*Corresponding author: Dept. of Statistics, Mathematics and Insurance, College of Commerce, Benha University, Egypt. Email. drmervat.mahdy@fcom.bu.edu.eg. Tel: +20-122-068-2460, fax: +20-13-323-0860.
Corresponding Author
Mervat Mahdy
Received 24 March 2016, Accepted 19 November 2017, Available Online 31 March 2018.
Stochastic orders; Mean inactivity time class; Strong mean inactivity time class; Variance inactivity time; Preservation; Risk measure
Abstract
The paper opens up a new aging class and a new stochastic orders which is depend on risk class, that plays vital role in the reliability theory, finance topics, stochastic orders, and the economic theory. The article presents some new interesting implications and characterizations concerning this class. In addition, we list a series of inequalities that provide bounds for strong risk and some aging classes. Furthermore, a sufficient condition for a probability distribution to have a new class is provided. In addition, The paper demonstrates the preservation properties of a new stochastic order under some reliability operations such as mixture, and convolution. Moreover, some new reliability concepts based on discrete lifetime random variable are studied.
Consider a probability density function f(t) for a lifetime random variable X with distribution function F (t) and survival function
F¯(t)=1−F(t)
, t ∈ ℝ+. In addition, we assume that the mean life
μX=∫0∞F¯(u)du
and the variance
σX2
are finite. Likewise, let Y be the second lifetime random variable with the density function g (t), distribution function G (t) and survival function
G¯(t)=1−G(t)
; t ∈ ℝ+. Furthermore, the mean life
μY=∫0∞G¯(u)du
and variance
σY2
are both assumed to be finite. Let lX = inf {t ∈ ℝ+ : F (t) > 0}, uX = sup {t ∈ ℝ+ : F (t) < 1}, ΩX = (lX, uX), lY = inf {t ∈ ℝ+ : G (t) > 0}, uY = sup {t ∈ ℝ+: G (t) < 1} and ΩY = (lY, uY). Let X has reversed hazard rate function
r˜F(t)=f(t)/F(t)
, t > lX and Y has reversed hazard rate function
r˜G(t)=g(t)/G(t)
, t > lY. Then, the mean inactivity lifetime functions, and variance inactivity lifetime functions respectively are defined by
Let X1 and X2 be two non-negative and absolutely continuous random variables, with the distribution functions F1 (.) and F2 (.), density functions f1 (.) and f2 (.), reversed hazard rate functions
r˜F1(.)
and
r˜F2(.)
, mean past lifetime functions mF1 (.) and F2 (.), and the variance past lifetime functions
σF1(x)2(.)
and
σF2(x)2(.)
, respectively. Hence, X1 is smaller than or equal X2 in the following cases:
(i)
A reversed residual lifetime ordering (X1 ≤RHR X2), if
r˜F1(x2)≤r˜F2(x2)
, for all x2 > 0 or {F1 (x1) / F1 (x2)} ≥ F2 (x1) / F2 (x2)}, for all x1 ≤ x2.
(ii)
A mean past lifetime order (X1 ≤MP X2), if mF1 (x2) ≥ mF2 (x2), for all x2 > 0.
(iii)
A variance past lifetime order (X1 ≤VP X2) if
σF1(x)2(x2)≥σF2(x)2(x2)
, for all x2 > 0.
(iv)
A strong mean past lifetime order (X1≤SMP X2), if
∫0x2xF1(x)dx/F1(x2)≥∫0x2xF2(x)dx/F2(x2)
for all x2 ∈ ℝ+.
(v)
A likelihood ratio order (X1 ≤LR X2), if f1 (u) f2 (v) ≥ f1 (v) f2 (u), for all u ≤ v.
Recently, Mahdy (2012) shown that, if X1 ≤VP X2, we can be get that X1 (φ) ≤VP X2 (φ) for any concave transformation φ. In addition, X1 (φ) ≤VP X2(φ) ⇒ X1 ≤VP X2 for any strictly increasing function (φ).
In insurance, if we represent the distribution by the appropriate random variable X, and let θ present the risk measure functional, then
θ:X→ℝ.
