(α),…,A_m(α),A¯m+1(α), A¯m+2(α),…A¯n(α), fA¯1(α),A¯2(α),…A¯m(α),A_m+1(α), A_m+2(α),…,A_n(α)

The result of evaluating nonlinear functions with TFNs is not a TFN. In this way, [24] proposed a TFN approximation for any real-valued function, derivable and increasing (decreasing) respect to the first (last) m (n − m) variables, built up from the first-order Taylor polynomial expansion from the 1-cut to any α-cut. It can be demonstrated that in Equation (5) B˜≈B,lB,rB where, naming the vector that comprises the centers of A˜j, j=1,2,…,n, AC=(A1,A2,…,An)

(6)B=fAClB=∑j=1m∂fAC∂xjlAj−∑j=m+1n∂fAC∂xjrAjrB=∑j=1m∂fAC∂xjrAj−∑j=m+1n∂fAC∂xjlAj

Arithmetic operations between real numbers can be extended to FNs by using the appropriate real-valued function. Since this work uses TFNs, when this function is linear the result of the arithmetic operation will also be a TFN. Otherwise, the result will be approximated by using Equation (6). So, it is obtained:

Addition:
B˜=∑j=1nA˜j=∑j=1nAj,∑j=1nlAj,∑j=1nrAj
Scalar multiplication:
B˜=kA˜=kA,klA,krAk≥0kA,|k|rA,|k|lAk<0
Product of two positive TFNs (i.e., their supports are contained within ℜ+):
B˜=A˜1⋅A˜2=A1A2,A2lA1+A1lA2,A2rA1+A1rA2
Division of two positive TFNs:
B˜=A˜1A˜2=A1A2,lA1A2+A1rA2A22,rA1A2+A1lA2A22
Exponential function:
B˜=expA˜ ≈expA,expAlA,expArA
Logarithmic function:
B˜=lnA˜ ≈lnA,lAA,rAA

3.2. FR Model with Asymmetric Coefficients

This paper uses the FR model of [21] that combines the least squares method with the minimum fuzziness principle in [22]. This type of FR method has been used in financial and actuarial applications like fitting options volatility smile [25,26] or calculating claim reserves [27]. In the actuarial field, a wide survey of FR models can be found in [28].

Let us suppose that for the j-th observation of the sample, j=0,1,…,n, the pair of the dependent variable (that may be a FN) and the independent variables (that we suppose crisp) is y˜j,xjj=1,2,…,n, where xj=x1,j,x2,j,…,xm,j, y˜j=yj,lyj,ryj, xi,j∈ℜ. Likewise, we suppose a linear relationship and also that the coefficients of the linear function are TFNs a˜i=ai,lai,rai,i=0,1,…,m. So

y˜j=a˜0+∑i=1ma˜ixi,j

and then

yj,lyj,ryj=a0,la0,ra0+∑i=1mai,lai,raixi,j

where

yj=a0+∑i=1maixi,j

lyj=la0+∑i=1xi,j≥0m|xi,j|lai+∑i=1xi,j<0n|xi,j|rai

ryj=ra0+∑i=1xi,j≥0m|xi,j|rai+∑i=1xi,j<0n|xi,j|lai

The final objective is obtaining the estimates of a˜i=ai,lai,rai, i=0,1,…m, that will be denoted by a˜i*=ai*,lai*,rai*. Following [21], we implement the following steps:

Step 1. By taking the centers of the dependent variable, yj, j=0,1,…,n, we fit the centers of the fuzzy coefficients a˜i*, ai, i=0,1,…m, by using OLS on the expression yj=a0+∑i=1maixi,j. In such a way, we obtain the estimates a0*,a1*,…,am*. To solve this step we can use Equations (3a–3c).

Step 2. We fit the spreads of parameters applying the minimum fuzziness criterion in [22]. So, spread estimates must minimize the uncertainty of the estimated outputs and simultaneously these estimated outputs have to contain the real observations, with a membership level of at least α. If we symbolize the estimates of the spreads as lai* and rai* i=0,1,…m, the estimated output for y˜j will be y˜j*=yj,*,lyj*,ryj*, where yj*=a0*+∑i=1mai*xi,j.

Considering, as in [22], that y˜j ⊆α y˜j*⇔yjα⊆yjα*, the spreads lai* and rai*, i=0,1,…m, minimize for a prefixed level α

minlai,raiz=∑j=1nlyj+∑j=1nryj

And accomplish the constraints

yjα⊆yjα*,j=1,2,…,n

(7)lai*,rai*≥0, i=0,1,…,m

[29] proposed a rule to choose α when the observations on inputs are crisp. α must reach a compromise between containing observed outputs in y˜j* reasonably well but, likewise, y˜j* must be narrow enough in order to be a useful prediction.

If we name as y˜j*α=yj*α,lyj*α,ryj*α the estimate of the jth observation of the dependent variable at a given α, we can define the credibility level cjα as

cjα=μy˜j*α(yj)lyj*α+ryj*α

Thus, the credibility for the entire sample c α is cα=∑j=0nμy˜j*αyjlyj*α+ryj*α. [29] showed that maximizing c α is equivalent to solve the following quadratic linear programming problem

max c α=−p0α2+p0−c0α,α∈0,1

where

p0=∑j=0n1−μy˜j*0yjlyj*0+ryj*0c0=∑j=0nμy˜j*0yjlyj*0+ryj*0

being the solution of this problem

(8)α=121−c0p0c0<p0 0otherwise

Therefore, the process that we follow to fit the fuzzy coefficients consists in implementing Step 2 for α=0. Once the spreads lai*0 and rai*0, i=0,1,…,m, have been obtained, the optimal value of α, α′, will be calculated by using the expression Equation (8). Following [30], the final value of lai* and rai*, i=0,1,…m is, simply

lai*=lai*α′=lai*01−α′ and rai*=rai*α′=rai*01−α′

4. FUZZY-RANDOM APPROACH OF THE LC MODEL

4.1. Fuzzy-Random Fitting of the LC Model

Our fuzzy-random approach of the LC model considers two different sources of uncertainty:

It is supposed that historical influences of each specific age are due to fuzziness in the model structure. As a consequence, both, the coefficient that describes the average age-specific pattern of mortality and the coefficient which reflects the variation in the central death across time, turn into FNs and so a˜x=ax,lax,rax and b˜x=bx,lbx,rbx.
The mortality index kt follows an ARIMA stochastic process, that is, kt is an outcome of a RV k_t.

Under these assumptions, once the average pattern of mortality, a˜x, and the decline in mortality, b˜x, have been estimated we can obtain for an outcome of the RV k_t, kt, the central rate of mortality (and its logarithm) as

(9a)lnm˜x,t=a˜x+B˜xkt

where

(9b)lnm˜x,t=lnmx,t,llnmx,t,rlnmx,t

being

(9c)lnmx,t=ax+bxkt

(9d)llnmx,t=lax+ktlbxfor kt>0lax−ktrbxfor kt≤0

(9e)rlnmx,t=rax+ktrbxfor kt>0rax−ktlbxfor kt≤0

In order to fit the estimates a˜x*=ax*,lax*,rax*, b˜x*=bx*,lbx*,rbx*, and kt*,t≤T, we follow the process described in Section 3.2 as follows:

Step 1. By taking the centers of lnm˜x,t, t=0,1,…,T and x=0,1,…,ω, we fit the centers of a˜x* and b˜x* and the outcomes of the RVs k_t, kt*, as we described in Section 2. In this step, it is necessary to point out that the observed values of the central rate of mortality (and its logarithms) in which we will base our work are crisp. So, lnm˜x,t=lnmx,t.

Step 2. We have to calculate the values lax*0,lbx*0, rax*0 and rbx*0 by solving the linear problem Equation (7) for α=0, that is,

minlax,lbx,rax,rbxT+1∑xlax+rax+∑t|kt| ∑xlbx+rbx

subject to

ax*+bx* kt*+rax+ktrbx≥lnmx,t for kt>0

ax*+bx* kt*+rax−ktlbx≥lnmx,t for kt≤0

ax*+bx* kt*−lax−ktlbx≤lnmx,t for kt>0

ax*+bx* kt*−lax+ktrbx≤lnmx,t for kt≤0

bx*−lbx≥0 if bx*≥0

bx*+rbx<0 if bx*<0

lax,lbx,rax,rbx≥0, x=0,1,2,…,ω

Let us remark that the constraints bx*−lbx≥0 and bx*+rbx<0, ensure that the estimate of the center of b˜x, b˜x*, will have clearly defined its sign. It will make easier to fit fuzzy-probabilistic confidence intervals for out-of-sample predictions.

Step 3. We obtain the optimal value α′ from Equation (8). Finally, the spreads lax*,lbx*, rax* and rbx* are obtained, simply, dividing lax*0,lbx*0, rax*0 and rbx*0 by 1−α′.

By using a˜x*, b˜x*, and kt*, it is possible to have a fuzzy estimate for the observed central mortality rates. In fact, from Equations (9a–9e) we obtain

(10a)lnm˜x,t*=lnmx,t*,llnmx,t*,rlnmx,t* =ax*,lax*,rax*+kt*bx*,lbx*,rbx*

where

(10b)lnmx,t*=ax*+bx*kt*

(10c)llnmx,t*=lax*+kt*lbx*for kt*>0lax*−kt*rbx*for kt≤0

(10d)rlnmx,t*=rax*+kt*rbx*for kt*>0rax*−kt*lbx*for kt*≤0

and, consequently, m˜x,t*=expa˜x*+b˜x*kt*. Using the results in Section 3.1., m˜x,t* can be approximated by a TFN

(11)m˜x,t∗≈(mx,t∗,lmx,t∗,rmx,t∗)=(lin(mx,t∗)exp(ax∗+bx∗kt∗),exp(ax∗+bx∗kt∗)llnmx,t∗,exp(ax∗+bx∗kt∗)rlnmx,t∗)

4.2. Forecasting with the Fuzzy-Random LC Model

To forecast future central mortality rates and related variables, it is necessary projecting the values of the index kt. In our approach, these values are the outcomes of the RVs k_t for each year t>T that actuarial literature commonly fits by an ARIMA(p, 1, q) on the data set kt*, t≤T. Subsequently, these projections must be combined with Equations (10a–10d) and (11).

