EXPONENTIATED WEIBULL DISTRIBUTION: BAYESIAN ESTIMATION USING PROGRESSIVE TYPE I INTERVAL CENSORING Текст научной статьи по специальности «Математика»

EXPONENTIATED WEIBULL DISTRIBUTION: BAYESIAN ESTIMATION USING PROGRESSIVE TYPE I

INTERVAL CENSORING

M. KUMAR, K P ASWATHI •

Department of Mathematics, National Institute of Technology Calicut, 673601, Kerala, India mahesh@nitc.ac.in, asw athichithra01@gmail.com

Abstract

A three-parameter distribution known as the Generalized Weibull (GW) or Exponentiated Weibull distribution is studied in this work. We construct Baye's estimators for the unknown parameters and present reliability function using progressive type I interval censoring data. Two different loss functions, namely, squared error loss and general entropy loss functions are applied to derive Baye's estimators. It is observed that there is no closed-form solution for Baye's estimators as well as for MLE. Hence, Lindley's approximation procedure is applied to obtain Bayesian estimator of unknown parameters, and Newton Rapson method is employed to obtain MLE's numerically. The corresponding reliability function is derived. Monte Carlo simulation is used to obtain MLE. Further, the performance ofMLE and Bayes estimators are compared in terms of their respective MSE and Relative errors. It is noted by numerical computation that MLE's performs better than Bayes estimators. In addition to this, Bayes estimators obtained using Squared error loss function and general entropy loss function are compared. It is observed through numerical computation that general entropy loss function is better in terms of MSE.

Keywords: Bayesian infer ence, Exponentiated Weibull distribution, Lindle y's appr oximation, Maximum likelihood function, Monte Carlo simulation, Relativ e error.

1. Introduction

When it comes to analyzing data and adapting it to practical situations, statistical distributions are crucial. Weibull or Gamma distributions are typically employed to fi the data in real-world scenarios. In survival analysis, the Gamma distribution has more major applications than all other distributions. But the main drawback of Gamma distribution is that the survival function cannot be obtained in closed form unless the shape parameter is an integer. This makes Weibull distribution mor e popular than Gamma distribution. Its sur viv al function and failur e rate are simple and easy to analyze. And this distrib ution is easy to handle the censoring data because of that, in recent years Weibull distribution is more popular in analyzing lifetime data. The Exponentiated Weibull distribution (EW) or Generalized Weibull distribution, was firs described by [24] as a way to extend the Weibull family of two parameters by one more shape parameter . This distribution yields better fit than classic models such as exponential, gamma, Weibull, and log-nor mal distribution. Owing to its flexibilit in modeling a wide range of industrial data, the EW distribution may be widely and efficientl applied in reliability applications. The fundamental featur e of this family is that it supports bathtub-shaped as well as unimodal hazar d rates, in addition to numer ous monotone hazar d rates. The applications of this distribution were firs developed by [24]. Using fi e different classical failure data sets obtained for the Bus-motor

system, [25] demonstrated the potential unfulness and flexibilit of EW distribution. It is a submodel of a generic class of exponentiated distributions suggested by [11]. Generalized Weibull distribution was used by [26] to model survival data. The reliability and survival functions of this distribution were studied by [23]. Further statistical featur es and the importance of this distribution are addressed by [29] and [28]. The moments of the EW distribution were determined by [8]. EW distribution was compared on two-parameter Weibull and Gamma distributions in [32] study with regar d to the failur e rate. Exponentiated Weibull family distributed lifetime data obser ved under Type I progressiv e interval censoring with random removals were analyzed by [6]. Bayesian estimate and prediction for the EW distribution using both informative and non-informative priors was examined by [21]. After fittin a Weibull distribution and an EW distribution to the wind speed data and deter mining the mean and variance, [9] estimated the parameter using the MLE method. The non-Bayesian estimators methods for parameters of EW distribution studied by [4].The discrete case of EW distribution studied by [30]. The entropy and stress-strength model of EW distribution studied by [3]. Numerical estimation of parameters of EW distribution based on generalized progressiv e hybrid censoring scheme studied by [10]. In recent years, estimation of EW distribution under progressiv e type II censor ed data studied by [22].

The fundamental featur e of this family is that it supports bathtub-shaped as well as unimodal hazar d rates, in addition to numer ous monotone hazar d rates. The EW distribution is define in the following way. It has distribution function given by

F(x; a, p, A) = (1 - e-(Ax)p )a, x > 0 and a, p, A > 0 (1)

and therefore its probability density function is of the form

f (x; a,p, A) = apApx(p-1)e-(Ax)p((1 - e-(Ax)p)a-1) (2) The corresponding reliability function is given by

R(x; a, A) = 1 - (1 - e-(Ax)P )a (3)

and the hazar d rate is

h(x)=,x >0 (4)

Note here that, the shape parameters are a and p, and the scale parameter is A. Several well known distributions are particular cases of the EW distribution. For example, the Exponential distribution is the case when a = 1and p = 1, the Weibull Distribution is define with a = 1, Rayleigh Distribution with a = 1 and p = 2, p = 1 Generalized Exponential (GE) Distribution studied by [12], [13], [15], [17] [18], [37] and [39]. p = 2 Two parameter Burr Type X or Exponentiated Rayleigh(ER) or Generalized Rayleigh(GR) Distribution studied by [2], [36], [16], [14], [43], [38], [5] and [27] among others. Fig.(1) and Fig.(2) represents the many forms of these distributions graphically .

