estimating the parameters of a burr distribution under progressive type ii censoring

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Statistical Methodology 9 (2012) 381–391 Contents lists available at SciVerse ScienceDirect Statistical Methodology journal homepage: www.elsevier.com/locate/stamet Estimating the parameters of a Burr distribution under progressive type II censoring Manoj Kumar Rastogi, Yogesh Mani Tripathi Department of Mathematics, Indian Institute of Technology Patna, Patna-800013, India article info Article history: Received 13 October 2010 Received in revised form 8 October 2011 Accepted 8 October 2011 Keywords: Bayesian estimates Lindley approximation Maximum likelihood estimates Progressive type II censoring Reliability function abstract The problem of estimating unknown parameters and reliability function of a two parameter Burr type XII distribution is considered on the basis of a progressively type II censored sample. Several Bayesian estimates are obtained against different symmetric and asymmetric loss functions such as squared error, linex and general entropy. These Bayesian estimates are evaluated by applying the Lindley approximation method. Using simulations, all Bayesian estimates are compared with the corresponding maximum likelihood estimates numerically in terms of their mean square error values and some specific comments are made. Finally, two data sets are analyzed for the purpose of illustration. © 2011 Elsevier B.V. All rights reserved. 1. Introduction In the statistical literature two parameter Burr distribution was introduced by Burr [6]. This distribution has found its application in the study of biological, industrial, reliability and life testing, and several industrial and economic experiments. One may refer to Dubey [9], Evans and Simons [11], Wingo [18,19], and Gupta et al. [12] for detail exposition on such situations. The probability density function and the cumulative distribution function of a random variable X having two parameter Burr type XII distribution are respectively given by f X (x) = αβ x β1 (1 + x β ) +1) , x > 0, (1.1) F X (x) = 1 (1 + x β ) α , x > 0, (1.2) where α> 0 and β> 0 are the parameters. In this paper, such a distribution will be denoted by X Burr (α,β). Corresponding author. Tel.: +91 6122552015. E-mail address: [email protected] (Y.M. Tripathi). 1572-3127/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.stamet.2011.10.002

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Statistical Methodology 9 (2012) 381–391

Contents lists available at SciVerse ScienceDirect

Statistical Methodology

journal homepage: www.elsevier.com/locate/stamet

Estimating the parameters of a Burr distribution underprogressive type II censoringManoj Kumar Rastogi, Yogesh Mani Tripathi ∗Department of Mathematics, Indian Institute of Technology Patna, Patna-800013, India

a r t i c l e i n f o

Article history:Received 13 October 2010Received in revised form8 October 2011Accepted 8 October 2011

Keywords:Bayesian estimatesLindley approximationMaximum likelihood estimatesProgressive type II censoringReliability function

a b s t r a c t

The problem of estimating unknown parameters and reliabilityfunction of a two parameter Burr type XII distribution is consideredon the basis of a progressively type II censored sample. SeveralBayesian estimates are obtained against different symmetricand asymmetric loss functions such as squared error, linex andgeneral entropy. These Bayesian estimates are evaluated byapplying the Lindley approximation method. Using simulations,all Bayesian estimates are compared with the correspondingmaximum likelihood estimates numerically in terms of their meansquare error values and some specific comments are made. Finally,two data sets are analyzed for the purpose of illustration.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

In the statistical literature two parameter Burr distribution was introduced by Burr [6]. Thisdistribution has found its application in the study of biological, industrial, reliability and life testing,and several industrial and economic experiments. Onemay refer to Dubey [9], Evans and Simons [11],Wingo [18,19], and Gupta et al. [12] for detail exposition on such situations. The probability densityfunction and the cumulative distribution function of a random variable X having two parameter Burrtype XII distribution are respectively given by

fX (x) = αβxβ−1(1 + xβ)−(α+1), x > 0, (1.1)

FX (x) = 1 − (1 + xβ)−α, x > 0, (1.2)

where α > 0 and β > 0 are the parameters. In this paper, such a distribution will be denoted byX ∼ Burr(α, β).

∗ Corresponding author. Tel.: +91 6122552015.E-mail address: [email protected] (Y.M. Tripathi).

1572-3127/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.stamet.2011.10.002

382 M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391

The reliability function of this distribution is

R(t) = (1 + tβ)−α, t > 0, (1.3)

and the hazard function is given by

h(t) = αβtβ−1(1 + tβ)−1, t > 0. (1.4)

The parameters α and β are known as shape parameters. In this paper, both parameters are assumedto be unknown.