[31] developed a framework for predictions that mixes conventional regression and fuzzy parameters. Following those developments, the predicted values of k_t, t>T, that may be point values or statistical confidence intervals with a linked significance level, will allow obtaining predictions for the central rate of mortality. The values for the central rates of mortality that we obtain from a point prediction of k_t are FNs given the fuzziness of a˜x and b˜x. If we use a probabilistic confidence interval of k_t, the prediction of the central rate of mortality is a fuzzy-probabilistic confidence interval, that is, a probabilistic interval whose lower and upper bounds are FNs.

We can use three different estimates for k_t, t>T, and so, forecasted central rates of mortality change:

If we use the mathematical expectation, E*kt, the mathematical expectation of the logarithm of the central rate of mortality, lnm˜x,t, is denoted by E˜*lnm˜x,t, and
(12a)E˜*lnm˜x,t=E*lnmx,t,lE*lnm˜x,t,rE*lnm˜x,t =ax*,lax*,rax*+E*ktbx*,lbx*,rbx*
that can be calculated with Equations. (10a–10d). So, the central rate of mortality obtained from E*kt,E˜*m˜x,t, is
(12b)E˜*m˜x,t=exp E˜*lnm˜x,t
which can be approximated by a TFN using Equation (11).
If we estimate k_t by its ε-percentile, kt*,ε, the fuzzy forecast of the logarithm of the central rate of mortality, lnm˜x,t, is denoted by lnm˜x,t*ε, being
- –
  If bx*−lbx*≥0
  (13a)lnm˜x,t*ε=lnmx,t*ε,llnmx,t*ε,rlnmx,t*ε =ax*,lax*,rax*+kt*,εbx*,lbx*,rbx*
- –
  If bx∗+rbx∗<0:
  (13b)lnm˜x,t*ε=lnmx,t*ε,llnmx,t*ε,rlnmx,t*ε =ax*,lax*,rax*+kt*,1−εbx*,lbx*,rbx*
  
  And it can be implemented with Equations (10a–10d). The central rate of mortality obtained from kt*,ε, m˜x,t*,ε, is
  (13c)m˜x,t*,ε=explnm˜x,t*ε≈mx,t*,ε,lmx,t*,ε,rmx,t*,ε
  and this FN can also be approximated with Equation (11).
  
  Of course, with this procedure, we maintain the fuzzy uncertainty of a˜x* and b˜x* but the probabilistic uncertainty of k_t is reduced to a point estimation.
If we take for k_t its probabilistic 1−ε confidence interval, kt*,ε︷=kt*,ε2,kt*,1−ε2, following [31], the 1-ε confidence interval prediction for lnm˜x,t− or m˜x,t−, that we denote lnmx,t*ε︷ - or m˜x,t*,ε︷- is a fuzzy-probabilistic confidence interval.
If bx*−lbx*≥0, the lower bound of lnmx,t*ε︷ is the FN lnm˜x,t*ε2=ax*,lax*,rax*+kt*,ε2bx*,lbx*,rbx*, whereas the upper bound of lnmx,t*ε︷ is lnm˜x,t*1−ε2=ax*,lax*,rax*+kt*,1−ε2bx*,lbx*,rbx*.
If bx*+rbx*<0, the lower bound of lnmx,t*ε︷ is lnm˜x,t*ε2=ax*,lax*,rax*+kt*,1−ε2bx*,lbx*,rbx* and the upper bound of lnmx,t*ε︷ is lnm˜x,t*1−ε2=ax*,lax*,rax*+kt*,ε2bx*,lbx*,rbx*

In both cases, we have to apply Equations (13a–13b).

To calculate the bounds of the fuzzy-probabilistic confidence interval m˜x,t*,ε︷ we implement

m˜x,t*,ε2=explnm˜x,t*ε2 and m˜x,t*,1−ε2=explnm˜x,t*1−ε2

as we do in Equation (13c).

Let us remark that [1] did not take into account the uncertainty of ax and bx but only that from k_t. In our model, it is the particular case where a˜x and b˜x have null spreads and the expressions of the fuzzy-probabilistic 1−ε confidence intervals of the logarithm of the central rate of mortality turn into conventional confidence intervals

(14a)lnmx,t*ε︷=ax*+kt*,ε2bx*,ax*+kt*,1−ε2bx* if bx*≥0

(14b)lnmx,t*ε︷=ax*,+kt*,1−ε2bx*,ax*+kt*,ε2bx* if bx*<0

4.3. An Empirical Application of the Fuzzy-Random LC Model: The Case of Spanish Male Population

We apply our extension of the LC model to Spanish male population within the period 1970–2000 and we test its out-of-sample performance during 2001–2012. Central mortality rates have been collected from the “Human Mortality Database,” [32] (http://www.mortality.org). Ages are grouped in 5-year intervals, except for ages less than 1 year, for ages from 1 to 5 years, and for ages greater or equal to 110 years. The values of the estimates a˜x* and b˜x* are in Table 1, whereas the estimates of the behavior of k_t are in Figure 1.

	ax*			bx*
Age	Center	Left Spread	Right Spread	Center	Left Spread	Right Spread
[0, 1)	−4.49273	0.30688	0.25300	0.17351	0.00000	0.00000
[1, 5)	−7.48194	0.20455	0.18860	0.12731	0.00000	0.00000
[5, 10)	−8.10376	0.21022	0.22686	0.11147	0.00000	0.00000
[10, 15)	−8.11329	0.12371	0.11770	0.08472	0.00000	0.01665
[15, 19)	−7.17041	0.12666	0.21927	0.03932	0.00000	0.02229
[20, 24)	−6.77416	0.12234	0.33523	0.02428	0.00000	0.02060
[25, 29)	−6.64539	0.19804	0.43116	0.00113	0.00000	0.00113
[30, 34)	−6.47171	0.30079	0.29775	−0.01338	0.01338	0.00000
[35, 39)	−6.25015	0.17167	0.12575	0.00356	0.00356	0.00000
[40, 44)	−5.89617	0.07598	0.04676	0.02483	0.00243	0.00000
[45, 49)	−5.45921	0.06226	0.02487	0.03075	0.00000	0.00520
[50, 54)	−5.00591	0.05166	0.05161	0.03864	0.00000	0.00000
[55, 59)	−4.55867	0.06395	0.05231	0.04121	0.00000	0.00087
[60, 64)	−4.10372	0.08045	0.05451	0.04445	0.00000	0.00000
[65, 69)	−3.64283	0.07091	0.04687	0.04724	0.00000	0.00523
[70, 74)	−3.15519	0.07888	0.05236	0.05065	0.00030	0.00000
[75, 79)	−2.66456	0.09569	0.05943	0.04685	0.00210	0.00000
[80, 84)	−2.18259	0.07938	0.05492	0.04257	0.00203	0.00001
[85, 89)	−1.72857	0.06846	0.04214	0.03342	0.00580	0.00000
[90, 94)	−1.32104	0.06281	0.04782	0.02257	0.00465	0.00028
[95, 99)	−0.97328	0.04833	0.02706	0.01436	0.00810	0.00692
[100, 104)	−0.68435	0.05068	0.02508	0.00759	0.00636	0.00759
[105, 109)	−0.46013	0.04953	0.03142	0.00277	0.00000	0.00277
[110, ∞)	−0.31708	0.04852	0.03221	0.00017	0.00000	0.00017

Table 1

Parameters a˜x* and b˜x* for Spanish male population for the period 1970–2000.

The unit root test [33] on kt*t=1970,1971,…,2000, suggests that it is I1. We cannot reject the null hypothesis of one unit root on the level (the Students' t is −0.202) but we reject that null hypothesis on the first difference because the Students' t is −7.563.

In Table 2, Ljung and Box Q-statistic suggests that a pure random walk for the first difference is acceptable. So, we model k_t as Δkt*=−0.375+εt*, where the estimate for the standard deviation of εt* is 0.68. Figure 2 represents the evolution that we predict for k_t for years 2001–2012 which has been elaborated by using the bootstrapping procedure for ARIMA time series described in [34].

Lag	Autocorrelation	Partial Autocorrelation	Q-Statistic	p-Value
1	−0.305	−0.305	30.004	0.120
2	0.145	0.005	38.019	0.149
3	−0.01	0.054	38.053	0.283
4	−0.101	−0.109	41.836	0.382
5	0.174	0.123	53.422	0.376
6	−0.023	0.092	53.638	0.498
7	−0.002	−0.004	53.927	0.612
8	−0.112	−0.174	59.353	0.654
9	−0.333	−0.437	11.001	0.276
10	0.014	−0.055	12.450	0.256
11	0.22	−0.152	14.886	0.188
12	0.11	−0.023	15.530	0.215

Table 2

Autocorrelation statistics for the time series kt*t=1970,1971,…,2000.

