On a Generalized Burr Life-Testing Model: Characterization, Reliability, Simulation, and Akaike Information Criterion

M. Ahsanullah; M. Shakil; B. M. Golam Kibria; M. Elgarhy

doi:10.2991/jsta.d.190818.001

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Volume 18, Issue 3, September 2019, Pages 259 - 269

On a Generalized Burr Life-Testing Model: Characterization, Reliability, Simulation, and Akaike Information Criterion

Authors

M. Ahsanullah¹, M. Shakil²^{, *}, B. M. Golam Kibria³, M. Elgarhy⁴

¹Rider University, New Jersey, USA

²Miami Dade College, Hialeah, Florida, USA

³Florida International University, Florida, USA

⁴University of Jeddah, Jeddah, KSA

^*Corresponding author. Email: mshakil@mdc.edu

Corresponding Author

M. Shakil

Received 22 October 2018, Accepted 22 March 2019, Available Online 30 August 2019.

DOI: 10.2991/jsta.d.190818.001 How to use a DOI?
Keywords: Akaike information criterion; Characterization; Reliability; Truncated moment
Abstract: For a continuous random variable X, M. Shakil, B.M.G. Kibria, J. Stat. Theory Appl. 9 (2010), 255–282, introduced a generalized Burr increasing, decreasing, and upside-down bathtub failure rate life-testing model. In this paper, we provide some characterizations of this life-testing model by truncated first moment, order statistics and upper record values. We also investigate the reliability, simulation, and Akaike information criterion for this model.
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

For a continuous positive random variable X, the following increasing, decreasing, and upside-down bathtub failure rate life testing five-parameter family of continuous probability density function (pdf) in terms of beta function has been introduced by Shakil and Kibria [1]:

(1)fX(x)=p(α)v−μp(β)μpxμ−1B(μp,ν−μp)α+βxpν,

where x>0, α>0, β>0, μ>0, ν>0, p>0, and where, in view of the definition of beta function, the parameters are chosen as such that ν>μp. For details, the interested readers are referred to Shakil and Kibria [1], and references therein. The cumulative distribution function (cdf) is given by

(2)FXx=PrX≤x=pβαμpBμp,ν−μp∫0xtμ−11+βαtpνdt,

(3) =pβαμpBμp,ν−μp∑k=0∞−1kΓν+kΓνβαkxμ+pk(k!)μ+pk,

(4) =pβαμpBμp,ν−μp∑k=0∞−1kβαkνkxμ+pkk!μ+pk.

(5)=pμβαμpxμB(μp,ν−μp)2F1(ν,μp;μp+1;−βαxp),

where 2F1(.) and νk denote the Gauss hypergeometric and Pochhammer functions.

The kth moment, αk, for some integer k>0, is given by

(6)αk=EXk=∫0∞xkfXxdx=αβkpBμ+kp,ν−μ+kpBμp,ν−μp.

The first moment, when k = 1 in (6), is given by

(7)α1=EX=αβ1pBμ+1p,ν−μ+1pBμp,ν−μp,

which is obviously a finite and fixed number for α>0, β>0, μ>0, ν>0, p>0, and ν p>μ. The possible shapes of the pdf (1) are provided for some selected values of the parameters in Fig. 1 (a and b) below.

The effects of the parameters can easily be seen from the above graphs. From these graphs, it is observed that the above-said distribution is right skewed. Also, since the mode is the value of x for which the probability density function fXx defined in (1) is maximum, therefore differentiating (1), we have

(8)dfXxdx=αμ−1−βν p−μ+1xpfXxxβxp+α,

which, when equated to 0, gives the mode of the pdf (1) as

(9)x=αμ−1βν p−μ+11p=δsay, α>0, β>0, μ>0, ν>0, p>0,

which exists provided μ>1, and ν p>μ. Differentiating (8) and using (9), it can easily be seen by simple arguments that d2fX(δ)dx2<0. Thus the maximum value of the probability density function (1) is given by fXδ, where δ is given by (9). It follows that the probability density function (1) is unimodal.