In recent years, several authors have considered the problem of estimating parameters of aBurr(α, β) distribution under various loss functions. Papadopoulos [15] considered the problem ofestimating the parameter α and the reliability function under squared error loss function whenβ is known. He obtained Bayes estimates for these unknown quantities and suggested that thisdistribution could be used as a failure model. Evans and Ragab [10] obtained Bayes estimates of α andthe reliability function based on type II censored sample. Based onprogressive type II censored sample,Ali Mousa and Jaheen [2] obtained the maximum likelihood and Bayes estimates for the parametersα, β and the reliability function. Authors evaluated these Bayesian estimates using the approximationmethod developed by Tierney and Kadane [17]. Finally, authors compared these estimates in terms oftheir risk values. Soliman [16] investigated properties of Bayesian estimates of reliability and hazardfunctions of a Burr(α, β) model. One may also refer to the papers by Al-Hussaini and Jaheen [1],Asgharzadeh and Valiollahi [3] for further results in this direction.

In this paper, we consider the problem of estimating unknown parameters α, β and the reliabilityfunction R(t) of a Burr type XII distribution under various loss functions. In Section 2, concept ofprogressive type II censoring is discussed briefly and maximum likelihood estimates (MLE’s) of α, βand R(t) are obtained. Bayesian estimates for unknown parameters and the reliability function areobtained in Section 3 for different loss functions such as squared error, linex and general entropy. InSection 4, the Lindley’s approximation method is discussed to evaluate these Bayesian estimates. InSection 5, using simulations a numerical comparison is made between various estimates in terms oftheir mean square error values. To illustrate our proposed methods of estimation two data sets, onenumerical and one real, are analyzed in Section 6.

2. Progressive type II censoring and maximum likelihood estimation

In this section, first we briefly discuss the progressive type II censoring scheme. Suppose that nindependent and identically distributed units taken from a continuous distribution are placed ona life test experiment. Let a censoring scheme (r1, r2, . . . , rm) be prefixed in such a manner thatimmediately after the first failure, r1 surviving units are removed from the experiment randomly.Similarly when the second failure occurs, r2 surviving units are removed from the experiment, againrandomly. The test is continued until themth (1 ≤ m ≤ n) failure takes place. By this time remainingunits rm, where rm = n − m − r1 − r2 − · · · − rm−1 are removed from the test. We notice that whenr1 = r2 = · · · = rm−1 = 0 and rm = n−m, this censoring scheme reduces to the type II censoring andif r1 = r2 = · · · = rm−1 = rm = 0, then it reduces to the complete sampling case. As a consequence,we note that the complete sample case and type II censoring are particular cases of progressive typeII censoring scheme. For deep insight on this censoring scheme one may refer to the treatise byBalakrishnan and Aggarwala [4]. Let (X1:m:n, X2:m:n, . . . , Xm:m:n), (1 ≤ m ≤ n) be a progressively typeII censored sample observed from a life test involving n units taken from a Burr(α, β) distributionand (r1, r2, . . . , rm) being the censoring scheme. The probability density function and the cumulativedistribution function of a Burr(α, β) distribution are given in (1.1) and (1.2) respectively. Then thejoint probability density function of (X1:m:n, X2:m:n, . . . , Xm:m:n) is (details are given in [4]) obtained as

f(X1:m:n,X2:m:n,...,Xm:m:n)(x1:m:n, x2:m:n, . . . , xm:m:n) = Cm∏i=1

fXi:m:n(xi:m:n){1 − FXi:m:n(xi:m:n)}ri , (2.1)

where C is a constant defined as, C = n(n− r1 − 1)(n− r1 − r2 − 2) · · · (n− r1 −· · ·− rm−1 −m+ 1).

M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391 383

Several researchers have investigated inference problems under progressive type II censoring forvarious life time distributions under different loss functions. The focus of investigations have beenthrough classical and Bayesian perspectives. For comprehensive insight on progressive censoringreaders may refer to the treatise by Balakrishnan and Aggarwala [4] and references cited therein.