We now check the capability of our extension of the LC model to fit the central rate of mortality, mx,t, into the sample used to adjust the coefficients, t=1970,1971,…,2000 but also its performance in out-of-sample predictions at t = 2001, 2002,…, 2012. We measure this capability with the membership level that the actual central mortality rate mx,t has in its fuzzy estimate m˜x,t*, μm˜x,t*mx,t. Figure 3 shows the average of grades of membership, for all age groups, for the period 1970–2000, that is, μ¯t=∑xμm˜x,t*mx,tN, with N the number of age groups that have been considered N=24. Likewise, Figure 4 represents, for t=2001,…,2012, the values of μ¯t=∑xμE˜*m˜x,tmx,tN, where the central rate of mortality has been forecasted by using E^*(k_t), and so, with Equations (12a–12b).

Figure 3 shows that the mean grade of membership until the middle of the 80s oscillates, depending on the year, between 0.4 and 0.8. Subsequently, μ¯t remains always around 0.6. In Figure 4, where we also predict central mortality rates with Equations (12a–12b), we can check that with the exception of 2003 and 2005, the average grade of membership of the real observed central rates of mortality in E˜*m˜x,t is at least 0.4. Therefore, it can be said that the capability of the model to fit the central mortality rates in the sample as well as to extrapolate them for a period of more than 10 years is reasonably good.

Table 3 shows the TFN predictions for the central mortality rates of year 2010 that come from E*kt and from the bounds of kt*,10%︷=kt*,5%,kt*,95%, that is, we forecast the mathematical expectation and the 10% fuzzy-probabilistic confidence interval of mx,2010. For example, if we consider the age group [30, 34)

E˜*m˜30,34,2010=0.00176,0.00053,0.00087

which means that the forecasted mean of the central rate of mortality in the year 2010 is approximately 0.00176, although it can vary between 0.00123 and 0.00263. Likewise, m˜30,34,2010*,10%︷ has as lower and upper bounds, m˜30,34,2010*,5% and m˜30,34,2010*,95%, the TFNs (0.00168, 0.00051, 0.00075) and (0.00184, 0.00055, 0.00100), respectively, that is, the real central mortality rate in the year 2010 is contained, with a probability of 90%, between approximately 0.00168, in the most optimistic scenario, and approximately 0.00184, in the most pessimistic scenario. If we were using the basic LC model, which only takes into account the uncertainty related to the index k_t, the results would not be FNs but the real numbers 0.00176 for the mathematical expectation and [0.00168, 0.00184], for its 90% confidence interval.

	E˜*m˜x,t			m˜x,t*,5%			m˜x,t*,95%
Age	Center	Left Spread	Right Spread	Center	Left Spread	Right Spread	Center	Left Spread	Right Spread
[0, 1)	0.00206	0.00063	0.00052	0.00115	0.00035	0.00029	0.00379	0.00116	0.00096
[1, 5)	0.00016	0.00003	0.00003	0.00011	0.00002	0.00002	0.00025	0.00005	0.00005
[5, 10)	0.00010	0.00002	0.00002	0.00007	0.00001	0.00002	0.00015	0.00003	0.00003
[10, 15)	0.00013	0.00004	0.00002	0.00010	0.00003	0.00001	0.00018	0.00004	0.00002
[15, 19)	0.00052	0.00018	0.00011	0.00046	0.00019	0.00010	0.00060	0.00016	0.00013
[20, 24)	0.00090	0.00029	0.00030	0.00083	0.00033	0.00028	0.00098	0.00025	0.00033
[25, 29)	0.00129	0.00037	0.00055	0.00128	0.00038	0.00055	0.00129	0.00037	0.00056
[30, 34)	0.00176	0.00053	0.00087	0.00168	0.00051	0.00075	0.00184	0.00055	0.00100
[35, 39)	0.00186	0.00032	0.00047	0.00184	0.00032	0.00049	0.00189	0.00032	0.00045
[40, 44)	0.00216	0.00016	0.00015	0.00199	0.00015	0.00016	0.00235	0.00018	0.00015
[45, 49)	0.00315	0.00036	0.00008	0.00284	0.00037	0.00007	0.00351	0.00033	0.00009
[50, 54)	0.00460	0.00024	0.00024	0.00403	0.00021	0.00021	0.00526	0.00027	0.00027
[55, 59)	0.00701	0.00051	0.00037	0.00610	0.00046	0.00032	0.00810	0.00056	0.00042
[60, 64)	0.01070	0.00086	0.00058	0.00922	0.00074	0.00050	0.01251	0.00101	0.00068
[65, 69)	0.01651	0.00201	0.00077	0.01409	0.00197	0.00066	0.01949	0.00202	0.00091
[70, 74)	0.02602	0.00205	0.00144	0.02194	0.00173	0.00124	0.03107	0.00245	0.00169
[75, 79)	0.04410	0.00422	0.00352	0.03766	0.00360	0.00327	0.05196	0.00497	0.00377
[80, 84)	0.07445	0.00592	0.00556	0.06450	0.00513	0.00526	0.08642	0.00687	0.00584
[85, 89)	0.12816	0.00877	0.01264	0.11452	0.00784	0.01353	0.14409	0.00986	0.01129
[90, 94)	0.21414	0.01404	0.01994	0.19846	0.01321	0.02159	0.23176	0.01497	0.01781
[95, 99)	0.32848	0.03803	0.03485	0.31297	0.04353	0.04175	0.34543	0.03162	0.02684
[100, 104)	0.46845	0.05871	0.04225	0.45663	0.06889	0.05097	0.48107	0.04750	0.03266
[105, 109)	0.61439	0.05071	0.03279	0.60868	0.05592	0.03249	0.62038	0.04519	0.03311
[110, ∞)	0.72704	0.03900	0.04508	0.72661	0.03941	0.04506	0.72748	0.03858	0.04511

TFN, triangular fuzzy number.

Table 3

TFN approximations of the estimates of E˜*m˜x,t and the bounds of m˜x,t*,10%︷ m˜x,t*,5% and m˜x,t*,95% for year 2010.

Figure 5 depicts the membership level of true observed values of m0,1,t for the period 2001–2012 in the TFNs E˜*m˜0,1,t and m˜0,1,t*,95%. Such important age group is not well predicted when the mathematical expectation of the parameter k_t is used because from the year 2002 on the forecasted rates never contain the real values, that is, their grade of membership is always 0. Nevertheless, it does not happen when considering the upper bound of the fuzzy-probabilistic 90% confidence interval of m˜0,1,t*,95%. In this case, membership levels are never lower than 0.4. We statistically test the capability of E˜*m˜x,t to predict actual central mortality rates for each year t = 2001, 2002, …, 2012. Table 4 shows the results. Following [35], it is desirable that the observed rates mx,t attain membership levels of at least 0.5, in the fuzzy prediction E˜*m˜x,t i.e, μ E˜*m˜x,tmx,t≥0.5. So, for each year t = 2001, 2002, …, 2012 we implement a Wilcoxon rank test for the null hypothesis that the median value of μ E˜*m˜x,tmx,t is 0.5. It can be seen in Table 4 that only in years 2003 and 2012 the median of μ E˜*m˜x,tmx,t is under 0.5 and the null hypothesis is rejected at standard significant levels.

Year	W	Median	Mean	Year	W	Median	Mean
2001	92^*	0.602	0.656	2007	93	0.414	0.381
2002	141	0.488	0.643	2008	141	0.438	0.256
2003	77^**	0.181	0.439	2009	59^**	0.534	0.246
2004	140	0.454	0.431	2010	122	0.462	0.233
2005	117	0.165	0.337	2011	67^***	0.521	0.255
2006	61^***	0.528	0.320	2012	62^***	0.202	0.182

Notes: (1) W stands for the value of Wilcoxon rank test statistic. (2) “*,” “**” and “***” stand for the rejection of the null hypothesis that the median value of μ E˜*m˜x,tmx,t is 0.5 with a significance level of 10%, 5%, and 1%, respectively.

Table 4

Assessment of the capability prediction of E˜*m˜x,t with a Wilcoxon rank test in 2001–2012.

5. FORECASTING LIFE EXPECTANCY WITH THE FUZZY-RANDOM LC MODEL

5.1. Calculating Life Expectancy from Fuzzy Estimates of Central Mortality Rates

We now compute probabilities of death or survival and life expectancies after calculating estimates of central mortality rates. Let us denote the width of the age group as nx years.

To obtain the probability that a person in the age group x, at calendar year t, does not reach the following age group, qnx x,t, we have to take into account that it is a function of mx,t.

(15)qnx x,t=nxmx,t 1+nx1−γx,tmx,t

where γx,t∈0,1 is the average fraction of the nx-year period lived by those who died in that period and we will suppose that this coefficient is fixed beforehand. Given that

∂qnx x,t∂mx,t =nx1+nx1−x,tmx,t 2>0

by considering Equation (15) and bearing in mind that q˜nx x,t*α is a prediction of a probability, that is, q˜nx x,t*α⊆0,1, we can obtain the α-cuts of q˜nx x,t*, q˜nx x,t*α

nxq˜x,tα*=qnx x,t*_α,qnx x,t*¯α =max0,nxmx,t*−lmx,t*1−α1+nx1−γx,tmx,t*−lmx,t*1−α, min1,nxmx,t*+rmx,t*1−α1+nx1−γx,tmx,t*+rmx,t*1−α

It may be useful to obtain a triangular approximation for q˜nx x,t*, q˜nx x,t*≈qnx x,t*,lqnx x,t*,rqnx x,t*, with

(16a)qnx x,t*=nxmx,t*1+nx1−γx,tmx,t*

In order to obtain the support of q˜nx x,t*, we have to take into account that it is a probability and so, its support must be within the interval [0, 1]. Then

(16b)lqnx x,t*=minqnx x,t*,nxlmx,t*1+nx1−γx,tmx,t 2

and

(16c)rqnx x,t*= min1−qnx x,t*,nxrmx,t*1+nx1−γx,tmx,t 2

To determine the probability that a person in the age group x, at calendar year t, reaches the following age group, pnx x,t, from the crisp relationship pnx x,t=1−qnx x,t, under fuzziness we state p˜nx x,t=1−q˜nx x,t where

pnx x,t*α=pnx x,t*_α,pnx x,t*¯α=1−qnx x,t*_α,1−qnx x,t*¯α

From Equations (15a–15c), p˜nx x,t*≈pnx x,t*,lpnx x,t*,rpnx x,t*

pnx x,t*=1−qnx x,t*; lpnx x,t*=lqnx x,t*; rpnx x,t*=rqnx x,t*

The life expectancy of a person in the age group x, at calendar year t, ex,t can be calculated with the expression

(17)ex,t =∑i≥x∏x≤j<i(1−nj qj,t )[ni−(ni−γi,t)ni qi,t ]

Since

∂ex,t∂niqi,t=−{∑s>i∏x≤j<sj≠i∏x≤j<i(1−nj qj,t )[ni−(ni−γi,t)]+∑s>i∏x≤j<sj≠i(1−nj qj,t )[ns−(ns−γs,t)ns qs,t ]}

it turns out that ex,t is a decreasing function of qni i,t (and so, of its linked central mortality rate).