It was discovered that the EW family is a very versatile family that may be utilized to describe many sorts of skewed lifetime data. In reliability analysis, censoring is quite prevalent. It occurs when specifi failur e times for a subset of test units in an experiment are detected. In industrial life testing and medical survival analysis, very often the object of interest is lost or withdra wn before failure or the object's lifetime is only known within an interval. Hence, the obtained sample is called a censor ed sample (or an incomplete sample). The most common censoring schemes are type-I censoring, type-II censoring and progr essiv e censoring. For type-I censoring, life testing ends at a pre-scheduled time and for type-II censoring, life testing ends whene ver the number of lifetimes is reached. In type-I and type-II censoring schemes, the tested items are allowed to be withdra wn only at the end-of-life testing. In the progressive censoring

Figure 1: Graph of EW distribution for different values of a, f and for fixed \ = 0.5

scheme, the tested items are allowed to be withdra wn at some time before the end-of-life testing. See [7] for more information about progressiv e censoring combined with type-I or type-II and their applications. Using the concepts of progressiv e censoring, type I censoring, and interval censoring, [1] developed progressiv e type I inter val censoring. Combining progr essiv e censoring and type-II censoring, [18] and [34] investigated Bayesian inference for Weibull distribution and generalized exponential (GE) distribution, respectiv ely. It should be emphasized that in many practical situations, unit lifetime is set on an interval, therefore type I interval censoring is beneficia in these instances (see,[ 1]). It may be noted that in real-life situations, the lifetime of units may not be recorded precisely due to some reasons, such as technical problems, nonavailability of experimental resources or due to some unknown human errors, or some cost-saving measur es emplo yed by the industr y. Thus such censor ed data generated can be used effectiv ely in analyzing the reliability characteristics of well-known distribution, such as the more general class of distribution, namely, EW distribution, which gained lots of importance in recent times. The importance of progr essiv e type-I interval censoring in handling practical problem has been studied by authors, namely, [6] and [19]. The concept of progressive type-I interval censoring to the Weibull distribution and compar ed many different estimation methods for two parameters in the Weibull distribution via simulation introduced by [31]. The recent study about progressive type I interval censoring is On inference and design under progressive type-I interval censoring scheme for inverse Gaussian lifetime model by [40]. A Study on the experimental design for the lifetime performance index of Rayleigh lifetime distribution under progressiv e type I interval censoring by [44]. Optimal design of accelerated life tests under progressive type I interval censoring with random removals by [46], and experimental design for progressive type I interval censoring on the lifetime performance index of Chen lifetime distribution by [45]. All the works available in the literatur e aims at obtaining estimators of parameters of EW distribution based upon, either data obtain from complete censoring or from type I censoring, type II censoring, hybrid censoring, etc. No work in the literatur e addresses the estimation of parameters of EW distribution based upon progressiv e type I inter val-censor ed data. Therefor ewe

Figure 2: Graph ofEW distribution for different values of a, f and for fixed A = 1

consider in the next sections the deriv ation of MLE and Bayes estimators from data obtained via progressive type I interval censoring for EW distribution. Section 2 provides a brief fundamental requir ed for obtaining estimators based on censor ed data. Some simulation results and discussion based upon the results obtained are presented in Section 3. The conclusion and future scope of resear ch are given in Section 4.

2. Bayesian estimation using progressive type I interval censored data

In this section, we discuss the brief overview of the terms used in this paper and the procedur e of obtaining Baye's estimators for Parameters and reliability function of EW distribution.

2.1. Progressive type I interval censored data and the likelihood function

Statistical inference for exponential distributions using progressiv e type I inter val censor ed data and pioneered type I interval censoring in a progressive censoring scheme developed by [1]. Under progressive type I interval censoring, observations are only known within two successiv ely pre-scheduled timeframes, and items may be allowed to be deleted at pre-scheduled time points. The pr ogr essiv ely type I inter val censor ed sample may be generated in the follo wing manner: Let n units be put on a life testing platfor m simultaneously at time t0 = 0 and under examination at m pre-specifie time periods t1 < t2 < ... < tm wher e tm is the predeter mined time to end the experiment. The number of failures Xi within (ti-1, ti ] is recorded and Ri surviving items are randomly removed from the life testing at the ith inspection time, ti, for i = 1,2,..., m. Because the number of surviving items, Yi, is an random variable and the precise number of items removed at time schedule ti should not be larger than Yi, Ri might be calculated by a pre-specifie percentage of the remaining surviving units at ti for given i = 1, 2,..., m.

For example, given certain pre-specifie percentage values say, p1, p2,..., pm-1 and pm = 1, Ri can be determined by using Ri = floor[piYi] at each inspection time ti, where floor[x] yields

x's biggest integer. Therefore, a progressive type-I interval censored sample with size n, can be

m

denoted as D = (Xi, Ri, ti)m, i = 1,2,..., m. If Ri = 0, i = 1, 2,..., m — 1 and Rm = n — £ Xi, then

i=1

the type-I interval-censor ed sample gradually shrinks to the typical interval-censor ed sample. Given the progressively type-I censored data, D = (Xi, Ri, ti )m of size n, from a continuous lifetime distribution with CDF F(t; k), then the likelihood function is given as follows

m

L(D | k) aH[F(ti; K) — F(ti—1; K)]Xi [1 — F(ti; K)]r, (5)

i=1

where t0 = 0 and 6 is the parameter vector. The more details of progressive type I interval censoring can be seen in [33].

For the EW (a, A, ß) , the likelihood function (5) can be define in the following manner:

m

L(D | a, A, ß) a ^[(1 — e—(Ati)ß)a — (1 — e—(At—)ß)a]X [1 — (1 — e—(At)ß )a]R. (6)

i=1

The log-likelihood function is thus given by

m

¡(a, A,ß) a £ XM(1 — e—(Ati)ß)a — (1 — e—(At—)ß)a] + RJn[1 — (1 — e—(At')ß)a]. (7) i=1

2.2. Maximum likelihood function

In this section, we discuss the Maximum likelihood estimation to estimate unknown parameters a, A, ft, and the reliability function R(t) for EW distribution define in (l) using the numerical method.