The paper is organized as follows. In Section 2, we present some characterization results. We provide reliability results in Section 3. The Akaike information criterion (AIC), etc., are discussed in Section 4. Finally, some concluding remarks are presented in Section 5.

2. CHARACTERIZATION RESULTS

Since the truncated distributions arise in practical statistics where the ability of record observations is limited to a given threshold or within a specified range, there has been a great interest, in recent years, in the characterizations of probability distributions by truncated moments. The development of the general theory of the characterizations of probability distributions by truncated moment began with the work of Galambos and Kotz [2], Kotz and Shanbhag [3], Glänzel et al. [4], and Glänzel [5]. For recent developments, we refer our readers to Ahsanullah [6], and references therein. In this section, we provide the proposed characterizations of the said distribution by truncated moment, order statistics, and upper record values. For this, we shall need the following assumption and lemmas.

2.1. Assumption and Lemmas

Assumption 2.1.1.

Suppose the random variable X is absolutely continuous with the cumulative distribution function Fx and the probability density function fx. We assume that γ=infx|Fx>0, and δ=supx|Fx<1. We also assume that fx is a differentiable for all x, and EX exists.

Lemma 2.1.1.

Under the Assumption 2.1.1, if EX|X≤x=gxτx, where τx=fxFx and gx is a continuous differentiable function of x with the condition that ∫γxu−g′ugudu is finite for all x, γ<x<δ, then f(x)=ce∫γxu−g′(u)g(u)du, where c is determined by the condition ∫γδfxdx=1.

Proof:

See Ahsanullah and Shakil [7].

Lemma 2.1.2.

Under the Assumption 2.1.1, if EX|X≥x=gxrx, where rx=fx1−Fx and gx is a continuous differentiable function of x with the condition that ∫xδu+g′ugudu is finite for all x, γ<x<δ, then f(x)=ce−∫xδu+g′(u)g(u)du, where c is determined by the condition ∫γδfxdx=1.

Proof.

See Ahsanullah and Shakil [7].

2.2. Characterization by Truncated First Moment

We provide the proposed characterizations in Theorems 2.2.1 and 2.2.2.

Theorem 2.2.1.

If the random variable X satisfies the Assumption 2.1.1, with γ=0 and δ=∞, then EX|X≤x=gxfxFx, where

(10)g(x)=−x(α+βxp)νB(μ+1p,1)3F2(ν,μp,μ+1p;μp+1,μ+1p+1;−βαxp)μαν,

if and only if X has the distribution with the pdf (1), where Bμ+1p,1 represents the beta function and 3F2(ν,μp,μ+1p;μp+1,μ+1p+1;−βαxp) represents the generalized hypergeometric function (see, for example, Abramowitz and Stegun [8], Gradshteyn and Ryzhik [9], and Prudnikov et al. [10], among others).

Proof.

Suppose that EX|X≤x=gxfxFx. Then, since EX|X≤x=∫0xu fuduFx, we have gx=∫0xu fudufx. Now, if the random variable X satisfies the Assumption 2.1.1 and has the distribution with the pdf (1), then we have

g(x)=∫0xuf(u)duf(x)=−u(1−F(u))|0xf(x)+∫0x(1−F(u))duf(x) =−xf(x)+∫0x[1−pμβαμpuμB(μp,ν−μp)2F1(ν,μp;μp+1;−βαup)]duf(x) =−(α+βxp)ν∫0x[uμ2F1(ν,μp;μp+1;−βαup)]duμανxμ−1 =−x(α+βxp)νB(μ+1p,1)3F2(ν,μp,μ+1p;μp+1,μ+1p+1;−βαxp)μαν,

which follows from Prudnikov et al., [10], Vol. 3, Eq. 2.21.1.4, Page 314.

Conversely, suppose that

g(x)=−x(α+βxp)νB(μ+1p,1)3F2(ν,μp,μ+1p;μp+1,μ+1p+1;−βαxp)μαν.