Let an insurance contract in a specified period (0, a) and let Ω be the state space. If none of the risks specified in the policy contract happen during the policy term, then the policy holder has no monetary compensation for the paid premiums. Then the loss that the difference between amount of compensation (that denoted by ℂ) and total losses resulting from achieve state α occurs (that denoted by φ (X(α)) = X2 is convex function) is [ℂ – φ (X(α)) | X(α) ≤ ℂ], where X is premiums state α. Suppose we associate a “risk” as Var[ℂ – φ (X(α)) | X(α) ≤ ℂ], that is convex. We called it a strong risk function, SR, it is natural to demand that a risk function has this non-decreasing. Then the variance of random variable [ℂ – φ (X(α)) | X(α) ≤ ℂ] base on convex function, φ (X(α)), can use both in studying the effects of investor and analyst beliefs on securities trading. Also, we can use SR in measure of dispersion of returns of investment portfolio. In addition, SR measures the variability of a security’s returns relative to market index or a particular benchmark. If Var[ℂ – φ (X(α)) | X(α) ≤ ℂ] is greater than that of the benchmark, then the financial instrument is thought to be more perilous than the benchmark price. Low Var[ℂ – φ (X(α)) | X(α) ≤ ℂ] may mean lower risk.
As a result, the objective study is to achieve two aims. The first aim is consider a new nonparametric class of distributions, depended on SR with introduce some characterizations, and some preservation. The second aim is suggest a new technique that improves the comparison between two distributions with some characterizations, preservation results, and applications for strong risk order. Section 2 provides some properties and application of new class. Furthermore, the behavior of SR is presented. In addition, we listed a series of inequalities that provide bound for SMP and SR functions. Also, a sufficient conditions for a probability distribution to have a new class are studied. In Section 4, the relationships between the strong risk order and others stochastic orders will be discussed. Furthermore, useful properties and characterization of the strong risk order is studied. In addition, we establish closure properties of strong risk order under relevant reliability operations such as convolution, mixture and transformation. Finally, Section 5 discusses the useful applications in statistical reliability theory involving the strong risk order and the increasing strong risk class.
2 Characterizations of SR Class
Let a random variable X have the density function f (.), distribution function F (.), and reversed hazard rate
r˜F1(.)
. Setting
u(y)=∫0yF(x)dx
, and
E[U2|κ]=∫0∞u2dF[u|κ]dκ
, and by using integration by parts one has SR function we obtain:
θF(κ)=(κ(1−κ))2+4∫0κx(κ−x2)F(x)dxF(κ)−SF2(κ).
where
SF(κ)=∫0κxF(x)dx/F(κ)
, it is called strong mean past lifetime(SMP). Then, after some algebraic calculations, we obtain
Suppose we refer to strong risk with the symbol S, an increasing function with abbreviate I and for decreasing function with symbol D, we can define the following two classes of lifetime:
(i)
SD = (F ∈ ℒ : θF (κ), for all κ, is a D),
(ii)
SI = (F ∈ ℒ : θF (κ), for all κ, is a I).
Clearly SD and SI form a pair of dual classes base on the distinguishing θF (κ). In current section, we introduce and explain the properties and applications of the SD and SI classes. The sufficient conditions for F (.) to have the SD and SI are provided by next theorem.
Theorem 2.1:
Let T denote the lifetime of an equipment with SF(κ),
πF3(κ)
,
r˜F(κ)
and
SF\(κ)=∂SF(κ)/∂κ
. If
As mentioned earlier, the SR classes is important in reliability theory. The next example illustrated the application of theorem 2.1 in reliability theory whereas Theorem 1 in recognizing SI class.
Example 2.1:
Suppose Y be an Weibull random variable with the density function g (y) = βθβyβ−1 exp(−(θ(y)β), for y > 0, and β, θ > 0. One can easily prove that:
where ϖ(j) = (((j/β) – 1) ((j/β) – 2) … ((j/β) – k)). By using (2.4) it can easily check that
SG(y)(SG\(y)−1)−ySG\(y)<y(1−3y)+(r˜G(y)πG3(y))/2.
It follows from Theorem (2.1) that Y ∈ SI. Also, according to Y ∈ DRHR we obtain Y ∈ SI.
Introduce the notation
G¯(r)(κ)=∫κ∞G¯(r−1)dI(<∞),forr=1,2,…
where
G¯(0)≡G¯
, and Φr ≡ E(Yr). When normalized, these are the survival functions corresponding to the distribution function G(r) of Smith (1959, pp. 6), furthermore, by using Hall and Wellner (1981, pp. 173) we can prove that
Φr=r!G¯(r)(0)
, for r = 1, 2, … with
G¯(r)
finite if and only if Фr is finite. Hence
dG¯(r)/dκ=−G¯(r−1)
.