By setting the deriv ativ es of the log likelihood function with respectiv e to a, A or ft to zer o, the MLEs of a, A and ft are the solutions to the following likelihood equations

and

£

i=1

m £m

i=1

m £m

i=1

Xi

Xi

dF da dF— 1 \ 1 da 1 m = £m =1 R ( dFi da

Fi — Fi—1 J \1 — Fi

BFi dA dF— 1 \ 1 dA 1 m = £m =1 R ( dFi dA

Fi — Fi—1 )_ \1 — F

Xi

'dF- dFi—1 dß dß

FF

1 r 1 r

i1

£

i=1

Ri

dF dß 1 - Fi

There is no closed form of the solution to the above equations and numerical methods can be used to obtain the MLEs from the above likelihood equations. Since there is no closed form of the MLE, Newton-Raphson method is introduced as follows for findin the MLEs of a, A and ft. One of the most used methods for optimization in statistics is the Newton-Raphson method(or Newton™s rule). Assume that l only involves a one-dimensional parameter and that d is our current best guess on the maximum of 1(d). l(d) can be approximated by employing a Taylor series expansion around d. Hence we have

(d) = l(d) +l'(d)(d - d) + 2 f (d)(d - d)2.

m

m

When û is close to û, the difference l(û) — l(û)(û) is small. The maximum value of l(j)(û) is closer

to the maximum value of l(û) than l(û). The gradient of l(j)(û) at û is

ïm(û) = l' (û) + l'' (û)(û — û) and the Hessian or second deriv ativ e is

1^(d) = l'' (û).

At the point û, l(û) and l(û)(û) have equal firs and second derivatives. In the case of log likelihood

function Hessian is same as the minus of observed information evaluated at û = û, l''(û) = —J(û). In the optimum point of the approximation, I($)(&) has a gradient equal to zero, giving the following equation:

l'' (â)(û — û) = —l' (û).

Solving with respect to û, we get

û=û—HQ. l'' («)

This gives a procedur e for optimizing l($)(û). An iterative procedur e for optimizing l(û) is given by

^(s+1) _ fi(s) _

l' (â(s)) l' '(№)

which is the Newton-Raphson Method. The procedur e is run until there is no significan difference between d(s) and d(s+1).

When 1(d) is a log likelihood function, this algorithm can be written as

^(s+1) _ fl(s) _

s(û(s))

J(û(s))

wher e s(û) is the score function while J (û) is the observed information matrix.

2.3. Bayesian Estimation

In this section, we discuss the Bayesian technique to estimate unknown parameters a, A, f, and the reliability function R(t) using the Squared error loss and general entropy loss functions. Assume that all parameters, namely , a, A and f of EW distributions are unkno wn and independent. We addr ess the problem of constructing Baye's estimators for these parameters. We assume non-informative priors for a and f, and conjugate prior for A. The reason for choosing these prior forms is duo to their simplicity of in obtaining mathematically treatable posterior distributions. We observe that such priors are successfully applied by many authors, namely, [ [33] and [35]]. The following equations give respectiv e definition of prior densities.

ni (a)

-, a > 0 a

(8)

n(A) =

r(a)

A

a-1,

-bA

A > 0, a, b > 0

(9)

b

a

and

n (ft) = 1, ft > 0 (10)

respectiv ely wher e r(.) is the gamma function.

We consider two different form of loss functions in estimating the parameters of EW density. The firs one is a symmetric loss function, the squar ed error loss function(SEL), which is given by

L (Z, Z) = (Z - Z)2, (11)

wher e Z is the estimate of parameter Z. Then the Bayesian estimate of any function q = q(a, A, ft) is obtained by considering following equation

a fa ¡A Ift q(a A ft)l(a A ft)n1 (a)n2 (A)n (ft)dadMft

q= (q 1 ) = fa fA fft l(a, A, ft)m (a)n2(A)n3(ft)dadAdft (12)

The second loss function, is the generalization of the Entropy loss used by several authors ([41] and [42]). The General Entropy loss(GEL) is defin as:

L2(Z,Z) "(f)-cl°g\ - 1, (13)

wher e Z is an estimate of parameter Z. It may be noted that when c > 0, a positiv e error causes more serious consequences than a negativ e error. On the other hand, when c < 0, a negativ e error causes more serious consequences than a positive error. Then the Bayesian estimator of q(a, A, ft) under this general entr opy loss function is

qGEL =[E(q-c)]-1, (14)

provided that E(q-c) exists and is finite It can be shown that, when c = 1 , the Bayes estimate (12) coincides with the Bayes estimate under the weighted squar ed-err or loss function. Similarly, when c = -1 the Bayes estimate (14) coincides with the Bayes estimate under squared error loss function. The equations (12) and (14) cannot be solved for obtaining closed form solutions. Hence, we resort to well known Lindle y approximation [20] procedur e to evaluate the ratio of integrals involved in (12) and (14). Note that the Lindle y appr oximation procedur e is successiv ely employed by authors, such as [18] to obtain Bayesian estimators. Next, the Bayesian posterior expection function of a parameter vector 7, say k(q) is obtained by using the following equation

t M ^\ ¡n k(7)l(7)n(7)d7 , , kg = E(k(n) | D) = 7 -, (15)

In l(7)n(7)d7

Recall that in the above expression 1(7) denotes log likelyhood function, n(q) denotes prior density and D denotes the data obtained using progressive type I interval censoring. By [20], if n, the sample size is sufficientl large, every ratio of the integral of the form,

k = E[v(n1,72, 73)]