Then, differentiating gx with respect to x, using Lemma 2.1.1, and simplifying, we have

g/(x)=x−g(x)(μ−1x−βpxp−1α+βxp),

from which we obtain

x−g/(x)g(x)=μ−1x−βpxp−1α+βxp.

Since, by Lemma 2.1.1, we have

x−g/(x)g(x)=f/(x)f(x),

it follows that

f/(x)f(x)=μ−1x−βνpxp−1α+βxp.

On integrating the above expression with respect to x and simplifying, we obtain

ln fx=lnc xμ−1α+β xpν,

or,

fx=c xμ−1α+β xpν,

where c is the normalizing constant to be determined. Thus, on integrating the above equation with respect to x from x = 0 to x = ∞, using the condition ∫0∞fxdx=1, and following Gradshteyn and Ryzhik [9], Equation 3.194.3, Page 285, we easily obtain

c=p ανBμp,ν−μpβα−μp,

and, hence, we have

fXx=pαν−μpβμpxμ−1Bμp,ν−μpα+β xpν,

which is the required pdf (1). This completes the proof of Theorem 2.2.1.

Theorem 2.2.2.

If the random variable X satisfies the Assumption 2.1.1, with γ=0 and δ=∞, then EX|X≥x=g~xfx1−Fx, where

g~x=∫x∞u fudufx=EXfx+∫0xu fudufx=EXfx+gxfx,

if and only if X has the distribution with the pdf (1), where EX, the first moment, is given by (7), gx is given by (10), and fx represents the pdf (1)

Proof.

The proof is omitted, since it is similar to the Theorem 2.2.1, and easily follows from Lemma 2.1.2.

2.3. Characterization by Order Statistics

Suppose that X1,X2,…,Xn are n independent random variables with the continuous cumulative distribution function Fx and probability density function fx. We assume that X1,n≤X2,n≤…≤Xn,n are the corresponding order statistics. Then Xj,n|Xk,n=x, for 1≤k<j≤n, is distributed as the j−kth order statistics from n−k independent observations from the random variable V, and has the probability density function fVv|x, where fVv|x=fv1−Fx, 0≤v<x, see, for example, Ahsanullah et al. [11], chapter 5, or Arnold et al. [12], chapter 2. Further, Xi.,n|Xk,n=x,1≤i<k≤n, is distributed as ith order statistics from k independent observations from the random variable W having the pdf fWw|x where fWw|x=fwFx,w<x. Let Sk−1=1k−1X1,n+X2, n+…+Xk−1,n, and Tk,n=1n−kXk+1,n+Xk+2,n+…+Xn,n.

Now, based on the order statistics, we will provide the characterizations in Theorem 2.3.1 and 2.3.2 below.

Theorem 2.3.1.

Assume that the random variable X satisfies the Assumption 2.1.1, with γ=0 and δ=∞, then ESk−1|Xk,n=x=gx τx, where τx=fxFx, and

g(x)=−x(α+βxp)νB(μ+1p,1)3F2(ν,μp,μ+1p;μp+1,μ+1p+1;−βαxp)μαν,

if and only if fXx=pαν−μpβμpxμ−1Bμp,ν−μpα+βxpν.

Proof.

Since ESk−1|Xk,n=x=EX|X≤x, see David and Nagaraja [13], the proof of Theorem 2.3.1 easily follows from Theorem 2.2.1.

Theorem 2.3.2

Assume that the random variable X satisfies the Assumption 2.1.1, with γ=0 and δ=∞. Then, ETk,n|Xk,n=x=g~xrx, where rx=fx1−Fx and

g~x=∫x∞u fudufx=EXfx+∫0xu fudufx=EXfx+gxfx,

if and only if fXx=pαν−μpβμpxμ−1Bμp,ν−μpα+βxpν.

Proof.

Since ETk,n|Xk,n=x=EX|X≥x, see David and Nagaraja [13], the proof of Theorem 2.3.2 easily follows from Theorem 2.2.2.