Now, we obtain maximum likelihood estimates of α, β and R(t). Utilizing Eqs. (1.1) and (2.1), thelikelihood function of α and β is given by

L(α, β | x) ∝ αmβmm∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1), (2.2)

and the corresponding log likelihood function is

log L(α, β | x) ∝ m logα + m logβ + (β − 1)m−i=1

log xi:m:n

m−i=1

{α(ri + 1) + 1} log(1 + xβ

i:m:n), (2.3)

where x = (x1:m:n, x2:m:n, . . . , xm:m:n). Consequently, likelihood equations of α and β are obtained as

∂ log L∂α

=mα

m−i=1

(ri + 1) log(1 + xβ

i:m:n) = 0, (2.4)

and

∂ log L∂β

=mβ

+

m−i=1

log xi:m:n −

m−i=1

{α(ri + 1) + 1}xβ

i:m:n log xi:m:n

(1 + xβ

i:m:n)= 0. (2.5)

Maximum likelihood estimates of α and β , say α and β respectively, can be obtained by solving thesetwo likelihood equations. We have employed numerical technique to evaluate α and β . Finally, usingthe invariance property, the MLE of R(t) is obtained as

R(t) = (1 + t β)−α, t > 0.

3. Bayesian estimation

In the Bayesian estimation unknown parameters are assumed to behave as random variableswith distributions commonly known as prior probability distributions. In practice, usually quadraticloss function is taken in to consideration to produce Bayesian estimates. However, under this lossfunction overestimation and underestimation are equally penalized which is not a good criteria frompractical point of view. As an example, in reliability estimation overestimation is considered to bemore serious than the underestimation. Due to such restrictions various asymmetric loss functionshave been introduced in the literature such as linex and entropy loss functions. These loss functionshave been proved useful for performing Bayesian analysis in different fields of reliability estimationand life testing problems. Onemay refer to papers by Basu and Ebrahimi [5], Canfield [8], Calabria andPulcini [7] for further details in this direction.

In this section, our interest lies in deriving Bayes estimates for parameters α, β and the reliabilityfunction of a Burr(α, β) distribution under symmetric as well asymmetric loss functions. A very wellknown symmetric loss function is the squared error (a particular case of quadratic loss function)whichis defined as L1(d(µ), d(µ)) = (d(µ)−d(µ))2 with d(µ) being an estimate of d(µ). Here d(µ) denotessome parametric function of µ. For this situation the Bayesian estimate, say dSB(µ), is given by theposterior mean of d(µ). One of the most commonly used asymmetric loss function is the linex lossfunction which is defined by

L2(d(µ), d(µ)) = eh(d(µ)−d(µ))− h(d(µ) − d(µ)) − 1, h = 0.

384 M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391

For this loss function, underestimation is heavily penalized when h is negative and overestimationis considered to be more serious for positive h. Further interesting properties of L2 can be found inthe paper by Zellner [20]. The Bayes estimate of d(µ) for the loss function L2 can be obtained asdLB(µ) = −

1h log{Eµ(e−hµ

| x)}, provided Eµ(.) exists. Another useful asymmetric loss function isthe general entropy loss given by

L3(d(µ), d(µ)) ∝

d(µ)

d(µ)

q

− q log

d(µ)

d(µ)

− 1, q = 0.

In this case Bayes estimate of d(µ) is obtained as

dEB(µ) = (Eµ(µ−q| x))

−1q

provided the above expectation exists.Let X1:m:n, X2:m:n, . . . , Xm:m:n be a progressively type II censored order statistics of a random sample

of n units with censoring scheme (r1, r2, . . . , rm), drawn from a Burr(α, β) distribution. The priordistributions for β and α are taken to be Gamma(b, a) and Gamma(c, d) respectively. So, the jointprior distribution of α and β is of the form

π(α, β) ∝ αc−1e−dαβb−1e−aβ α > 0, β > 0, a > 0, b > 0, c > 0, d > 0. (3.1)

Then the posterior distribution of α and β is obtained as (using (2.2) and (3.1)),

π(α, β | x) =αm+c−1βm+b−1

ke−dαe−aβ

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

, (3.2)

where x = (x1:m:n, x2:m:n, . . . , xm:m:n) and

k =

∫∞

0

∫∞

0αm+c−1βm+b−1e−dαe−aβ

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ.

Now,we obtain Bayesian estimates ofα, β and R(t) against the squared error loss function L1, the linexloss function L2 and the entropy loss function L3 when the prior distribution is taken to be π(α, β).