_ W v(n 1,72,73)el(ni'n2'n3)+G(n1,n2,n3)d(m, 72,73) = e1^,7?273H^17273)d(m, 72,73)

wher e

v(7) = v(71,72,73) is a function of 71,72 or 73 only, l (71,72,73) is log of likelihood function, and G(71,72,73) is log joint prior of 71,72 and 73, can be evaluated as

k = v(71, 72, 73 ) + (V1 «1 + V2«2 + V3«3 + «4 + «5 ) + J [A(V1 ^11 + V2O12 + V3O13 ) +

B(V1021 + V2022 + V3O23 ) + C(V1 O31 + V2O32 + V3O33 )]

wher e

i/i, J/2 and i/3 are the MLE of t]i, and rj3 respectiv ely.

ai = pi°ii + P2°i2 + P3°i3, i = i, 2, 3, «4 = V12 °i2 + V13 °i3 + V23 °23,

«5 = 2 (Vii°ii + V22°22 + V33°33 )»

A = 011 ¿iii + 2^121121 + 2oi3 ¿131 + 2023 ¿23i + °22 ¿22i + °33 l33l»

B = °ii 1112 + 2°121122 + 2°131132 + 2°23 '232 + °22 ¿222 + °33 ¿332 »

C = °ii I113 + 2°12 ¿123 + 2°131133 + 2°231233 + °22 '223 + °33 ¿333

and subscripts 1,2,3 on the right-hand sides refer to t] i, n2, n3 respectiv ely and,

p = dp v = dv(ni,V2,V3) i =

Pi = dm, v = dm '

v. = 32^ff,n3), i,/ = i, 2, 3, J dtji dtjj

= d^^, i,j = ^,3, (i6) ' dt] i dtjj

= d3l(ni,tj2,n3) , . k = ( )

and 0/ is the (i,j)th element of the inverse of the matrix {¿j}, which is given by

- _dH __dH_ d2l 1

da2 dadA dadß

I (a, A, ß) = d2l d2l d2l

dAda dA2 dAdß

d2l d2l d2l

_ dadß dßdA dß2 .

Now by equations, (8), (9) and (10), by using independence of a, A, ß, the joint prior distribution of ther e thr ee parameters is giv en by

ha Aa—i e-bA

n(a, A, ß) = ßaT(a) , a, A, ß > 0, a, h > 0. (18)

Let

p = ln n(a, A, ß)

= a ln h + (a - 1) ln A - hA - ln ß - ln a - ln r(a). (19)

Differentiating (19) with respect to a, A,ß respectiv ely, we have

1 a- 1 1

P1 = - a' P2 = ~r - h, P3 = - ß ■

Observe that while performing progressive type I interval censoring, there are ' m' pre-specifie time periods, say, t1 < t2 < ... < tm, wher e tm is pre-specifie stopping time of experiment. Now let us defin the pdf for EW distribution for 1 < i < m as Fi = (1 - e-(Ax)ß)a i = 1,2,3,..., m. Now from the expression (5) we have

m

I {Xi ln [Ft - Fi-1 ]+ Ri ln [1 - Ft]} i=1

Then,

h h

E

i=1

m

E

i=1

m Em

i=1

Xi-

Xi

X-

■dfL_ dFi-1 ' da_da

Fi — Fi—1

dfL_ BFj—1 ' dA dA

Fi — Fi—1

■dfL_ BFj—1 dß dß

Fi — Fi-1

-Ri

-Ri

-Ri

E

da

1 — F

E

dA

1 — F,

EL dß

1 - F-

From equation (16), the values of ljj, (i, j = 1,2,3) can be obtained as follows

in = E {xt

i=1

(F _F ) ( &ÏL- d2 Fi—1 )_( El_ E—1x2

(Fj Fj—1 ) y da2 da2 J ^ da da

(Fj — F, — 1 )2

Ri

d2 F' ( BF-^ 2

(1 —Fj ) dOF + ( di (1— F )2

I12 = E \ Xi

i=1

(F _F ) ( d2Fi__ d2 Fj— ! )_( dF_ dF—1 )( dF_ dF—1 (Fj Fj—1 ^ dadA dadA ) \da da J \ dA dA

(Fi — F- — 1 )

2

Ri

(1 _ F) +(EL\ [EL

(1 Fj) daA + ^ da ) \dA

(1— Ft )2

l21,

I13 = E { Xr

i=1

(F.-F. d?FL_ a2F—A _ (EL- dF—i

(Fj Fj—1 ^ dadß dadß J \da da

E. - dFj—1 dß dß

Ri

(F- —F, — 1 )

daß + ^ da) (dß

2

(1 _ F) + (E\ iE

(1 Fj ) daß + l da \ dß

131,

l22 = E { Xi

i=1

(1— Fr )2

(F-F \ ( d2A- VF—1 ) _ ( El_ dFi—1 ) 2 (Fj Fj—1M dA2 dA2 \dA dA

Ri

(Fi —Fi — 1 )2

(1—f ) B+( dA )2

l23 = E \ Xi

i= 1

(Fj Fj — 1 M dXd'ß

(1— Fi )2

d2Fi d2Fj_, ) fdFi Ei=L\ (dFi dFj_,

1 ) ^dXdß dAdß ) \dA dA ) \dß dß

Ri

(Fi —Fi — 1 )2 dXß + ^bâ) (dß

(1 _ F ) d^Fi + [Ei\ (dFi

(1 Fj ) dAß + l dA \ dß

(1— F )2

3

m

I33 = E \ X

i=1

(f Fi-i ) ^ dß2 dß2 j ^ dß dß (Fi - F- )2

-Ri

(i - f) dß+(f42

(1 - Fi)2

Similarly , from equation (i7), the values for j(i, j, k = i, 2, 3) can be obtained. Now we proceed to obtain Bayes estimators of the parameters a, A, ft of EW distribution function, and the reliability function R(t) under squared error loss function. Recall that v(as, As, fts) denotes a function MLE's for a, A, ft. Hence we present here the Bayes estimators of a, A, ft and R(t) via following equations:

• v(a, A, ft) = a then

- 1 a - 1 - bk 1 1r„ „

a = a - - +-j-^12 - + - [ A On + B021 + C031 ],

a k ß 2

(20)

v(a, k, ß) = k then

л 1 a - 1 - bk 1 1 r . „ , . .

ks = k - T O21 +--7-O22 - -JO23 + - [AO12 + BO22 + CO32] , (21)

k

ß 2

v(a, k, ß) = ß then

л 1 a -1 - bk 1 1 r . „ ,

ßs = ß - -O31 +-т-O32 - -5O33 + - [AO13 + BO23 + CO33] ,

a k ß 2

(22)

v(&,k,ß) = R(x) then

Rs = R + (R 1 a1 + R2a2 + R3a3 + a4 + a5) + ^ [A(R" 1 on + R2+ R3O13 ) +B(R 1021 + R2O22 + R3O23) + C(R 1031 + R2O32 + R3O33)] ,

(23)

wher e,

R 1 =

dR да

- (1 - e-kxfjjl0g(1 - e-(ix)^

r = dR

2 dk

aßx(-e-(kx)ß) (kx)ß-^1 - e

_e-(kx)ß

a-1

R3 = —Г

dR dß

=а

(-e-(kx)ß)(kx)ß log (kx) (1 - e-(kx)ß)

a-1

Next, we present Baye's estimators using GEL function. Let tig, kg, ftg and Rg denote Baye's estimators of a, k, ft and R(t) respectiv ely. The following steps, for various choice of v(a, k, ft) Bayes estimators for a, k, ft and R(t) respectiv ely,

2

M. Kumar and K P Asw athi

EWD:BAYESIAN ESTIMATION USING PROGRESSIVE TYPE I RT&A, No 1 (77)

INTERVAL CENSORING Volume 19, March 2024

• v(a, A, ß) = a-c then

% = a-c - ca-(c+1) aou + (- bj au - ß(24)

i / r \ \ ca-(c+i) + 2 [c(c + i)a-(c+2)a-ii)--2-[Aaii + Ba^ + Co3i]

• v(a, A, ß) = A-c then

Ag = A-c - cA-(c+1) -jalx + (^ - bj an - ßa^ (25)

+2 (c(c + i)A-(c+2)an\ - Aai2 + Ba^ + ^]

• v(a, A, ß)= ß-c then

ßg = ß-c - cß-(c+1) ^ + (^ - bj a32 - ^33) (26)

1 / , . \ c ß-(c+1) +1 [c(c + i)ß-(c+2)a33) - -[Aai3 + Ba^ + Co33]

• v(a, A, ß) = R-c then

Rg = R-c + (R i «i + RA2 «2 + R3 «3 + «4 + «5 ) + 1 [ A(R 1 aii + R au + R3 ) ^ +B(R 1 an + R2022 + R3O23) + C(R 1031 + R2032 + R3a33)] ,

wher e

Ri = jrr, i = i, 2,3 and (t]\, tj 2, n3) = (a, A, ft).

Observe that all equations define above depends upon MLEs of a,A and ft. The detailed procedur e for obtaining MLE is discussed in Section 2.2. Moreover, these MLEs don't have closed form studies. Note that we resorted to using Newton Raphson method for solving equations for obtaining MLEs numerically . Then next Section present the simulation study to obtain Bayes estimators for various parameters of EW distribution and the reliability function R(t).

3. Simulation

In this Section, The results obtained in previous section, are illustrated by means of simulation. The data simulated by using R programming language are used to obtain Baye's estimators of parameters of EW distribution, namely, a,A,ft and R(t). Further, the performance of these estimators are studied by computing their respectiv e mean squar e error and standar d deviation. The following subsection will describe the details of simulation procedure.

3.i. Simulation Algorithm

Let us assume that prior distribution for a ~ U(0,i), A ~ Gamma(a, b) and ft ~ U(0,i) are chosen at random.

- A log (1 - U

If the random variable U follows a uniform distribution in (0, i), then X

follows the GW(a, A, ft). Next, progressiv e type-I inter val censor ed sampling data, D = (Xi, Ri, ti )m of the GW (a, A, ft), are generated as follows. First, the random variables, Ui, U2,..., Un, n < m, are generated from U(0, i), and then GW(a, A, ft) data t[, t2,..., tk,..., t'n are calculated by inverting

— Xlog 1 — U

1

ft

. Now,the number, Xi, of failures within (t(i-1), ti ] are generated and

Ri surviving items are randomly removed from the testing based on the pre-specifie inspection times t1 < ... < tm and the pre-specifie percentage p = (p1, p2,..., pm-1,1), respectiv ely. The specifi steps are as given below.(see, Aggar wala [?])

• Set X0 = 0 and R0 = 0 and for i = 1,2,..., m

Xi | Xi-1,..., X0,R(i-1),..., R0 - rbinom (n - j (Xj + Rj), ^¡I^ R

Ri | Xj,..., Xo, R(l—1),..., Ro = floor [pt * (n — j Xj — E— Rj)

wher e rbinom( n,p) generates a random variable from the binomial distribution with parameters n and p.