2.4. Characterization by Upper Record Values

Suppose that X1,X2,…, is a sequence of independent and identically distributed absolutely continuous random variables with cumulative distribution function Fx and probability density function fx. Let Yn=maxX1,X2,…,Xn for n≥1. We say that Xj is an upper record value of Xn,n≥1 if Yj>Yj−1,j>1. The indices at which the upper records occur are given by the record times Un>minj|j>Un+1,Xj>XUn−1,n>1 and U1=1. We will denote the nth upper record value as Xn=XUn. For details on record values, see Ahsanullah and Nevzorov [14], among others. In the following theorem, we will provide the characterization based on upper record values.

Theorem 2.4.1.

Assume that the random variable X satisfies the Assumption 2.1.1, with γ=0 and δ=∞. Then, EXn+1|Xn=x=g~xrx, where rx=fx1−Fx and

g~x=∫x∞u fudufx=EXfx+∫0xu fudufx=EXfx+gxfx,

if and only if

fXx=pαν−μpβμpxμ−1Bμp,ν−μpα+β xpν.

Proof.

Since EXn+1|Xn=x=EX|X≥x, see Nevzorov [15], the proof of Theorem 2.4.1 easily follows from Theorem 2.2.2.

3. RELIABILITY ANALYSIS

The reliability analysis of lifetime distributions plays an important role in modelling many real world phenomena in the fields of biological, economics, engineering, physical, and other pure and applied sciences. For a nonrepairable population, we define the failure rate as the instantaneous rate of failure for the survivors to time t during the next instant of time. We investigate some reliability properties of the said model. The corresponding survival (or reliability) and the hazard (or failure rate) functions of the said model are respectively given by

(11)S(x)=R(x)=1−FX(x)=1−pμβαμpxμB(μp,ν−μp)2F1(ν,μp;μp+1;−βαxp),

and

(12)h(x)=fX(x)1−FX(x)=p(α)ν−μp(β)μpxμ−1B(μp,ν−μp)(α+βxp)ν1−pμβαμpxμB(μp,ν−μp)2F1(ν,μp;μp+1;−βαxp),

where x>0,α>0,β>0,μ>0,ν>0,p>0, and ν p>μ. Differentiating (12) with respect to x, we have

(13)h/(x)=f/(x)f(x)h(x)+[h(x)]2 =(α(μ−1)−β(νp−μ+1)xp)h(x)x(βxp+α)+[h(x)]2,

for x>0. In order to discuss the behavior of the failure rate function, h x, let h/x=0. We observe that the nonlinear equation h/x=0 does not have a closed form solution, but could be solved numerically by using some mathematical software such as Maple, Mathematica, and R. It is obvious from (13) that h/x is positive irrespective of the values of the parameters. This confirms the Increasing Failure Rate (IFR) property of said model. For some special values of the parameters, the graphs of the hazard function (12) are illustrated in Fig. 2 (a and b) below.

The effects of the parameters are obvious from these figures. The increasing, then decreasing, and upside-down bathtub shape behaviors of the failure rate function, hx, are observed from these figures. Further, it is also sometimes useful to find the average failure rate function (AFR), over any interval, say, 0,t, that averages the failure rate over the interval, 0,t, see, for example, Barlow and Proschan [16]. Thus, the AFR) of the distribution of the said model, over the interval 0,t, is given by

AFR=−ln(R(t))t=−1tln(1−pμβαμptμB(μp,ν−μp)2F1(ν,μp;μp+1;−βαtp)),

which in view of the expansion of logarithmic function as a power series, is seen to be positive irrespective of the values of the parameters, and hence the distribution of the said model is increasing failure rate on average, that is, Increasing Failure Rate on Average (IFRA). Also, recall that a life distribution F . is NBU (new better than used) if Rx+y≤RxRy,∀x,y≥0, and NWU (new worse than used) if the reversed inequality holds, see, for example, Barlow and Proschan [16]. We note that, for the said model, since

R(x+y)=1−pμβαμp(x+y)μB(μp,ν−μp)2F1(ν,μp;μp+1;−βα(x+y)p),

and

R(x). R(y)= {1−pμβαμp(x)μB(μp,ν−μp)2F1(ν,μp;μp+1;−βα(x)p)} ×{1−pμβαμp(y)μB(μp,ν−μp)2F1(ν,μp;μp+1;−βα(y)p)},

it is easily seen that Rx+y≤Rx. Ry, which implies that the distribution of the said model has the property of NBU.