Bayesian estimates of α and β against the loss function L1 are respectively obtained as,

αSB = E[α | x]

=1k

∫∞

0

∫∞

0αm+cβm+b−1e−dαe−aβ

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ, (3.3)

and

βSB = E[β | x]

=1k

∫∞

0

∫∞

0αm+c−1βm+be−dαe−aβ

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ. (3.4)

Similarly, for the loss function L2 we have

αLB = −1hlog{E(e−hα

| x)}, h = 0, (3.5)

where

E[e−hα| x] =

1k

∫∞

0

∫∞

0αm+c−1βm+b−1e−(d+h)αe−aβ

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ,

M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391 385

and

βLB = −1hlog{E(e−hβ

| x)}, h = 0, (3.6)

where

E[e−hβ| x] =

1k

∫∞

0

∫∞

0αm+c−1βm+b−1e−dαe−(a+h)β

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ.

Finally, the Bayesian estimate of α for the loss function L3 is

αEB = {E(α−q| x)}

−1q , (3.7)

where

E[α−q| x] =

1k

∫∞

0

∫∞

0αm+c−q−1βm+b−1e−dαe−aβ

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ,

and that of β is obtained as

βEB = {E(β−q| x)}

−1q , (3.8)

where

E[β−q| x] =

1k

∫∞

0

∫∞

0αm+c−1βm+b−q−1e−dαe−aβ

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ.

Next, the Bayesian estimate of the reliability function R(t) for the loss function L1 is obtained as

RSB(t) = E[R(t) | x] =1k

∫∞

0

∫∞

0αm+c−1βm+b−1e−dαe−aβ(1 + tβ)−α

×

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ. (3.9)

Against the loss function L2 we have

RLB(t) = −1hlog{E(e−h(1+tβ )−α

| x)}, h = 0, (3.10)

where

E[e−hR(t)| x] =

1k

∫∞

0

∫∞

0αm+c−1βm+b−1e−dαe−aβe−h(1+tβ )−α

×

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ,

and finally for the loss function L3 the Bayesian estimate of R(t) is obtained to be

REB(t) = (E((1 + tβ)αq | x))−1q (3.11)

where

E[(1 + tβ)αq | x] =1k

∫∞

0

∫∞

0αm+c−1βm+b−1e−dαe−aβ(1 + tβ)αq

×

m∏i=1

xβ−1i:m:n(1 + xβ

i:m:n)−(α(ri+1)+1)

dα dβ.

In the next section all estimators considered in this section, are obtained using a well knownapproximation method.

386 M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391

4. Lindley approximation method

In Section 3, based on a progressively type II censored sample we obtained several Bayesianestimates of α, β and the reliability function R(t) of a Burr(α, β) distribution. These Bayesianestimates are derived against squared error, linex and general entropy loss functions. It is easilyobserved that all these estimates are in the form of ratio of two integrals for which simplified closedforms are not available. Thus to evaluate these estimates in practice intensive numerical techniquesare required. Instead, one can apply approximation methods to evaluate these estimates. Here, weuse Lindley’s method (see [13]) to approximate all the Bayesian estimates discussed in the previoussection. For our estimation problem we briefly describe this method below. As noticed the Bayesianestimates involve the ratio of two integrals, we consider I(x) defined as

I(x) =

0

0 u(α, β)el(α,β|x)+ρ(α,β)dαdβ∞

0

0 el(α,β|x)+ρ(α,β)dαdβ, (4.1)

where u(α, β) is function of α and β only and l(α, β | x) is the log-likelihood (defined by the Eq. (2.3))and ρ(α, β) = logπ(α, β). Utilizing the Lindley’s method I(x) can be approximated as

I(x) = u(α, β) +12[(uαα + 2uαρα)σαα + (uβα + 2uβ ρα)σβα + (uαβ + 2uαρβ)σαβ

+ (uββ + 2uβ ρβ)σββ ] +12[(uα σαα + uβ σαβ)(lααασαα + lαβα σαβ + lβαα σβα

+ lββα σββ) + (uασβα + uβ σββ)(lβαα σαα + lαββ σαβ + lβαβ σβα + lβββ σββ)], (4.2)

where α and β are the MLE’s of α and β respectively. Also, uαα is the second derivative of the functionu(α, β) with respect to α and uαα is the same expression evaluated at (α, β). Other expressions canbe interpreted exactly in similar manner with following definitions,