The following a list series of inequalities, those provide bounds for SMP and SR functions.
Since
κ−SG(κ)=E[Y2|Y≤κ],
we have
{κ−SG(κ)}G(κ)=E[Y2.I(Y<κ)]=Φ2−E[Y2.I(Y≥κ)].
It is clear that
E[Y2.I(Y<κ)]≤κ2G(κ)
and
E[Y2.I(Y<κ)]≤Φ2,
and by using Jensen’s inequality, we get
E[Y2|Y≤κ]≤(κ−mG(κ))2.
Further, Hölder’s inequality implies that
E[Y2.I(Y<κ)]≤[E(Y2w)]1/w[G(κ)](1−1w),forw>1,
Similarly,
E[Y2.I(Y≥κ)]≤κ2G¯(κ)
is also true for all w > 1. In this way we obtain
E[Y2.I(Y≥κ)]≤[E(Y2ω)]1/w[G¯(κ)](1−1w),forw>1.
It means
κ2=(E(Y2w))1/w(G¯(κ))−1w.
Furthermore, we can prove that
θG(κ)=E[Y4|Y≤κ]−(E[Y2|Y≤κ])2.
We use below inequalities to obtain bound for SG(κ) and θG(κ).
Proposition 2.1.
If G is non-degenerate with Фr ≡ E(Yr) < ∞, then,
(a)
SG(κ) ≥ κ – (Ф2 / G (κ));
(b)
θG(κ) ≤ Ф4;
(c)
SG(κ)≥κ−[Φ2wG(κ)]1/w
, for all κ and any w > 1;
(d)
SG(κ)≥κ−{(Φ2−Φ2w1/w[G¯(κ)](1−1w))/G(κ)}
, for Y ≤ κ and any w > 1.
Proof.
Since E[Y2.I(Y < κ)] = Ф2 – E [Y2.I(Y ≥ κ)] , it implies that G (κ) (κ – SG(κ)) ≤ Ф2. Then, we finished the proof (a). If
We started by investigating the preservation properties of SR class closed under some reliability operations such as mixture.
For a family of distribution {M (θ | X), x ≥ 0} of random samples X and mixing distribution (prior distribution) D (θ), it is necessary to evaluate the mixture distribution (Ψ (.)). It obtained as follows
Ψ(κ)=∫0∞M(θ|κ)dD(θ),κ>0.
In order to verify the validity of SR class will be closed under mixture, we carried out the following theory.
Theorem 3.1.
Suppose that M (θ | X) ∈ SI for any θ ∈ ℝ+. Then Ψ (κ) ∈ SI.
Proof.
Without loss of generality, suppose that M (θ | X) = Exp (θ) and D (θ) = Exp (1), that is, M (θ | κ) = Mθ (k) = 1 – e–θκ, κ > 0, and D(θ) = 1 – e–θ, θ > 0, after some algebraic calculations, we have
Clearly, U ∈ unif (0, 1), while V ∈ Exp (1). Now, consider the following mixture of two distributions U and V :
Eθ(x)=θU(x)+θ¯V(x),whereθ∈(0,1)andθ¯=1−θ.
A random variable T is the lifetime of some devices with the distribution function Eθ. We started by investigating Eθ ∈ SI or Eθ ∈ SD. Hence, by using (2.5) we have
where
SC(κ)=∫0κxC(x)dx/C(κ)
,
r˜C(κ)=∂C(κ)/∂κC(κ)
, and
πC3(κ)=4∫0xy3C(y)dy/C(κ)
, εκ = (−κ exp (−κ) − exp (−κ) + 1), ρ(κ) = (1 − θ)2(24−24κe−κ − 16κ2e−κ −κ4e−κ − 24e−κ) and η (κ) = (2θ (1 – θ) – (1 – θ)2) (6 – 6κe–κ – 4κ2e–κ – 6e–κ) Then we can decide that ∂θC (κ) / ∂κ is I. Thus C ∈ SI. Therefore, T1 and T2 ∉ SI. But their convolution C is belong to SI. It mean SI is not closed under convolution, in general. Hence, the proof is complete.
4 Properties of SR Ordering Under Operations
In this section, we focus our attention on a new stochastic comparison defined in terms of SR function. The next theorem provides that ≤θ(t) order lies between ≤RHR and ≤ θ(t).