4. ESTIMATION, SIMULATION, AND AIC OF THE PERFORMANCE OF GENERALIZED BURR LIFE-TESTING MODEL

4.1. The Method of Moments

From the kth moment, EXk, in (6) of the said model, taking k=1,2,3,4,5, and evaluating the respective expressions numerically, we obtain the first five moments. Then, the moment estimations (MME's) of the parameters α>0,β>0,μ>0,ν>0, and p>0 can be determined by solving the system of five equations obtained from (6) by Newton–Raphson's iteration method, and using the computer package such as Maple, Mathematica 9, or R, MathCAD14, or other software.

4.2. The Method of Maximum Likelihood

Given a sample xi, i=1,2,3,…,n, the likelihood function of (1) is given by L=∏i=1nfxi. The objective of the likelihood function approach is to determine those values of the parameters that maximize the function L. Suppose that R=lnL=∑i=1nlnfxi. Then, upon differentiation, the maximum likelihood estimates (MLE) of the parameters α>0,β>0,μ>0,ν>0, and p>0 are obtained by solving the maximum likelihood equations ∂R∂α=0, ∂R∂β=0, ∂R∂μ=0, ∂R∂ν=0, and ∂R∂p, applying the Newton–Raphson's iteration method, and using the computer package such as Maple, Mathematica 9, or R, MathCAD14, or other software.

4.3. Simulation

In this section, we use simulation to compare the performances of the different methods of estimation mainly with respect to their biases, mean square errors (MSE), and variances for different sample sizes. A numerical study is performed using Mathematica 9 software. Different sample sizes are considered through the experiments at size n = 15, 20, 30, and 50 for different values of the parameters α>0,β>0,μ>0,ν>0, and p>0. The experiment is repeated 1000 times. In each experiment, the estimates of the parameters are obtained by the method of MLE. The means, MSEs and biases for the different estimators are reported from these experiments in Table 1 respectively below.

n	Par	Init	MLE	Bais	MSE	Init	MLE	Bais	MSE
	α	1.2	1.2536	0.0535	0.0219	1.4	1.4435	0.0435	0.0240
	β	1.2	1.1700	−0.0300	0.0195	1.4	1.3829	−0.0171	0.0252
10	ν	1.4	1.0709	−0.3291	0.1736	1.6	1.2408	−0.3593	0.3325
	µ	0.5	0.4269	−0.0731	0.0085	0.5	0.4348	−0.0652	0.0084
	p	0.8	1.5221	0.7221	0.9093	0.7	0.8870	0.1870	0.0993
	α	1.2	1.2437	0.0437	0.0101	1.4	1.4271	0.0271	0.0104
	β	1.2	1.1644	−0.0356	0.0102	1.4	1.3842	−0.0158	0.0128
20	ν	1.4	1.0480	−0.3520	0.1424	1.6	1.1871	−0.4129	0.1937
	µ	0.5	0.4240	−0.0760	0.0073	0.5	0.4344	−0.0656	0.0064
	p	0.8	1.4138	0.6138	0.5405	0.7	0.8703	0.1703	0.0618
	α	1.2	1.2390	0.0390	0.0069	1.4	1.4216	0.0215	0.0069
	β	1.2	1.1649	−0.0351	0.0075	1.4	1.3854	−0.0146	0.0087
30	ν	1.4	1.0382	−0.3618	0.1413	1.6	1.1794	−0.4206	0.1932
	µ	0.5	0.4241	−0.0759	0.0068	0.5	0.4343	−0.0657	0.0057
	p	0.8	1.3944	0.5944	0.4625	0.7	0.8622	0.1622	0.0488
	α	1.2	1.2386	0.0385	0.0044	1.4	1.4273	0.0273	0.0047
	β	1.2	1.1608	−0.0392	0.0049	1.4	1.3743	−0.0257	0.0057
50	ν	1.4	1.0354	−0.3646	0.1382	1.6	1.1835	−0.4165	0.1826
	µ	0.5	0.4223	−0.0777	0.0066	0.5	0.4297	−0.0703	0.0058
	p	0.8	1.3558	0.5558	0.3626	0.7	0.8410	0.1410	0.0331