lαα =∂2l∂α2

α=α

= −mα2

,

lββ =∂2l∂β2

α=α, β=β

= −m

β2−

m−i=1

{α(ri + 1) + 1}xβ

i:m:n(log xi:m:n)2

(1 + xβ

i:m:n)2

,

lααα =∂3l∂α3

α=α

=2mα3

, lββα =∂3l

∂β2∂α

α=α,β=β

= −

m−i=1

(1 + ri)xβ

i:m:n[log xi:m:n]2

(1 + xβ

i:m:n)2

,

lβββ =∂3l∂β3

β=β

=2m

β3−

m−i=1

(α(1 + ri) + 1)xβ

i:m:n[log xi:m:n]3(1 − xβ

i:m:n)

(1 + xβ

i:m:n)3

,

lβα =∂2l

∂β∂α

α=α,β=β

= lαβ =∂2l

∂α∂β

α=α, β=β

= −

m−i=1

(1 + ri)xβ

i:m:n log xi:m:n

(1 + xβ

i:m:n),

lβαα =∂3l

∂β∂α2

α=α, β=β

= 0, ρα =(c − 1)

α− d, ρβ =

(b − 1)

β− a,

σi,j = (i, j)th elements of matrix[−

∂2l∂α∂β

]−1

; i, j = 1, 2.

With the above defined expressions, we now obtain the approximate Bayesian estimates. Furthernoticing that

u(α, β) = α, uα = 1, uαα = uβ = uββ = uβα = uαβ = 0,

M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391 387

the Bayesian estimate of α against the loss function L1 becomes

αSB = E(α | x) = α + 0.5[2ρα σαα + 2ρβ σαβ + σ 2αα lααα + σαασββ lββα + 2σαβ σβα lαββ

+ σαβ σββ lβββ ]. (4.3)

Proceeding similarly, the Bayesian estimate of β under L1 is given by

(here u(α, β) = β, uβ = 1, uα = uαα = uββ = uβα = uαβ = 0),

βSB = E(β | x) = β + 0.5[2ρβ σββ + 2ρα σβα + σ 2ββ lβββ + 3σαβ σββ lαββ + σαασβα lααα]. (4.4)

Now, against the loss function L2 with the observation that in this case

u(α, β) = e−hα, uα = −he−hα, uαα = h2e−hα, uβ = uββ = uβα = uαβ = 0,

E(e−hα| x) = e−hα

+ 0.5[uαασαα + uα(2ρασαα + 2ρβ σαβ + σ 2αα lααα + σαασββ lββα

+ 2σαβ σβα lαββ + σαβ σββ lβββ)],

the Bayesian estimate of α is obtained to be

αLB = −1hlog{E(e−hα

| x)}. (4.5)

Similarly, for β we have

u(α, β) = e−hβ , uβ = −he−hβ , uββ = h2e−hβ , uα = uαα = uβα = uαβ = 0,

E(e−hβ| x) = e−hβ

+ 0.5[uββ σββ + uβ(2ρβ σββ + 2ρα σβα + σ 2ββ lβββ + 3σαβ σββ lαββ

+ σαασβα lααα)],

βLB = −1hlog{E(e−hβ

| x)}. (4.6)

Further, we evaluate the Bayesian estimates under the entropy loss function L3. For the parameter αwe have

u(α, β) = α−q, uα = −qα−(q+1), uαα = q(q + 1)α−(q+2),

uβ = uββ = uβα = uαβ = 0,

E(α−q| x) = α−q

+ 0.5[uαα σαα + uα(2ρα σαα + 2ρβ σαβ + σ 2αα lααα + σαασββ lββα

+ 2σαβ σβα lαββ + σαβ σββ lβββ)]

αEB = {E(α−q| x)}

−1q . (4.7)

For the parameter β we observe that

u(α, β) = β−q, uβ = −qβ−(q+1), uββ = q(q + 1)β−(q+2),

uα = uαα = uβα = uαβ = 0,

E(β−q| x) = β−q

+ 0.5[uββ σββ + uβ(2ρβ σββ + 2ρασβα + σ 2ββ lβββ + 3σαβ σββ lαββ

+ σαασβα lααα)].

Consequently, the Bayesian estimate of β against the loss function L3 is given by

βEB = {E(β−q| x)}

−1q . (4.8)

The Bayesian estimates for the reliability function R(t) can be evaluated exactly in a similar manner.Performance of all estimates are discussed in terms of their mean square error values in the next

section.