3.2. Example

Let the priors a — U(0,1), A — G«mm«(1,2) and ft — U(0,1) and a set of parameters a,A and ft are generated from these distributions. Let us assume that values for a = 0.4650936, A = 0.09790184, ft = 0.2090737 and R(t; a, A, ft)t=1 = 0.1592157 are selected from this set as true values. Let us assume that m=8.Then, the randomly generated data are chosen from the Uniform distribution U(0,1) as follows:

U=(0.8716594, 0.6916711, 0.3129649, 0.3065460, 0.7183383, 0.3928726, 0.4819814, 0.6090094)

To generate the inspection time set of the gradually type-I interval censored sample by appling

1

? • • K

is given by,

- xlog ^ - Ua

T=(0.4273016, 0.5336827, 6.341113, 10.02617, 63.84012, 108.4094, 223.2485, 595.9245)

To create distinct progressiv e type-I interval censor ed samples, four group sample sizes n=10,15,20,25,30,35,40,45 and fi e pre-specifie percentages p:p(1) and p(2) are consider ed, wher e

P(1) = (0, 0, 0, 0, 0, 0, 0, 1), P(2) = (0.1, 0, 0, 0, 0, 0, 0, 1)

In Tables 1 and 2, for specifie p(1) and p(2) in progressive type I interval censoring, relative error (Re) and mean square error (MSE) of Bayesian estimators under SEL function (Bg) and Linex Loss function (Bl) with c = 0.5, are permited. Note that Re is given by

Re = Ü-Ü g

and MSE is given by

1 n

MSE = n E (& - gi )2, i=1

wher e g denote the MLEs or Bayesian estimates of g.

After an extensiv e study of the results thus obtained, conclusions are drawn regarding the behavior of the errors of estimators, which are summarized below graphically(see Figur e 3-Figur e 14) .

Table 1: RE and MSE of the Example for fixed p = p(1)

RE

MSE

Item n

n a A ß R a A ß R

10 0.9149 0.7544 0.2869 0.7365 0.0902 0.0098 0.0023 0.0154

15 0.9703 0.0017 0.3649 0.9462 0.0231 0.0000 0.0027 0.0076

20 0.9341 0.1479 0.3482 0.8148 0.0536 0.0002 0.0029 0.0097

25 0.0909 0.8181 0909 0.1052 0.0000 0.1202 0.0005 0.0007

30 0.5389 0.1793 0.5890 0.6659 0.0004 0.0022 0.0098 0.0161

35 0.9808 0.1949 0.3735 0.9434 0.1066 0.0000 0.0102 0.0114

40 0.4328 0.3296 0.6769 0.6602 0.0024 0.0061 0.0113 0.0176

45 0.9366 0.0389 0.3552 0.9200 0.0833 0.0000 0.0085 0.0047

10 0.5954 0.5185 0.8065 0.5529 0.0382 0.2128 0.0179 0.0087

15 1.0388 0.2083 1.2265 0.2629 0.0265 0.0336 0.3157 0.0589

20 0.8271 1.2446 1.7592 0.5632 0.0420 0.0123 0.3435 0.0047

25 0.4623 0.7895 0.4622 0.8032 0.0162 0.1044 1.2557 0.0381

30 0.1860 0.2859 0.0094 0.0636 0.0000 0.5612 0.0000 0.0002

35 0.6569 1.4812 0.2202 0.0294 0.0478 0.0007 0.0262 0.0000

40 0.3758 0.6661 0.2903 0.0621 0.0000 0.0244 0.0021 0.0002

45 0.0828 0.4764 1,5272 0.1365 0.0007 0.0000 0.1572 0.0001

10 0.1253 1.7028 1.1261 0.5510 0.0017 0.0498 0.0349 0.0086

15 0.3227 0.3916 1.4387 0.2632 0.0026 0.1188 0.0434 0.0591

20 0.0519 1.9679 1.2674 0.5606 0.0002 0.0699 0.0390 0.0046

25 1.1834 0.2332 0.4580 0.2034 0.1062 0.0091 0.0123 0.2439

30 0.0138 0.3522 0.7898 0.7545 0.0000 0.0084 0.0343 0.0207

35 0.7894 1.8737 0.3078 0.2895 0.0690 0.0005 0.0069 0.0000

40 1.1188 0.4942 1.2519 1.3603 0.0161 0.0137 0.0386 0.0745

45 0.0184 0.4489 0.3615 0.1628 0.0000 0.0000 0.0088 0.0001

MLE

sample size

Figure 3: Relative Error of MLE for p(1)

Figure 4: Relative Error of Bs for p(1 )

B

s

B

g

Table 2: RE and MSE of the Example for fixed p = p(2)

RE

MSE

Item n

n a À ß R a À ß R

10 0.9009 0.3422 0.3621 0.7676 0.0837 0.0003 0.0066 0.0098

15 0.9481 0.3285 0.3743 0.8974 0.0134 0.0004 0.0003 0.0070

20 0.9152 0.4706 0.2872 0.7947 0.0629 0.0172 0.0014 0.0081

25 0.9561 0.0317 0.3609 0.8741 0.0663 0.0000 0.0016 0.0085

30 0.6986 0.4347 0.8745 0.7925 0.0252 0.0160 0.0822 0.0247

35 0.9852 0.9342 0.6004 0.8572 0.0715 0.0013 0.0333 0.0127

40 0.9431 0.3594 0.2985 0.8779 0.0284 0.0033 0.0007 0.0073

45 0.9467 0.6701 0.2837 0.8926 0.0244 0.0007 0.0014 0.0059

10 0.0605 0.4409 0.3096 1.2529 0.0004 0.0005 0.0048 0.0262

15 1.9421 0.1121 1.5398 1.9250 0.0563 0.0000 0.0249 0,0744

20 0.1961 0.0578 0.4542 0.7478 0.0029 0.0003 0.0035 0.0072

25 0.5377 0.2024 0.4913 0.2352 0.0210 0.0338 0.0030 0.0006

30 0.3550 0.0855 0.6875 0.1301 0.0065 0.0999 0.0508 0.0006

35 1.5559 1.5346 0.5596 1.3814 0.1783 0.3580 0.0289 0.0330

40 0.512 0.7987 0.4375 1.0897 0.0084 0.0165 0.1403 0.0413

45 0.4568 0.8569 0.1748 1.8877 0.0057 0.0314 0.0005 0.0265

10 0.0173 0.4638 0.5745 1.2778 0.0000 0.0005 0.0166 0.0272

15 0.7190 1.6573 1.0796 1.1535 0.0077 0.0840 0.0086 0.0403

20 0.1677 0.2676 1.7243 0.7769 0.0021 0.0056 0.0501 0.0078

25 0.1207 0.3234 1.1491 0.2385 0.0011 0.0729 0.0582 0.0006

30 0.5552 0.2137 0.0639 0.2737 0.0159 0.0038 0.0004 0.0000

35 1.2341 1.1229 0.1248 1.3814 0.1122 0.1917 0.0014 0.0329

40 0.4512 1.2013 0.3081 1.3114 0.0065 0.0372 0.0712 0.0466

45 0.5546 2.6507 1.6666 1.9275 0.0083 0.0978 0.0488 0.0276

MLE

Figure 5: Relative Error of Bg for p(l)