MLE, maximum likelihood estimate; MSE, mean square errors.

Table 1

The parameter estimation of the generalized burr life-testing model using MLE.

If we review Table 1, it is observed that as the sample size increases, both absolute bias and MSE decrease and converge to close to true values of parameters.

4.4. Akaike Information Criterion

As pointed out by Emiliano et al. [17], the choice of the best model is crucial in modeling data. Akaike [18,19] developed an entropy-based information criterion to test the applicability, relative measures, and performance of lifetime distributions to real life data, known in the literature as the AIC. Later, various information criteria for testing the goodness of fit of lifetime probability models were developed by different authors and researchers, among them the corrected Akaike information criterion (CAIC) (Sugiura [20]), the Bayesian information criterion (BIC) (Schwarz [21]), the generalized information criterion (GIC) (Konishi and Kitagawa [22]), and Hannan–Quinn information criterion (HQIC) (Hannan and Quinn [23]), are notable. For details on these, the interested readers are referred to Emiliano et al. [17], and references therein. In this section, we investigate the performance of the generalized Burr life-testing model by using various criteria such as −2lnL, AIC, BIC, the CAIC, and the HQIC. For this, we consider some health related data as described in Example 1 below.

Example 1.

This example considers the data for average annual percent change in private health insurance premiums (All Benefits: Health Services and Supplies), Calendar Years 1969–2007 (SOURCE: Centers for Medicare & Medicaid Services, Office of the Actuary, National Health Statistics Group), which are provided in Table 2. Based on this example, we test the goodness of fit of the generalized Burr life-testing model and compare it with the Burr's type XII, log-logistic, gamma, and lognormal distributions.

14.4, 14.0, 15.4, 9.4, 11.7, 15.0, 24.9, 20.7, 12.5, 14.9, 12.6, 16.7, 13.8, 11.0, 12.9, 10.1, 1.9, 8.5, 16.5, 15.3, 13.3, 9.8, 8.4, 7.9, 3.7, 5.1, 4.6, 4.4, 5.4, 6.1, 8.0, 10.0, 11.2, 10.1, 6.4, 6.7, 5.7, 5.8

Table 2

Average annual percent change in private health insurance premiums.

The mean, median, skewness, and kurtosis of this data are 10.7, 10.1, 0.603, and 0.557 respectively. We can see that the data is right skewed. The estimators of the unknown parameters of each distribution is obtained by the maximum likelihood method. In order to compare the generalized Burr life-testing model with other models, we use −2lnL critrion, AIC, BIC, the correct Akaike information criterion (CAIC), Hannan information criterion (HQIC), which are defined as follows:

AIC=2k−2lnL, CAIC=AIC+2kk+1n−k−1,

BIC=klnn−2lnL, and HQIC=2klnlnn−2lnL,

where k is the number of parameters in the statistical model, n is the sample size, and lnL is the maximized value of the log-likelihood function under the considered model. The “best” distribution corresponds to the smallest values of −2lnL, AIC, BIC, CAIC, and HQIC. Table 3 shows the MLEs of the model parameters and its standard error (S.E) (in parentheses) for the data set in Table 2.