388 M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391

5. Numerical comparisons

In Sections 2 and 3 several estimates of α, β and R(t) of a Burr(α, β) distribution are obtainedusing progressively type II censored sample. Bayesian estimates of Section 3 are obtained againstdifferent symmetric and asymmetric loss functions such as squared error, linex and general entropy.Approximate Bayesian estimates are derived in Section 4. In this section, performance of all estimateshave been compared numerically in terms of their mean square error (MSE) values. For comparisonpurpose different combinations of n,m and censoring schemes are considered. We notice that riskexpression of none of these estimates can be evaluated in closed form. So, we performed simulationto evaluate MSE values of all estimates. In each case, for a particular censoring scheme, we evaluatedrisk values based on 5000 simulations of progressively type II censored sample of size m from aBurr(α, β) distribution. In Tables 5.1 and 5.2, risk values of estimates α, β, αSB, βSB, αLB, βLB, αEB andβEB are presented for different choices of n,m and censoring schemes. In Tables 5.3 and 5.4, risk valuesof estimates R(t), RSB(t), RLB(t) and REB(t) of the reliability function R(t) are presented for different

Table 5.1Risk values of all estimates of α and β for different choices of n,m, α = 0.75, β = 0.25, a = 4, b = 1, c = 3, d = 4.

n m Scheme α αSB αLB αEB β βSB βLB βEB

20 15(5, 0∗14) 0.062924 0.026039 0.024354 0.028981 0.007319 0.005438 0.004106 0.003549(0∗14, 5) 0.063019 0.026467 0.024759 0.031091 0.008195 0.005798 0.004243 0.003750(0∗5, 1∗5, 0∗5) 0.062742 0.027218 0.025552 0.036398 0.008504 0.005811 0.004373 0.003879

20 (0∗20) 0.046136 0.024790 0.022829 0.026001 0.006252 0.004324 0.003633 0.003112

30

15

(15, 0∗14) 0.062142 0.026138 0.024852 0.028897 0.004748 0.003876 0.003669 0.003143(0∗14, 15) 0.077470 0.024405 0.023590 0.026215 0.005721 0.004236 0.004029 0.003561(1∗15) 0.061402 0.025695 0.024276 0.026929 0.004814 0.003777 0.003613 0.003219(0∗6, 5∗3, 0∗6) 0.059062 0.025560 0.024113 0.026879 0.004763 0.003725 0.003563 0.003184

20

(10, 0∗19) 0.042749 0.024121 0.022348 0.024765 0.004127 0.003184 0.003035 0.002647(0∗19, 10) 0.043092 0.024057 0.022591 0.024090 0.003896 0.003128 0.002997 0.002647(0∗9, 5∗2, 0∗9) 0.039498 0.023506 0.021732 0.023643 0.003857 0.002982 0.002869 0.002569(0∗5, 1∗10, 0∗5) 0.039859 0.023414 0.021843 0.023464 0.003961 0.003160 0.003027 0.002676

25

(5, 0∗24) 0.034985 0.022273 0.020787 0.022415 0.003737 0.002891 0.002769 0.002429(0∗24, 5) 0.032436 0.021376 0.020025 0.021639 0.003501 0.002764 0.002653 0.002351(0∗10, 1∗5, 0∗10) 0.033417 0.021971 0.020683 0.022234 0.003736 0.002986 0.002863 0.002528(0∗12, 5, 0∗12) 0.032606 0.021435 0.020070 0.021489 0.003273 0.002629 0.002523 0.002232

30 (0∗30) 0.030644 0.020791 0.019750 0.020956 0.003279 0.002575 0.002475 0.002196

Table 5.2Risk values of all estimates of α and β for different choices of n,m, α = 1, β = 0.5, a = 4, b = 2, c = 4, d = 4.

n m Scheme α αSB αLB αEB β βSB βLB βEB

20 15(5, 0∗14) 0.100882 0.032906 0.032265 0.038915 0.020972 0.012545 0.011140 0.010582(0∗14, 5) 0.115373 0.033487 0.032379 0.039143 0.022184 0.012674 0.011692 0.011159(0∗5, 1∗5, 0∗5) 0.119871 0.033903 0.032607 0.039579 0.021934 0.012885 0.011957 0.011417