	E˜e˜0,1,t			e˜0,1,t*,5%			e˜0,1,t*,95%
Year	Center	Left Spread	Right Spread	Center	Left Spread	Right Spread	Center	Left Spread	Right Spread
2001	76.31	1.18	1.28	75.73	1.17	1.28	76.96	1.20	1.28
2002	76.49	1.19	1.28	75.72	1.17	1.28	77.20	1.21	1.29
2003	76.66	1.19	1.28	75.75	1.17	1.28	77.54	1.22	1.29
2004	76.84	1.20	1.28	75.82	1.18	1.28	77.85	1.23	1.30
2005	77.01	1.20	1.29	75.89	1.18	1.28	78.07	1.24	1.30
2006	77.18	1.21	1.29	75.94	1.18	1.28	78.29	1.25	1.30
2007	77.35	1.21	1.29	75.94	1.18	1.28	78.60	1.27	1.31
2008	77.52	1.22	1.29	75.97	1.18	1.28	78.80	1.28	1.31
2009	77.69	1.23	1.29	76.10	1.18	1.28	79.05	1.29	1.32
2010	77.85	1.23	1.30	76.13	1.18	1.28	79.28	1.31	1.32
2011	78.02	1.24	1.30	76.26	1.18	1.28	79.50	1.32	1.33
2012	78.19	1.25	1.30	76.56	1.19	1.28	79.75	1.34	1.33

TFN, triangular fuzzy number.

Table 5

TFN approximation of the estimates of E˜*e˜0,1,t* and e˜[0,1),t*,10%︷, with lower and upper bounds e˜[0,1),t*,5% and e˜[0,1),t*,95%, in 2001–2012.

By evaluating Equation (17) with q˜ni i,t*, we will obtain a fuzzy estimate for ex,t , e˜x,t* . Moreover, it is straightforward to see that its α–cuts, ex,t* α, are

ex,tα∗=[ex,t∗_(α),ex,t∗¯(α)]=[(1− nj qj,t _(a))∑i≥x∏x≤j<i(1− nj qj,t ¯(α))[ni−(ni−γi,t) nj qj,t ¯(α)],∑i≥x∏x≤j<i(1− nj qj,t _(α))[ni−(ni−γi,t) ni qi,t _(α)]]

A TFN approximation of e˜x,t*, e˜x,t*≈ex,t*,lex,t*,rex,t*, can be obtained by using Equation (6)

(18a)ex,t∗=∑i≥x∏x≤j<i(1− nj qj,t∗)ni−(ni−γi,t)niqi,t*

(18b)lex,t∗=∑i≥x{∑s>i∏x≤j<sj≠i∏x≤j<i(1−nj qj,t∗)(ni−γi,t)+∑s>i∏x≤j<sj≠i(1−nj qj,t∗)[ns−(ns−γs,t)ns qs,t∗]}r ni qi,t∗

(18c)rex,t∗=∑i≥x{∑s>i∏x≤j<sj≠i∏x≤j<i(1−nj qj,t∗)(ni−γi,t)+∑s>i∏x≤j<sj≠i(1−nj qj,t∗)[ns−(ns−γs,t)ns qs,t∗]}l ni qi,t∗

Of course, if in Equations (18a–18c) we take as a prediction of the index k_t its mathematical expectation, kt*=E*kt, we will obtain a fuzzy estimate of life expectation that we symbolize as E˜*e˜x,t*.

If the prediction of the mortality trend comes from its probabilistic confidence interval, kt*,ε︷=kt*,ε2,kt*,1−ε2, we can built up a fuzzy-probabilistic confidence interval of the life expectancy ex,t*,ε︷=ex,t*,ε2,ex,t*,1−ε2. In the common case where the sensitivity of the central rate of mortality respect to changes in the index k_t is strictly positive, that is, b˜x*>0 bx*−lbx*≥0, kt*,ε2 will determine ex,t*,1−ε2, whereas kt*,1−ε2 will define the lower life expectancy ex,t*,ε2.

	E˜e˜65,69,t			e˜65,69,t*,5%			e˜65,69,t*,95%
Year	Center	Left Spread	Right Spread	Center	Left Spread	Right Spread	Center	Left Spread	Right Spread
2001	17.06	0.48	0.64	16.67	0.46	0.63	17.51	0.51	0.65
2002	17.18	0.49	0.64	16.66	0.46	0.63	17.68	0.52	0.65
2003	17.30	0.49	0.64	16.68	0.46	0.63	17.92	0.53	0.66
2004	17.42	0.50	0.65	16.73	0.46	0.63	18.15	0.55	0.66
2005	17.54	0.51	0.65	16.78	0.46	0.63	18.31	0.56	0.67
2006	17.67	0.52	0.65	16.82	0.47	0.63	18.47	0.57	0.67
2007	17.79	0.52	0.66	16.82	0.47	0.63	18.70	0.58	0.67
2008	17.91	0.53	0.66	16.83	0.47	0.63	18.85	0.59	0.68
2009	18.03	0.54	0.66	16.92	0.47	0.63	19.04	0.61	0.68
2010	18.15	0.55	0.66	16.94	0.47	0.64	19.21	0.62	0.68
2011	18.27	0.55	0.67	17.03	0.48	0.64	19.38	0.63	0.69
2012	18.40	0.56	0.67	17.24	0.49	0.64	19.57	0.64	0.69

TFN, triangular fuzzy number.

Table 6

TFN approximation of the estimates of E˜*e˜65,69,t* and e˜[65,69),t*,10%︷, with lower and upper bounds e˜[65,69),t*,5% and e˜[65,69),t*,95%, in 2001–2012.

5.2. Predicting Life Expectancies of Spanish Male Population in 2001–2012

Tables 5 and 6 show the estimates for the mean value and the 90% confidence fuzzy-probabilistic interval of the life expectancy for the age groups [0, 1) and [65, 69) during the period 2001–2012. These ages are significantly important because they are considered in order to quantify life expectancy at birth and at retirement, respectively. If only the centers of the fuzzy life expectancies are considered, predictions that come from the basic LC method are found. So, for example, the point estimate for e0,1,2012 is 78.19 years and the 90% confidence probabilistic interval is [76.56, 79.75] years. The fuzzy-random extension of the LC model allows introducing the fuzziness into the coefficients ax and bx and, as a consequence, point predictions and probabilistic interval predictions, as well, are fuzzified. So, the projection of e0,1,2012 is the TFN (78.19, 1.25, 1.30) years, whereas the lower and upper bounds of the 90% confidence fuzzy-probabilistic interval are respectively (76.56, 1.19, 1.28) and (79.75, 1.34, 1.33).

Figures 6 and 7 represent the membership level of true observed values for life expectancies, ex,t , into their estimates E˜*e˜x,t*, e˜x,t*,5% and e˜x,t*,95%. Concretely, we take the estimates calculated in Tables 5 and 6. We can check that e0,1,t is fitted quite accurately by E˜*e˜0,1,t* from 2001 to 2005. On the other hand, from 2006 to 2012, μ E˜*e˜0,1,t*e0,1,t decreases to values near 0. However, we can also remark that in those years the membership level of e0,1,t into the TFN e˜0,1,t*,95% stands clearly up to 0.5. Likewise, for [65, 69), Figure 7 shows that E˜*e˜65,69,t* fits e65,69,t , from 2006 to 2012, with clearly satisfactory membership levels that are usually up 0.8. It is true that μ E˜*e˜65,69,t* e65,69,t has low values in the years 2003 and 2005 but they are compensated by the greater membership levels of μe˜65,69,t*,5% e65,69,t . In a similar way as in Subsection 4.3., we statistically test the capability of fuzzy mean life expectancies to predict actual life expectations at t = 2001, 2002, …, 2012. Again, for each year t = 2001, 2002, …, 2012 we implement a Wilcoxon rank test with the null hypothesis that the median value of μ E˜*e˜x,t* mx,t=0.5. Table 7 shows that, except for the year 2003, in the years where the median of μ E˜*m˜x,t mx,t is under 0.5, we cannot reject the null hypothesis. On the other hand, in the years 2001, 2007, 2010, and 2011, there are statistical evidences that the median is above 0.5.