Model		MLEs and S. E
New Modelα,β,ν,μ,p	3.291 1.92311	1.561×10−3 5.411×10−3	15.23811.285	16.57110.362	2.3320.627
Burr XIIa,b	0.0630.02923	4.9161.945	-	-	-
Gammaθ,γ	0.7760.17506	19.9692.2	-	-	-
Log Logistic λ,k	0.040.0081	7.8510.073	-	-	-
Log normal η,σ	3.2230.03234	0.22 6.77×10−3	-	-	-

Table 3

The MLEs and S.E of the model parameters for data set in Table 2.

The variance covariance matrix Iλ^−1 of the MLEs of the generalized Burr life-testing model, with the pdf (1), is computed for the data set in Table 2 as follows:

3.6980.035 183.332 −49.764−12.657 0.0352.928×10−3 0.079 −0.022 −5.476×10−3 183.3320.079 127.361 −94.167 −6.765 −49.764−0.022 −94.167 107.3716.125 −12.657−5.476×10−3 −6.765 6.1250.393 .

The following Table 4 shows the values of AIC, BIC, CAIC, and HQIC statistics for the data set in Table 2.

Distribution	−2ln L	AIC	BIC	CAIC	HQIC
New Model	250.804	260.804	258.814	262.569	263.857
Burr XII	431.625	435.625	434.829	435.949	436.846
Gamma	504.553	508.553	507.757	508.877	509.774
Log Logistic	682.54	439.075	438.279	439.4	440.297
Log normal	473.113	477.113	476.317	477.437	478.334

AIC, Akaike information criterion; BIC, Bayesian information criterion; CAIC, corrected Akaike information criterion; HQIC, Hannan–Quinn information criterion.

Table 4

The AIC, BIC, CAIC, and HQIC statistics for the data set in Table 2.

In Fig. 3 below, we provide the plots of estimated densities of the fitted generalized Burr life-testing model, Burr's type XII, log-logistic, gamma and lognormal models for the data set in Table 2.

In our above analysis, the Table 3 shows the MLE's of the parameters of the fitted generalized Burr life testing and other distributions to the data set in Table 2. Further, Table 4 shows the values of AIC, BIC, CAIC, and HQIC. The values in Tables 3 and 4 indicate that the generalized Burr life-testing distribution is a strong competitor to other distributions used here for fitting to the data set in Table 2. A density plot in Fig. 3 compares the fitted densities of the models with the empirical histogram of the observed data. The fitted density for generalized Burr life-testing distribution is the closest to the empirical histogram of the other models.

5. CONCLUDING REMARKS

Since the truncated distributions arise in practical statistics where the ability of record observations is limited to a given threshold or within a specified range, we investigate, in this paper, some characterizations of the five-parameter continuous life-testing continuous probability distribution of Shakil and Kibria [1] by truncated first moment, order statistics, and upper record values. We also investigate some reliability properties, and Akaike (AIC) and other information criterion to test the measure of the goodness of fit of the said model. It is hoped that the findings of this paper will be quite useful for researchers in the fields of reliability, probability, statistics, and other applied sciences.

CONFLICT OF INTEREST

The authors declare that they have no Conflict of Interests or Competing Interests.

AUTHORS' CONTRIBUTIONS

All the authors have fully contributed to the article, and have read and approved the final manuscript.

Funding Statement

There are no sources of funding for the research.

ACKNOWLEDGMENT

The authors are thankful to the editor-in-chief and the referees for their valuable comments and suggestions, which certainly improved the presentation of the paper.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 18 - 3
Pages: 259 - 269
Publication Date: 2019/08/30
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.190818.001 How to use a DOI?
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/).

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ris enw bib

TY  - JOUR
AU  - M. Ahsanullah
AU  - M. Shakil
AU  - B. M. Golam Kibria
AU  - M. Elgarhy
PY  - 2019
DA  - 2019/08/30
TI  - On a Generalized Burr Life-Testing Model: Characterization, Reliability, Simulation, and Akaike Information Criterion
JO  - Journal of Statistical Theory and Applications
SP  - 259
EP  - 269
VL  - 18
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190818.001
DO  - 10.2991/jsta.d.190818.001
ID  - Ahsanullah2019
ER  -

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