Figure 6: Mean Squared Error of MLE for p(l)

B

s

B

g

Figure 7: Mean Squared Error of Bs for p(1 )

Figure 8: Mean Squared Error of Bg for p(1)

Figure 9: Relative Error of MLE for p(2)

Figure 10: Relative Error of Bs for p(2)

sanrplo size samplo si2G

Figure 11: Relative Error ofBgfor p(2) Figure 12: Mean Squared Error of MLE for p(2)

Figure 13: Mean Squared Error ofBsfor p(2)

Figure 14: Mean Squared Error of Bg for p(2)

4. C onclusion

In this article, the perfor mance of the proposed Bayes estimators has been compar ed to the maximum likelihood estimator of the EWD(a, A, ft) under the progressiv e type-I interval censoring based on the squared error loss function and general entropy loss function using Lindle y's appr oximation. The simulation result indicates that this appr oach is better suited for small sample sizes. MLE is the best choice when compared to Bayesian estimators. From Table 1, it is observed that the general entropy loss function in Bayesian estimation is better as compared to the squared error loss function in terms of MSE. From Table 2, it is noted that the squared error loss function in Bayesian estimation is better as compar ed to the general entropy loss function in terms of MSE. It can be seen from Figures 4, 5,10 and 11 that the RE of Bayes estimators show fluctuatio trend, and one can not see continuously decreasing or increasing trend for RE.

It is observed in practice, especially while modeling lifetime of electronic products, this three-parameter EW distribution describes the lifetime in the best possible way as compar ed to commonly used lifetime distributions such as Exponential distribution or Weibull distribution. Moreover, practically progressive type I interval censoring is the most convenient way of obtaining data of lifetimes as compar ed to traditional censoring schemes such as type I or type II or hybrid censoring. Further, the results obtained in this paper can be used for applications in the fiel of economics or analysis of clinical data in the medical field

The results obtained in this paper use the appr oximation process such as Lindle y appr oximation to obtain Bayes estimators of parameters of EW distribution. As futur e scope of resear ch an analytical solution for deriving Bayes estimators can be consider ed by using suitable choice of prior distributions.

R eferences

[1] Aggar wala, R. (2001). Progressive interval censoring: Some mathematical results with applications to inference. Communications in Statistics-Theory and Methods, 30(8-9):1921-1935.

[2] Ahmad Sartawi, H. and Abu-Salih, M. S.(1991). Bayesian prediction bounds for the burr type x model. Communications in Statistics-Theory and Methods, 20(7):2307-2330.

[3] Al-Noor, N. H., Abid, S. H. and Boshi, M. A. A. (2019). On the exponentiated weibull distribution. In Alp conference proceedings, volume 2183, page 110003. AIP Publishing LLC.

[4] ALkanani, I. H. and Abbas, M. S. (2014). The non-bayesian estimators methods for parameters of exponentiated weibull (ew) distribution. International Journal of Mathematics and Statistical Studies,Vol.2, No.5, pp. 81-94.

Aludaat, K. M., Alodat, M. T. and Alodat, T. T. (2008). Parameter estimation of Burr type X distribution for grouped data. Applied Mathematical Sciences, 2(9), 415-423. Ashour, S. K. and Afif , W. M. (2007). Statistical analysis of exponentiated Weibull family under type I progressiv e interval censoring with random removals. Journal of Applied Sciences Research, 3(12), 1851-1863.

Balakrishnan, N. and Aggar wala, R. Progressive censoring: theory, methods, and applications. Springer Science & Business Media. 2000.

Choudhur y, A. (2005). A simple deriv ation of moments of the exponentiated Weibull distribution. Metrika, 62, 17-22.

Datta, D. and Datta, D. (2013). Comparison of Weibull distribution and exponentiated Weibull distribution based estimation of mean and variance of wind data. International Journal of Energy, Information and Communications, 4(4), 1-12.

Elshahhat, A. (2017). Parameters estimation for the exponentiated Weibull distribution based on generalized progr essiv e hybrid censoring schemes. American Journal of Applied Mathematics and Statistics, 5(2), 33-48.

Gupta, R. C., Gupta, P. L. and Gupta, R. D. (1998). Modeling failure time data by Lehman alter nativ es. Communications in Statistics-Theory and methods, 27(4), 887-904. Gupta, R. D. and Kundu, D. (1999). Theory & methods: Generalized exponential distributions. Australian & New Zealand Journal of Statistics, 41(2), 173-188.

Gupta, R. D. and Kundu, D. (2001). Generalized exponential distribution: different method

of estimations. Journal of Statistical Computation and Simulation, 69(4), 315-337.

Gupta*, R. D. and Kundu, D. (2004). Discriminating betw een gamma and generalized

exponential distributions. Journal of Statistical Computation & Simulation, 74(2), 107-121.

Gupta, R. D. and Kundu, D. (2007). Generalized exponential distribution: Existing results

and some recent developments. Journal of Statistical planning and inference, 137(11), 3537-3547.