Capability Prediction of E˜e˜x,t Per Years
Year	W	Median	Mean	Year	W	Median	Mean
2001	92^*	0.602	0.718	2007	93	0.414	0.589
2002	141	0.488	0.656	2008	141	0.438	0.586
2003	77^**	0.181	0.338	2009	59^***	0.534	0.555
2004	140	0.454	0.653	2010	122	0.462	0.445
2005	117	0.165	0.440	2011	67^***	0.521	0.443
2006	61^***	0.528	0.699	2012	62^***	0.202	0.409

Capability Prediction of E˜e˜0,1,t and E˜e˜65,69,t on Life Expectancy at Birth and Retirement
		Median		Mean		W

E˜e˜0,1,t		0.628		0.567		37
E˜e˜65,69,t		0.655		0.609		19

Notes: (1) Each year has 24 predictions on life expectancies, one per age group. (2) Each age group has 12 predictions available, one for each assessed year. (3) W stands for the value of Wilcoxon rank test statistic. (4) “*,” “**,” and “***” stand for the rejection of the null hypothesis that median value of μ E˜*e˜x,t* ex,t is 0.5 with a significance level of 10%, 5%, and 1%, respectively.

Table 7

Assessment of the capability prediction of E˜*e˜x,t* with a Wilcoxon rank test for the period 2001–2012.

Due to the interest in actuarial analyses in life expectancy both at birth and at retirement, we test the quality of the prediction by E˜*e˜x,t* for age gropus [0, 1) an [65, 69). The results are also collected in Table 7. We can check that the median and mean membership levels of observed life expectancies in the period 2001–2012 are consistently above 0.5. However, the Wilcoxon rank test does not reject in both age groups that μ E˜*e˜x,t*ex,t =0.5.

6. EMPIRICAL ASSESSMENT OF THE FUZZY-RANDOM LC MODEL IN EIGHT WESTERN EUROPEAN COUNTRIES¹

6.1. Methodological Considerations

In this section we make a comparative assessment on the prediction capability of our proposed fuzzy-random extension of the LC model (FRLC) with both the basic LC (BLC) in [1] and the pure fuzzy LC version by Koissi and Shapiro in [14], (FKSLC). Let us remark that BLC only considers random uncertainty of coefficients kt. On the other hand, FKSLC introduces fuzzy uncertainty in all the coefficients of the LC model by means of symmetrical TFNs. Likewise, FKSLC handles uncertainty with the weakest t-norm instead of the commonly used minimum operator.

To carry out the analysis, we use central mortality rates collected separately for men and women in eight Western Europe countries (i.e., we use 16 databases) from [32] (http://www.mortality.org). As we made in Sections 4 and 5 for the case of Spanish male population, we fit the model parameters by using central mortality rates in the period 1970–2000 and we test models out-of-sample performance during 2001–2012. Ages are again grouped in 5-year intervals, except for ages lower than 1 year, for ages from 1 to 5 years, and for ages greater or equal to 110 years.

We assess two aspects regarding the fitting quality of the models:

Item 1. We measure and compare models' performance to make point predictions on central mortality rates and life expectancies. This is made by using the conventional error measures: root mean squared error (RMSE), normalized mean squared error (NMSE), and mean absolut error (MAE). We consider these point predictions: the expectation for BLC, the core of the fuzzy expectation for FRLC and, finally, the core of the fuzzy prediction for FKSLC. Notice that point predictions by BLC and FRLC are the same by definition. So, in fact, we are making a comparison of a couple of predictive methods: BLC/FRLC versus FKSLC. Following [36], this pairwise comparison between techniques is made with both a sign test (Wins/Losses) and a Wilcoxon rank test.

Item 2. We evaluate the capability of BLC, FRLC, and FKSLC to predict future values of mx,t_, and ex,t by means of confidence intervals. In this second item, we measure the accuracy of a method as the rate of right predictions on mx,t or ex,t through confidence intervals estimates provided by the methods.

In this regard, let us make the following remarks:

BLC only considers random uncertainty of kt. So, after establishing a significance level ε, that in our numerical assessment will be 10%, BLC predicts life variables as a 1−ε confidence interval like in Equation (14).
FRLC estimates the lower and upper bounds of the 1−ε confidence interval by means of two TFNs. To obtain standard confidence interval predictions, we transform these estimates into a conventional confidence interval that comes from the convex hull, C·, of the expected intervals Equation (4b) corresponding to ε2 and 1−ε2 percentiles of fuzzy predictions (5% and 95% in our numerical application). So, for mx,t, the 1−ε confidence interval prediction is CeIm˜x,t*,ε2∪eIm˜x,t*,1−ε2. Analogously, the 1−ε confidence interval prediction of ex,t is CeIe˜x,t*,ε2∪eIe˜x,t*,1−ε2. For example, the life expectancy at birth of a Spanish man born in 2004 for ε=10% is built up from e˜0,1,2004* 5%=75.82,1.18,1.28 and e˜0,1,2004* 95%=77.85,1.23,1.30 (see Table 5). We easily find that eIe˜0,1,2004* 5%=75.23,76.46 and eIe˜0,1,2004* 95%=77.23,78,50. Then, the 90% confidence interval prediction for e0,1,2004 is (in years):
C75.23,76.46∪77.23,78.50 =75.23,78.50
FKSLC directly predicts mortality variables as FNs. The expected interval of the FN obtained from this method is taken as its confidence interval.

The analysis of both questions is developed in two levels:

In each population, we independently assess the predictive capability of each method. For a given population we must predict 24 variables for each of the 12 years that testing period 2001–2012 comprises. In each year we find the mean value of the accuracy measures and so, for each population, we have 12 available mean values of accuracy (one per year). The results that we find in this case are exclusive to the population studied.
We will use the mean results of the accuracy predictions within the whole period 2001–2012 of all populations to make an inter-population assessment. It may lead to extract more general conclusions about the method performance. In this case, we will work with a sample of 16 different goodness of fit measures.

Following [37] and [38], an adequate nonparametrical test to carry out this kind of analysis is the Friedman rank test (Friedman χ2 and Iman-Davenport F statistics) that may be completed by the pairwise comparisons that allow using Friedman ranks (Z-score). Likewise, given that FRLC and FKSLC are extensions of BLC, we will implement the multiple sign test described in [37] where the control technique is BLC.

6.2. Comparison of BLC, FRLC, and FKSLC for Each Population

We now show the adequacy of the three LC methods evaluated in 16 populations. We present in a more detailed way the results corresponding to Spanish male population (Tables 8a–8d) and a summary table for all the analyzed countries (Tables 9a–9e).

	RMSE		NMSE		MAE
Year	BLC/FRLC	FKSLC	BLC/FRLC	FKSLC	BLC/FRLC	FKSLC
2001	0.006	0.002	0.0039	0.0009	0.0031	0.0003
2002	0.007	0.011	0.0044	0.0051	0.0040	0.0097
2003	0.020	0.011	0.0108	0.0058	0.0280	0.0088
2004	0.012	0.023	0.0070	0.0091	0.0110	0.0392
2005	0.018	0.018	0.0101	0.0082	0.0240	0.0247
2006	0.011	0.020	0.0066	0.0111	0.0103	0.0297
2007	0.015	0.022	0.0084	0.0095	0.0168	0.0360
2008	0.016	0.022	0.0089	0.0115	0.0204	0.0361
2009	0.014	0.018	0.0078	0.0097	0.0151	0.0240
2010	0.013	0.016	0.0074	0.0089	0.0139	0.0202
2011	0.011	0.015	0.0066	0.0085	0.0110	0.0175
2012	0.022	0.026	0.0119	0.0129	0.0407	0.0498

	Wins/losses	10/2**	Wins/losses	9/3*	Wins/losses	10/3**

	W	8**	W	11**	W	15*

RMSE, root mean squared error; NMSE, normalized mean squared error; MAE, mean absolute error; BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “Wins/losses” stands for the number of cases in which BLC and FRLC point predictions are better/worse than FKSLC. (2) W stands for the value of the Wilcoxon rank test statistic. (3) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively.

Table 8a

Mean RMSE, NMSE, and MAE (per years) of central mortality rates point predictions for Spanish men by the evaluated methods (Item 1).

	RMSE		NMSE		MAE
Year	BLC/FRLC	FKSLC	BLC/FRLC	FKSLC	BLC/FRLC	FKSLC
2001	0.152	0.246	1.51E-04	2.45E-04	0.122	0.160
2002	0.194	0.232	1.91E-04	2.30E-04	0.149	0.155
2003	0.327	0.218	3.21E-04	2.16E-04	0.287	0.202
2004	0.190	0.400	1.83E-04	3.90E-04	0.173	0.287
2005	0.270	0.344	2.61E-04	3.35E-04	0.235	0.264
2006	0.348	0.647	3.30E-04	6.19E-04	0.235	0.484
2007	0.306	0.571	2.89E-04	5.44E-04	0.251	0.404
2008	0.443	0.754	4.14E-04	7.38E-04	0.308	0.561
2009	0.560	0.889	5.18E-04	8.30E-04	0.008	0.010
2010	0.712	1.060	6.51E-04	9.79E-04	0.007	0.009
2011	0.769	1.127	6.98E-04	1.03E-03	0.007	0.008
2012	0.703	1.056	6.36E-04	9.75E-04	0.012	0.013

	Wins/losses	11/1*	Wins/losses	11/1**	Wins/losses	11/1**

	W	9**	W	9**	W	5**

RMSE, root mean squared error; NMSE, normalized mean squared error; MAE, mean absolute error; BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “Wins/losses” stands for the number of cases in which BLC and FRLC point predictions are better/worse than FKSLC. (2) W stands for the value of the Wilcoxon rank test statistic. (3) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively.

Table 8b

Mean RMSE, NMSE, and MAE (per years) of life expectancy point predictions for Spanish men by the evaluated methods (Item 1).