Theorem 4.1.
Let X1 ≥ 0 and X2 ≥ 0 be two continuous random variables, with the distribution functions F1 (.) and F2 (.), and let
αi(t)=∫0tviF1(v)dv/F1(t)
and
βi(t)=∫0tviF2(v)dv/F2(t)
. Then we have
By (4.1) we can conclude that X1 ≤θ(t)X2 does not hold in this case. While in parallel, Kayid and Izadkhah (2014, pp.595) showed in their counterexample 1 that X1 ≤SMP X2. Also, (4.2) is increasing in t, and hence X1 ≤VP X2.
The next result provides useful characterization of the SR order.
Theorem 4.2.
Suppose X1 ∈ ℝ+ and X2 ∈ ℝ+ be two continuous random variables with distribution functions F1 and F2, respectively, and let
αi(x)=∫0xuiF1(u)du/F1(x)
and
βi(x)=∫0xuiF2(u)du/F2(x)
, hence X1 ≤SMP X2 holds, if X1 ≤θ(t)X2 holds, therefore
since X1 ≤SMP X2, Thus the following result is obtained:
∫0tu3F1(u)F2(t)−u3F2(u)F1(t)du≥0,
consequently,
tmF1(t)−tmF2(t)≥SF1(t)−SF2(t),
which is equivalent to the required statement.
In next result develop some preservation of SMP and SR orders and SI class.
Proposition 4.1.
Let X1 ∈ ℝ+ be continuous random variable with distribution function F1 and SMP function SF1 (t), hence SF1 (t) is increasing in t if X1 ≤SMP X1 + X2 for any continuous random variable X2 ∈ ℝ+ independent of X1.
Proof:
By apply Fubini’s theorem and using change of order double integration, it is obvious that
This is true for SF1 (t) is increasing in t ∈ ℝ+. Therefore, we shall write the above expression as
SX1+X2(t)≥SX1(t),
when
a=∫0tuF1(t−u)mX1(t−u)dF2(u)/∫0tF1(t−u)dF2u
and a ≥ SX1 + X2 (t). Hence, we have the required results.
Theorem 4.3.
Let X1 ∈ ℝ+ be continuous random variable with distribution function F1 and SR function θF1 (t), then X1 ∈ SI if X1 ≤ θ(t)X1 + X2 for any continuos random variable X2, that is independent of X1.
Proof:
Based on Fubini’s theorem and using change of order double integration, it is clear that
By apply proposition (4.1) we can prove that
SX1+X22(t)≥SX12(t)
. This completes the proof.
Theorem 4.4.
Suppose X1 ∈ ℝ+ and X2 ∈ ℝ+ are two random variables with distribution functions U1 and U2 respectively, and let ψ is strictly increasing and convex upwards and ψ (0) = 0. If X1 ≤ θ(t)X2, then we have ψ (X1) ≤ θ(t)ψ (X2).
Proof:
Let ψ is differentiable with derivative ψ\. If X1 ≤θ(t)X2 then for any t > 0, we have
where β(s) = ψ3 (s) ψ/ (s) / s3. Its obvious that, if ψ (s) is non-negative and decreasing, we can conclude that ψ3 (s) / s3 is decreasing in s, it is implies that β (s) is decreasing in s. Therefore, the complete proof is given by apply lemma 7.1(b) of Barlow and Proschan (1981). .
5 Reliability Applications of SR Ordering
In the following results, we explore the possibility of apply a new techniques in statistical reliability theory. Let U1, U2, …, Un denote the component lifetimes of the system and assume that U1:n, U2:n, …, Un:n represent the ordered lifetimes of the components.
5.1 Order Statistics
Theorem 5.1.
Let U1, U2, …, Un and V1, V2, …, Vn be independent and identically distributed (i.i.d) copies of U and V, with distribution functions ℍ1 and ℍ2 respectively. If Un:n ≤θ(t)Vn:n, then we have Ui ≤θ(t)Vi.