Jaheen, Z. F. and Al-Matrafi B. N. (2002). Bayesian prediction bounds from the scaled Burr

type X model. Computers & Mathematics with Applications, 44(5-6), 587-594.

Kundu, D. and Gupta, R. D. (2005). Estimation of P [Y < X] for generalized exponential

distribution. Metrika, 61, 291-308.

Kundu, D. and Gupta, R. D. (2008). Generalized exponential distribution: Bayesian estimations. Computational Statistics & Data Analysis, 52(4), 1873-1883.

Lin, C. T., Wu, S. J. and Balakrishnan, N. (2009).Planning life tests with progressively Type-I interval censored data from the log-normal distribution. Journal of Statistical Planning and Inference, 139(1), 54-61.

Lindle y, D. V. (1980). Appr oximate bayesian methods. Trabajos de estadstica y de investigacin operativa, 31, 223-245.

Madi, M. T. and Raqab, M. Z. (2009). Bayesian inference for the generalized exponential distribution based on progressiv ely censor ed data. Communications in Statistics-Theory and Methods, 38(12), 2016-2029.

Moradi, N., Panahi, H. and Habibirad, A. (2022). Estimation for the Three-Parameter Exponentiated Weibull Distribution under Progr essiv e Censor ed Dat. Journal of the Iranian Statistical Society, 21(1), 153-177.

Mudholkar , G. S. and Hutson, A. D. (1996). The exponentiated Weibull family: some properties and a floo data application. Communications in Statistics-Theory and Methods, 25(12), 3059-3083.

Mudholkar , G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing

bathtub failur e-rate data. IEEE transactions on reliability, 42(2), 299-302.

Mudholkar , G. S., Srivastava, D. K. and Freimer, M. (1995).The exponentiated Weibull family:

A reanalysis of the bus-motor -failur e data. Technometrics, 37(4), 436-445.

Mudholkar , G. S., Srivastava, D. K. and Kollia, G. D. (1996). A generalization of the Weibull

distribution with application to the analysis of survival data. Journal of the American Statistical

Association, 91(436), 1575-1583.

[27] Nadarajah, S. (2011). exponentiated exponential distribution: a survey. AStA Advances in Statistical Analysis, 95, 219-251.

[28] Nadarajah, S. and Gupta, A. K. (2005). On the moments of the exponentiated Weibull distribution. Communications in Statistics-Theory and Methods, 34(2), 253-256.

[29] Nassar , M. M. and Eissa, F. H. (2003). On the expo nentiated Weibull distribution. Communications in Statistics-Theory and Methods, 32(7), 1317-1336.

[30] Nekoukhou, V. and Bidram, H. (2015). The exponentiated discrete Weibull distribution. Sort, 39, 127-146.

[31] Ng, H. K. T. and Wang, Z. (2009). Statistical estimation for the parameters of Weibull distribution based on progressively type-I interval censored sample. Journal of Statistical Computation and Simulation, 79(2), 145-159.

[32] Pal, M., Ali, M. M. and Woo, J. (2006). Exponentiated weibull distribution. Statistica, 66(2), 139-147.

[33] Peng, X. Y. and Yan, Z. Z. (2013). Bayesian estimation for generalized exponential distribution based on progr essiv e type-I interval censoring. Acta Mathematicae Applicatae Sinica, English Series, 29(2), 391-402.

[34] Pradhan, B. and Kundu, D. (2009). On progressively censored generalized exponential distribution. Test, 18, 497-515.

[35] Preda, V., Panaitescu, E. and Constantinescu, A. (2010). Bayes estimators of modified- eibull distribution parameters using Lindle y™s appr oximation. Wseas transactions on mathematics, 9(7), 539-549.

[36] Raqab, M. Z. (1998). Order statistics from the Burr type X model. Computers & Mathematics with Applications, 36(4), 111-120.

[37] Raqab, M. Z. (2002). Inferences for generalized exponential distribution based on record statistics. Journal ofstatistical planning and inference, 104(2), 339-350.

[38] Raqab, M. Z. and Kundu, D. (2006). Burr type X distribution: revisited. Journal of probability and statistical sciences, 4(2), 179-193.

[39] Raqab, M. Z. and Madi, M. T. (2005). Bayesian infer ence for the generalized exponential distribution. Journal of Statistical computation and Simulation, 75(10), 841-852.

[40] Roy, S., Pradhan, B. and Puraka yastha, A. (2022). On inference and design under progressive type-I interval censoring scheme for inverse Gaussian lifetime model. International Journal of Quality & Reliability Management, 39(8), 1937-1962.

[41] Singh, P. K., Singh, S. K. and Singh, U. (2008). Bayes estimator of inverse Gaussian parameters under general entropy loss function using Lindle y's approximation. Communications in Statistics-Simulation and Computation®, 37(9), 1750-1762.

[42] Singh, S. K. (2011). Estimation of parameters and reliability function of exponentiated exponential distribution: Bayesian approach under general entropy loss function. Pakistan journal of statistics and operation research, 217-232.

[43] Surles, J. G. and Padgett, W. J. (2005). Some properties of a scaled Burr type X distribution. Journal of statistical planning and inference, 128(1), 271-280.

[44] Wu, S. F., Liu, T. H., Lai, Y. H. and Chang, W. T. (2022). A study on the experimental design for the lifetime performance index of Rayleigh lifetime distribution under progressiv e type I interval censoring. Mathematics, 10(3), 517.

[45] Wu, S. F. and Song, M. Z. (2023). Experimental Design for Progressive Type I Interval Censoring on the Lifetime Perfor mance Index of Chen Lifetime Distribution. Mathematics, 11(6), 1554.

[46] Yang, C. and Tse, S. K. (2005). Planning accelerated life tests under progr essiv e type I interval censoring with random removals. Communications in Statistics"Simulation and Computation®, 34(4), 1001-1025.