	Proportion of Successful Predictions			Test Results
Year	BLC	FRLC	FKSLC	Global comparison Friedman χ2 = 20.667* Iman-Davenport F Statistic = 31* Pairwise comparisons FRLC versus BLC Z score = 2.858 p-values: (a) 4.27E-03; (b) 0.013; (c) 0.009 FKSLC versus FRLC Z score = −4.491 p-values: (a) 7.10E-06; (b) 2.13E-05; (c) 2.13E-05 FKSLC versus BLC Z score = −1.633 p-values: (a) 0.102; (b) 0.307; (c) 0.102

2001	0.625	0.875	0.708
2002	0.625	0.875	0.750
2003	0.542	0.667	0.458
2004	0.625	0.833	0.458
2005	0.542	0.667	0.292
2006	0.625	0.792	0.375
2007	0.583	0.792	0.500
2008	0.583	0.708	0.333
2009	0.583	0.750	0.250
2010	0.583	0.750	0.292
2011	0.583	0.708	0.333
2012	0.542	0.583	0.167

Multiple sign test (the control method is BLC)

Wins/losses of FRLC against BLC: 12/0 + Wins/losses of FKSLC against BLC: 2/10 −

BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) Friedman χ2 follows a Squared-Chi with 2 grades of freedom and Iman–Davenport F follows a Snedecor F with 2(24) grades of freedom. (2) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively. (3) (a) indicates standard p-value and (b) and (c) Nemenyi and Holm p-value corrections for multiple pairwise comparisons. (4) “+” indicates that the evaluated method outperforms the control method with at least at 10% significance level, whereas “−” indicates that the evaluated method underperforms the control method with at least at 10% significance level.

Table 8c

Mean proportion of successful predictions on central mortality rates of BLC, FRLC, and FKSLC with confidence intervals (Item 2).

	Proportion of Successful Predictions			Test Results
Year	BLC	FRLC	FKSLC	Global comparison Friedman χ2 = 8.0417* Iman–Davenport F Statistic = 5.5431 Pairwise comparisons FRLC versus BLC Z score = 2.756 p-values: (a) 0.006; (b) 0.018; (c) 0.012 FKSLC versus FRLC Z score = −3.878 p-values: (a) 6.88E-05; (b) 2.06E-04; (c) 1.38E-04 FKSLC versus BLC Z score = −1.123 p-values: (a) 0.262; (b) 0.785; (c) 0.262

2001	0.792	0.958	0.958
2002	0.833	0.958	0.958
2003	0.667	0.792	0.667
2004	0.833	0.958	0.625
2005	0.750	0.833	0.458
2006	0.875	0.958	0.375
2007	0.833	0.917	0.583
2008	0.833	0.875	0.292
2009	0.875	0.917	0.292
2010	0.875	0.958	0.250
2011	0.833	0.958	0.250
2012	0.833	0.875	0.167

Multiple sign test (the control method is BLC)

Wins/losses of FRLC against BLC: 12/0 + Wins/losses of FKSLC against BLC: 3/9 −

BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) Friedman χ2 follows a Squared-Chi with 2 grades of freedom and Iman–Davenport F follows a Snedecor F with 2(24) grades of freedom. (2) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively. (3) (a) indicates standard p-value and (b) and (c) Nemenyi and Holm p-value corrections for multiple pairwise comparisons. (4) “+” indicates that the evaluated method outperforms the control method with at least at 10% significance level, whereas “−” indicates that the evaluated method underperforms the control method with at least at 10% significance level.

Table 8d

Mean proportion of successful predictions on life expectancies of BLC, FRLC, and FKSLC with confidence intervals (Item 2).

	RMSE		NMSE		MAE
	Wins/Losses	W	Wins/Losses	W	Wins/Losses	W
Austria (men)	12/0**	0***	12/0**	0***	12/0**	0***
Austria (women)	12/1**	0***	12/0**	0***	12/0**	0***
Belgium (men)	3/9*	23	5/7	25	2/10**	18
Belgium (women)	12/0**	0***	12/0**	0***	12/0**	0***
France (men)	11/1**	12**	11/1**	11**	11/1**	11**
France (women)	11/1**	11**	10/2**	22	11/1**	11**
Italy (men)	12/0**	0***	12/0**	0***	12/0**	0***
Italy (women)	11/1**	8**	8/4	34	11/1**	3***
Netherlands (men)	9/3*	15*	10/2**	11**	9/3*	16*
Netherlands (women)	8/4	31	8/4	28	8/4	30
Portugal (men)	10/2**	22	9/3*	24	10/2**	16*
Portugal (women)	4/8	23	3/9*	14**	4/8	22
Spain (men)	10/2**	8**	9/3*	11**	10/2**	15*
Spain (women)	12/0**	0***	12/0**	0***	12/0**	0***
United Kingdom (men)	11/1**	11**	12/0**	0***	11/1**	12**
United Kingdom (women)	5/7	27	6/6	38	5/7	31

RMSE, root mean squared error; NMSE, normalized mean squared error; MAE, mean absolute error; BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “Wins/losses” are accounted from the perspective of BLC/FRLC. (2) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively.

Table 9a

Results of sign and Wilcoxon tests on the difference between the accuracy of point estimates on central mortality rates by BLC/FRLC and FKSLC in the period 2001–2012 (Item 1).

	RMSE		NMSE		MAE
	Wins/Losses	W	Wins/Losses	W	Wins/Losses	W
Austria (men)	12/0**	0***	12/0**	0***	12/0**	0***
Austria (women)	0/12**	0***	0/12**	0***	0/12**	0***
Belgium (men)	0/12**	0***	0/12**	0***	0/12**	0***
Belgium (women)	12/0**	0***	12/0**	0***	12/0**	0***
France (men)	11/1**	12**	11/1**	12**	11/1**	12**
France (women)	9/3*	11**	9/3*	29	9/3*	28
Italy (men)	12/0**	0***	12/0**	0***	12/0**	0***
Italy (women)	4/8	30	4/8	30	3/9*	26
Netherlands (men)	0/12**	0***	0/12**	0***	0/12**	0***
Netherlands (women)	4/8	18	4/8	18	4/8	18
Portugal (men)	12/0**	0***	12/0**	0***	12/0**	0***
Portugal (women)	12/0**	0***	8/4	37	8/4	37
Spain (men)	11/1**	9**	11/1**	9**	11/1**	9**
Spain (women)	1/11**	12**	1/11**	12**	1/11**	12**
United Kingdom (men)	12/0**	11**	12/0**	0***	12/0**	0***
United Kingdom (women)	0/12**	0***	0/12**	0***	0/12**	0***

RMSE, root mean squared error; NMSE, normalized mean squared error; MAE, mean absolute error; BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “Wins/losses” are accounted from the perspective of BLC/FRLC. (2) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively.

Table 9b

Results of sign and Wilcoxon tests on the difference between the accuracy of point estimates on life expectancies by BLC/FRLC and FKSLC in the period 2001–2012 (Item 1).

	Pairwise Z Scores from Friedman Ranks			Friedman Test
	FRLC versus BLC	FKSLC versus FRLC	FKSLC versus BLC	Friedman χ2	Iman–Davenport F
Austria (men)	3.164***	−3.776***	−0.612	8.542**	6.079***
Austria (women)	3.062***	−4.082***	−1.021	14.083***	15.621***
Belgium (men)	2.654**	−4.491***	−1.837	18.375***	35.933***
Belgium (women)	2.654**	−4.695***	−2.041	22.167***	133.026***
France (men)	2.858**	−4.491***	−1.633	20.660***	68.042***
France (women)	2.449**	−4.695***	−2.245*	20.000***	55.000***
Italy (men)	2.654**	−4.491***	−1.837	16.420***	23.828***
Italy (women)	2.654**	−4.695***	−2.041	22.167***	133.026***
Netherlands (men)	2.858**	−4.491***	−1.633	20.667***	68.208***
Netherlands (women)	2.654**	−4.287***	−1.633	10.830***	9.046***
Portugal (men)	2.654**	−4.695***	−2.041	22.167***	133.026***
Portugal (women)	3.062***	−3.674***	−0.612	15.500***	20.059***
Spain (men)	2.858**	−4.491***	−1.633	20.667***	68.208***
Spain (women)	3.062***	−4.082***	−1.021	14.083***	15.621***
United Kingdom (men)	2.654**	−4.082***	−1.429	5.417*	3.207*
United Kingdom (women)	2.654**	−4.491***	−1.837	16.417***	23.815***

BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively. (2) Friedman χ2 follows a Squared-Chi with 2 grades of freedom and Iman–Davenport F follows a Snedecor F with 2(24) grades of freedom.

Table 9c

Results of Friedman rank tests and pairwise Friedman rank tests for the accuracy of the confidence interval predictions on central mortality rates by BLC, FRLC, and FKSLC in sample populations in the period 2001–2012 (Item 2).

	Pairwise Z Scores from Friedman Ranks			Friedman Test
	FRLC versus BLC	FKSLC versus FRLC	FKSLC versus BLC	Friedman χ2	Iman–Davenport F
Austria (men)	3.164***	−4.185***	−1.021	19.042***	42.247***
Austria (women)	2.449**	−3.266***	−0.816	7.750**	5.246**
Belgium (men)	2.143*	−4.287***	−2.143*	18.370***	35.892***
Belgium (women)	2.245*	−4.491***	−2.245*	20.100***	56.692***
France (men)	2.347*	−3.470***	−1.123	12.540***	12.037***
France (women)	1.123	−3.572***	−2.449**	9.375***	7.051***
Italy (men)	2.449**	−4.899***	−2.449**	24.000***	∞***
Italy (women)	2.041	−2.449**	−0.408	6.500**	4.086**
Netherlands (men)	4.695***	−2.041	2.654**	22.167***	133.026***
Netherlands (women)	2.347*	−3.470***	−1.123	12.542***	12.041***
Portugal (men)	2.245*	−3.572***	−1.327	13.040***	13.088***
Portugal (women)	1.429	−3.878***	−2.449**	11.420***	9.986***
Spain (men)	2.756**	−3.980***	−1.225	16.250***	23.065***
Spain (women)	2.960***	−2.449**	0.510	14.083***	15.622***
United Kingdom (men)	2.552**	−4.491***	−1.939	20.000***	55.000***
United Kingdom (women)	2.654**	−4.695***	−2.041	22.167***	133.026***

BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively. (2) Friedman χ2 follows a Squared-Chi with 2 grades of freedom and Iman–Davenport F follows a Snedecor F with 2(24) grades of freedom.