Suppose X ∈ ℝ+ and Y ∈ ℝ+ be two random variables, with distribution functions FX and FY, and discrete strong mean past lifetime Sd,X (x) and Sd,Y (x) respectively. Then, it is said that X is smaller than or equal Y in discrete SR order (X ≤d–θ(x)Y), if
Given a sequence of absolutely continuous non-negative random variables Z1, Z2, …are i.i.d. random variables with common cumulative distribution function FZ1 and a common density function fZ1 Let 𝕄1 ∈ N+ and 𝕄2 ∈ N+ be two random variables which are independent of the Zi’s. Denote Z1:𝕄j ≡ min{Z1, Z2, ..., Z𝕄j} and Z𝕄j:𝕄j ≡ max{Z1, Z2, ..., Z𝕄j}, j = 1, 2. Below we consider the SR order between two such extreme order statistics.
Theorem 5.2.
Let 𝕄1 ≤d–θ(z) 𝕄2, then we have Z𝕄1:𝕄1 ≰θ(z)Z𝕄2:𝕄2.
Proof.
The density function of Z𝕄j:𝕄j is given by
fZ𝕄j:𝕄j(z)=∑n=1∞nFZ1n−1(z)fZ1(z)Pr(𝕄j=n),
and the distribution function of Z𝕄j:𝕄j which is given by
where
ϕ(z,n)=∫0z∫0y{Fn(u)−y2Fn(y)}dudy
and in addition, τ(n, j) = Pr (𝕄j = n). Let
k(n,j)=∑i=n∞τ(i,j)
for each n ∈ N and j = 1, 2. We observe that 𝕄1 ≤LR 𝕄2, implies 𝕄1 ≤θ(z) 𝕄2 by using Theorem 1.C.1 on page 43 of Shaked and Shanthikumar (2007), and Li and Zhang (2008). If 𝕄1 ≤LR 𝕄2, and thus k(n, j) is TP2 in (n, j) ∈ N × {1, 2}. But it is not hard to see that φ(z, n) isn’t TP2 in (n, j) ∈ N × {1, 2}, where det [φ(z, n)] is negative for any strictly increasing convex function and
zFn(z)−∫0zufZ𝕄j:𝕄j(u)du≤z3Fn(z)
, for z ∈ R+. Also, therefore, by the basic composition formula (Theorem 5.1 on page 123 of Karlin (1968)) we can be seen that ψ(z, j) is not TP2 in z ≥ 0 and j ∈ {1, 2}. That is, Z𝕄1:𝕄1 ≰θ(z)Z𝕄2:𝕄2.
5.3 Inactivity time of random at a random
Let U and V be two the lifetimes of system C1 and system C2, respectively. Then the residual lifetime at random time V can be represented as
UV=U−V|U>V.
It represents the residual life of system C1 at a time when system C2 fails. Dequan and Jinhua (2000) established a number of stochastic orders for UV under a lot of assumptions of U and V. In addition, Misra et al. (2008) studied adequate permissions for log-concavity and log-convexity of the residual life at random time. Stochastic orders are conducted under particular permissions on the concerned total life and random time and their preservation properties are established by Li and Zuo (2004). Dewan and Khaledi (2014) presented now stochastic orders between multivariate residual lifetime at random time and studies some characterizations. Furthermore suppose G (.) be distribution function of random variable V and Vi’s hold for V as well. Moreover, let U and V are independent. The random variable U(V) = V – U |U ≤ V is called the inactivity time of U at a random time V. It has the distribution function
by using (5.1), we can decide that U(V) ≤SMP U. Further, let U(V) ≤SMP U holds for all value of y and let y equal to constant then U(t) ≤SMP U for all t ≥ 0.
Proposition 5.1.
U(V) ≤θ(t)U for any V that is d-independent of U iff U(t) ≤θ(t)U for all t ≥ 0.
Proof.
With same steps in theorem 8, and by using (2.4) and (5.1) and we can achieve its proof.
[3]WJ Hall and JA Wellner, M Csorgo, JNK Rao, and AKMdE Saleh (editors), Mean Residual Life, in: Statistics and Related Topics, North Holland, Amesterdam, 1981, pp. 169OE184.
[4]S Karlin, Total Positivity, Stanford University Press, Stanford, California, Vol. I, 1968.
TY - JOUR
AU - Mervat Mahdy
PY - 2018
DA - 2018/03/31
TI - On Some Aspects of Strong Risk Class and Associated Ordering
JO - Journal of Statistical Theory and Applications
SP - 172
EP - 192
VL - 17
IS - 1
SN - 2214-1766
UR - https://doi.org/10.2991/jsta.2018.17.1.13
DO - 10.2991/jsta.2018.17.1.13
ID - Mahdy2018
ER -