Table 9d

Results of Friedman rank tests and pairwise Friedman rank tests for the accuracy of the confidence interval predictions on life expectancies by BLC, FRLC, and FKSLC in sample populations in the period 2001–2012 (Item 2).

	Central Mortality Rates Predictions
	FRLC versus BLC			FKSLC versus BLC
	Wins	Losses	r_j	Wins	Losses	r_j
Austria (men)	12	0	0 **	3	8	3
Austria (women)	10	0	0 **	3	8	3
Belgium (men)	10	0	0 **	2	10	2 **
Belgium (women)	10	0	0 **	1	11	1 **
France (men)	11	0	0 **	2	9	2 **
France (women)	6	0	0 **	0	9	0 **
Italy (men)	12	0	0 **	0	12	0 **
Italy (women)	8	0	0 **	0	8	0 **
Netherlands (men)	12	0	0 **	12	0	0 **
Netherlands (women)	10	0	0 **	3	9	3
Portugal (men)	11	0	0 **	2	10	2 **
Portugal (women)	6	0	0 **	1	10	1 **
Spain (men)	12	0	0 **	2	9	2 **
Spain (women)	11	0	0 **	6	5	5 **
United Kingdom (men)	11	0	0 **	1	10	1 **
United Kingdom (women)	12	0	0 **	1	11	1 **

	Life Expectancy Predictions
	*FRLC versus* BLC**			*FKSLC versus* BLC**
	Wins	Losses	*r_j*	Wins	Losses	*r_j*

Austria (men)	12	0	0 **	3	8	3
Austria (women)	10	0	0 **	3	8	3
Belgium (men)	10	0	0 **	0	10	0 **
Belgium (women)	10	0	0 **	1	11	1 **
France (men)	11	0	0 **	2	9	2 **
France (women)	6	1	1 **	0	9	0 **
Italy (men)	12	0	0 **	0	12	0 **
Italy (women)	8	0	0 **	2	6	2 **
Netherlands (men)	12	0	0 **	12	0	0 **
Netherlands (women)	10	0	0 **	3	9	3
ortugal (men)	11	0	0 **	2	10	2 **
Portugal (women)	6	0	0 **	1	10	1 **
Spain (men)	12	0	0 **	2	9	2 **
Spain (women)	11	0	0 **	6	5	5 **
United Kingdom (men)	11	0	0 **	1	10	1 **
United Kingdom (women)	12	0	0 **	1	11	1 **

BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “**” stands for the rejection of the null hypothesis with a significance level of at least 5%. (2) “Wins/losses” stands for the number of wins/losses of the evaluated method over the control method. (3) r_j stands for the minimum between number of wins and losses of the evaluated method.

Table 9e

Results of multiple sign test for the confidence interval predictions of FRLC and FKSLC with a control method (BLC) (Item 2).

Regarding Item 1, we can check in Tables 8a and 8b that for Spanish male population, BLC/FRLC point predictions of mx,t and ex,t are, generally, more accurate than those by FKSLC and this best adjustment has a consistent statistical significance. Furthermore, Table 9a shows that in the studied populations, as in the case of Spanish men, point predictions of mx,t from BLC/FRLC are normally better than those from FKSLC and this fact has also statistical significance. We can appreciate three exceptions: Belgian male population, where FKSLC beats BLC/FRLC with a consistent statistical level and United Kingdom and Netherlands female populations where we do not appreciate any significant better method. Table 9c shows that in the prediction of ex,t, it is less clear that BLC/FRLC predictions are better than those by FKSLC. BLC/FRLC beats FKSLC with a clear statistical significance in eight populations but in five populations FKSLC works clearly better. Likewise, in three populations the possible superior performance of a given method has no statistical significance.

In regards to Item 2, in Spanish male population, we can check in Tables 8c and 8d that Friedman rank test rejects the homogeneity in the accuracy of the predictions over analyzed life variables by the three assessed methods. Pairwise comparisons lead us to conclude that FRLC makes better interval predictions than BLC and FKSLC. However, despite the fact that we can detect that BLC beats FKSLC, this superior performance has no statistical significance. In this sense, multiple sign test shows that our method clearly beats the control method and, on the other hand, the control method seems to be superior to FKSLC but without statistical significance. Tables 9c–9d show that those facts are common to all studied populations. So, Friedman χ2 and Iman–Davenport statistics always reject the homogeneity of the prediction capability by the three methods. This fact applies for mx,t and for ex,t. Pairwise Friedman ranks tests show that the prediction on central mortality rates by FRLC beats significantly those obtained by BLC and FKSLC. Likewise, we can also check that BLC usually makes more accurate interval predictions than FKSLC but that better performance, except for the case of French women, has not statistical significance.

In the analysis of life expectancy predictions, pairwise Friedman ranks tests (see Table 9d) reveal that FRLC predicts confidence intervals consistently better than other methods in most populations. In any case, it is also true that in French and Italy female populations and Portugal male population (Netherlands male population) the greater accuracy of FRLC over BLC (FKSLC over FRLC) is not statistically significant. We can also check that in most cases BLC includes more percentage of observed values of ex,t than FKSLC but it is only statistically relevant in five populations. However, in the case of Netherlands male population, FKSLC model predicts life expectancies better than BLC with a clear significance level. Results of multiple sign tests in Table 9e show that our method improves significantly BLC (the control method), whereas this clearly does not follow with FKSLC method.

6.3. A Global Comparison of BLC, FRLC, and FKSLC

In this section we show the results of testing BLC, FRLC, and FKSLC from a sample composed by the mean values of accuracy prediction measures within 2001–2012 of the 16 populations considered in this paper. They are summarized in Tables 10a–10d. Regarding Item 1, when evaluating predictions about central mortality rates, Table 10a shows that BLC and FRLC have greater accuracy than FKSLC method and it is significant. From Table 10b we can also indicate that point predictions on life expectancy by BLC/FRLC are more accurate than those by FKSLC but this better performance has not enough statistical significance.

	RMSE		NMSE		MAE
	BLC/FRLC	FKSLC	BLC/FRLC	FKSLC	BLC/FRLC	FKSLC
Austria (men)	0.0462	0.0538	0.0711	0.0698	0.0249	0.0295
Austria (women)	0.0037	0.0024	0.0003	0.0002	0.0017	0.0029
Belgium (men)	0.0207	0.0151	0.0295	0.0150	0.0096	0.0074
Belgium (women)	0.0150	0.0425	0.0208	0.1658	0.0076	0.0213
France (men)	0.0059	0.0178	0.0029	0.0245	0.0033	0.0096
France (women)	0.0056	0.0114	0.0043	0.0135	0.0030	0.0053
Italy (men)	0.0125	0.0204	0.0147	0.0327	0.0062	0.0105
Italy (women)	0.0083	0.0108	0.0086	0.0118	0.0042	0.0050
Netherlands (men)	0.0141	0.0150	0.0129	0.0146	0.0073	0.0086
Netherlands (women)	0.0080	0.0124	0.0067	0.0219	0.0043	0.0060
Portugal (men)	0.0117	0.0176	0.0130	0.0210	0.0063	0.0089
Portugal (women)	0.0124	0.0115	0.0075	0.0061	0.0126	0.0086
Spain (men)	0.0138	0.0170	0.0165	0.0247	0.0078	0.0084
Spain (women)	0.0055	0.0080	0.0032	0.0061	0.0029	0.0037
United Kingdom (men)	0.0101	0.0143	0.0080	0.0149	0.0049	0.0082
United Kingdom (women)	0.0104	0.0093	0.0114	0.0086	0.0052	0.0047

	Wins/losses	12/4**	Wins/losses	11/5	Wins/losses	13/3**

	W	50	W	62	W	30**

RMSE, root mean squared error; NMSE, normalized mean squared error; MAE, mean absolute error; BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “Wins/losses” stands for the number of cases in which BLC and FRLC point predictions are better/worse than FKSLC. (2) W stands for the value of the Wilcoxon rank test statistic. (3) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively.

Table 10a

Mean value of RMSE, NMSE, and MAE of the predictions on central mortality rates in 2001–2012 in sample populations (Item 1).

	RMSE		NMSE		MAE
	BLC/FRLC	FKSLC	BLC/FRLC	FKSLC	BLC/FRLC	FKSLC
Austria (men)	1.284	1.429	1.26E-03	1.42E-03	0.999	1.122
Austria (women)	0.3710	0.2432	2.97E-04	1.95E-04	0.3293	0.2158
Belgium (men)	0.6309	0.5018	6.31E-04	5.02E-04	0.5594	0.4443
Belgium (women)	0.1325	0.3647	1.06E-04	2.91E-04	0.1169	0.3298
France (men)	0.3806	0.5661	3.61E-04	5.44E-04	0.3155	0.4776
France (women)	0.1896	0.2127	1.41E-04	1.58E-04	0.1627	0.1818
Italy (men)	0.5940	0.9189	5.50E-04	8.65E-04	0.4933	0.7639
Italy (women)	0.1770	0.1739	1.34E-04	1.32E-04	0.1595	0.1558
Netherlands (men)	1.3555	1.2628	1.34E-03	1.25E-03	1.1806	1.1047
Netherlands (women)	0.3707	0.2834	3.00E-04	2.31E-04	0.3242	0.2526
Portugal (men)	0.6177	0.7797	6.27E-04	8.03E-04	0.4653	0.5970
Portugal (women)	1.2565	1.2897	2.34E-03	2.51E-03	0.5543	0.5793
Spain (men)	0.4146	0.6287	3.87E-04	5.95E-04	0.1495	0.2130
Spain (women)	0.2247	0.1671	1.68E-04	1.26E-04	0.1944	0.1433
United Kingdom (men)	0.9079	0.9679	8.82E-04	9.48E-04	0.7989	0.8514
United Kingdom (women)	0.5779	0.5256	4.74E-04	4.32E-04	0.4970	0.4516

	Wins/losses	9/7	Wins/losses	9/7	Wins/losses	9/7

	W	61	W	55	W	65

RMSE, root mean squared error; NMSE, normalized mean squared error; MAE, mean absolute error; BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) “Wins/losses” stands for the number of cases in which BLC and FRLC point predictions are better/worse than FKSLC. (2) W stands for the value of the Wilcoxon rank test statistic. (3) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively.

Table 10b

Mean value of RMSE, NMSE, and MAE of the predictions on life expectancies in 2001–2012 in sample populations (Item 1).

	Proportion of Successful Predictions			Test Results
	BLC	FRLC	FKSLC	Global comparison Friedman χ2 = 32* Iman-Davenport F Statistic = ∞32* Pairwise comparisons FRLC versus BLC Z score = 2.858 p-values: (a) 0.005; (b) 0.014; (c) 0.005 FKSLC versus FRLC Z score = −5.567 p-values:(a) 0.000; (b) 0.000; (c) 0.000 FKSLC versus BLC Z score = −2.828 p-values: (a) 0.005; (b) 0.014; (c) 0.009

Austria (men)	0.326	0.729	0.295
Austria (women)	0.524	0.847	0.431
Belgium (men)	0.434	0.576	0.236
Belgium (women)	0.576	0.760	0.354
France (men)	0.618	0.733	0.340
France (women)	0.587	0.767	0.396
Italy (men)	0.514	0.656	0.306
Italy (women)	0.681	0.802	0.476
Netherlands (men)	0.385	0.590	0.316
Netherlands (women)	0.618	0.806	0.483
Portugal (men)	0.566	0.705	0.257
Portugal (women)	0.583	0.778	0.451
Spain (men)	0.587	0.750	0.410
Spain (women)	0.556	0.792	0.486
United Kingdom (men)	0.458	0.688	0.358
United Kingdom (women)	0.698	0.858	0.451

Multiple sign test (the control method is BLC)

Wins/losses of FRLC against BLC: 16/0 + Wins/losses of FKSLC against BLC: 0/16 −

BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) Friedman χ2 follows a Squared-Chi with 2 grades of freedom and Iman–Davenport F follows a Snedecor F with 2(32) grades of freedom. (2) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively. (3) (a) indicates standard p-value and (b) and (c) Nemenyi and Holm p-value corrections for multiple pairwise comparisons. (4) “+” indicates that the evaluated method outperforms the control method with at least at 10% significance level, whereas “−” indicates that the evaluated method underperforms the control method with at least at 10% significance level.

Table 10c

Mean proportion of successful predictions on central mortality rates in 2001–2012 by BLC, FRLC, and FKSLC in sample populations (Item 2).

	Proportion of Successful Predictions			Test Results
	BLC	FRLC	FKSLC	Global comparison Friedman χ2 = 30.125* Iman-Davenport F Statistic = 241* Pairwise comparisons FRLC versus BLC Z score = 3.005 p-values: (a) 0.003; (b) 0.008; (c) 0.005 FKSLC versus FRLC Z score = −5.480 p-values: (a) 0.000; (b) 0.000; (c) 0.000 FKSLC versus BLC Z score = −2.475 p-values: (a) 0.013; (b) 0.004; (c) 0.013

Austria (men)	0.326	0.729	0.295
Austria (women)	0.524	0.847	0.431
Belgium (men)	0.434	0.576	0.236
Belgium (women)	0.576	0.760	0.354
France (men)	0.618	0.733	0.340
France (women)	0.587	0.767	0.396
Italy (men)	0.514	0.656	0.306
Italy (women)	0.681	0.802	0.476
Netherlands (men)	0.385	0.590	0.316
Netherlands (women)	0.618	0.806	0.483
Portugal (men)	0.566	0.705	0.257
Portugal (women)	0.583	0.778	0.451
Spain (men)	0.587	0.750	0.410
Spain (women)	0.556	0.792	0.486
United Kingdom (men)	0.458	0.688	0.358
United Kingdom (women)	0.698	0.858	0.451

Multiple sign test (the control method is BLC)

Wins/losses of FRLC against BLC: 16/0 + Wins/losses of FKSLC against BLC: 1/15 −

BLC, basic LC model; FRLC, Fuzzy-random extension of the LC model; FKSLC, fuzzy version by Koissi and Shapiro of the LC model.

Notes: (1) Friedman χ2 follows a Squared-Chi with 2 grades of freedom and Iman–Davenport F follows a Snedecor F with 2(32) grades of freedom. (2) “*,” “**,” and “***” stand for the rejection of the null hypothesis with a significance level of 10%, 5%, and 1%, respectively. (3) (a) indicates standard p-value and (b) and (c) Nemenyi and Holm p-value corrections for multiple pairwise comparisons. (4) “+” indicates that the evaluated method outperforms the control method with at least at 10% significance level, whereas “−” indicates that the evaluated method underperforms the control method with at least at 10% significance level.

Table 10d

Mean proportion of successful predictions on life expectancies in 2001–2012 by BLC, FRLC, and FKSLC in sample populations (Item 2).

Tables 10c and 10d reveal that Friedman rank test undoubtedly rejects that the three evaluated methods provide interval confidence predictions with homogenous accuracy. Likewise, we can observe in these tables that from the interval confidence prediction perspective, our method improves BLC and FKSLC. Also, that BLC provides better predictions than FKSLC. In this sense, multiple sign tests reveal that whereas FRLC improves BLC significantly, FKSLC performs poorer than the control method.

7. CONCLUSIONS AND FURTHER EXTENSIONS

This paper proposes a fuzzy-random approach of the LC model. A fuzzy version of the LC model was firstly proposed by [14], who considered two different formulations. In the first one, which was refined in the works [15,16], the authors introduced fuzziness in all the parameters of the model by using TFNs. Nevertheless, the model developed in this paper assumes, as it was done in the seminal paper [1] and its subsequent extensions, that the trend of mortality across time is captured with an ARIMA model.

This fuzzy-random approach of the LC model can also be used to derivate variables linked to central mortality rates as probabilities of death or survival and life expectancies. From these variables, it is possible to price life annuities or insurance contracts. It can be done by using directly fitted fuzzy probabilities, as in the framework exposed in [39] or, alternatively, by reducing these fuzzy probabilities to a crisp value with the use of a defuzzifiying method.

When applying this new model to Spanish male population within the period 1970–2012, it is found that the model is satisfactory when it comes to its capability of fitting outcomes in the estimation sample (1970–2000) and forecasting central mortality rates over a time horizon of more than 10 years (2001–2012).

Moreover, we have made a comparative assessment of our fuzzy-random methodology with seminal LC method [1] and fuzzy version of LC [14] and we have checked that, from interval confidence prediction perspective, our proposed methodology improves the models of these papers.

CONFLICT OF INTEREST

none

Funding Statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

ACKNOWLEDGMENTS

Authors thank helpful comments of two anonymous referees.

Footnotes

1

This section is especially benefited by the helpful suggestions of one anonymous referee.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 12 - 2
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TI  - A Fuzzy-Random Extension of the Lee–Carter Mortality Prediction Model
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International Journal of Computational Intelligence Systems

A Fuzzy-Random Extension of the Lee–Carter Mortality Prediction Model

1. INTRODUCTION

2. OVERVIEW OF THE LEE–CARTER MODEL

3. FNS AND FR

3.1. FNs and Their Arithmetic

3.2. FR Model with Asymmetric Coefficients

4. FUZZY-RANDOM APPROACH OF THE LC MODEL

4.1. Fuzzy-Random Fitting of the LC Model

4.2. Forecasting with the Fuzzy-Random LC Model

4.3. An Empirical Application of the Fuzzy-Random LC Model: The Case of Spanish Male Population

5. FORECASTING LIFE EXPECTANCY WITH THE FUZZY-RANDOM LC MODEL

5.1. Calculating Life Expectancy from Fuzzy Estimates of Central Mortality Rates

5.2. Predicting Life Expectancies of Spanish Male Population in 2001–2012

6. EMPIRICAL ASSESSMENT OF THE FUZZY-RANDOM LC MODEL IN EIGHT WESTERN EUROPEAN COUNTRIES1

6.1. Methodological Considerations

6.2. Comparison of BLC, FRLC, and FKSLC for Each Population

6.3. A Global Comparison of BLC, FRLC, and FKSLC

7. CONCLUSIONS AND FURTHER EXTENSIONS

CONFLICT OF INTEREST

Funding Statement

ACKNOWLEDGMENTS

Footnotes

REFERENCES

Cite this article

6. EMPIRICAL ASSESSMENT OF THE FUZZY-RANDOM LC MODEL IN EIGHT WESTERN EUROPEAN COUNTRIES¹