20 (0∗20) 0.078089 0.036008 0.033743 0.044978 0.018507 0.010853 0.010082 0.009619

30

15

(15, 0∗14) 0.100612 0.033543 0.032978 0.038872 0.014929 0.010975 0.010084 0.009359(0∗14, 15) 0.091366 0.028635 0.027679 0.031251 0.021359 0.014790 0.011778 0.010857(1∗15) 0.125484 0.032939 0.031457 0.036071 0.017061 0.011443 0.010718 0.010258(0∗6, 5∗3, 0∗6) 0.125617 0.033978 0.032584 0.037268 0.015640 0.010755 0.010091 0.009656

20

(10, 0∗19) 0.074765 0.035403 0.033171 0.036689 0.012337 0.009488 0.008815 0.008178(0∗19, 10) 0.073705 0.033118 0.032150 0.034939 0.014429 0.010679 0.009997 0.009443(0∗9, 5∗2, 0∗9) 0.072757 0.033429 0.031671 0.034169 0.012351 0.009520 0.008943 0.008448(0∗5, 1∗10, 0∗5) 0.071943 0.033662 0.032043 0.034861 0.012714 0.009648 0.009068 0.008591

25

(5, 0∗24) 0.055351 0.032058 0.029948 0.032536 0.010662 0.008248 0.007719 0.007205(0∗24, 5) 0.053891 0.031887 0.029683 0.031787 0.010548 0.008308 0.007821 0.007357(0∗10, 1∗5, 0∗10) 0.054112 0.031786 0.029757 0.031982 0.010691 0.008398 0.007895 0.007391(0∗12, 5, 0∗12) 0.054460 0.032152 0.030038 0.032267 0.010949 0.008553 0.008052 0.007575

30 (0∗30) 0.046201 0.029697 0.027838 0.029697 0.009636 0.007574 0.007145 0.006692

M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391 389

Table 5.3Risk values of all estimates of R(t) for different choices of n,m, t, α = 0.75, β = 0.25, a = 4, b = 1, c = 3, d = 4.

n m Scheme t = 1 t = 2R(t) RSB(t) RLB(t) REB(t) R(t) RSB(t) RLB(t) REB(t)

20 15(5, 0∗14) 0.009046 0.003946 0.003975 0.004519 0.009591 0.004315 0.004323 0.004910(0∗14, 5) 0.009038 0.004195 0.004206 0.005546 0.009210 0.004391 0.004419 0.005035(0∗5, 1∗5, 0∗5) 0.009073 0.004198 0.004229 0.005160 0.009287 0.004401 0.004413 0.005040

20 (0∗20) 0.007432 0.003867 0.003906 0.004316 0.007476 0.003996 0.004021 0.004469

30

15

(15, 0∗14) 0.008154 0.003862 0.003864 0.004291 0.008554 0.004127 0.004156 0.004674(0∗14, 15) 0.008714 0.003714 0.003706 0.041047 0.009392 0.004501 0.004585 0.051415(1∗15) 0.008522 0.003995 0.004002 0.005005 0.009028 0.004361 0.004379 0.005633(0∗6, 5∗3, 0∗6) 0.008101 0.003872 0.003887 0.004504 0.009053 0.004402 0.004418 0.004865

20

(10, 0∗19) 0.006710 0.003812 0.003838 0.004210 0.006911 0.003999 0.004019 0.004448(0∗19, 10) 0.006244 0.003742 0.003775 0.004103 0.006282 0.003851 0.003879 0.004287(0∗9, 5∗2, 0∗9) 0.005961 0.003601 0.003632 0.003954 0.006609 0.004012 0.004043 0.004471(0∗5, 1∗10, 0∗5) 0.006193 0.003735 0.003764 0.004084 0.006405 0.003894 0.003925 0.004536

25

(5, 0∗24) 0.005714 0.003587 0.003610 0.003883 0.005591 0.003549 0.003564 0.003854(0∗24, 5) 0.005221 0.003396 0.003413 0.003641 0.005419 0.003551 0.003576 0.003879(0∗10, 1∗5, 0∗10) 0.005197 0.003379 0.003396 0.003617 0.005370 0.003517 0.003541 0.003838(0∗12, 5, 0∗12) 0.005236 0.003403 0.003424 0.003659 0.005294 0.003472 0.003496 0.003795

30 (0∗30) 0.004549 0.003056 0.003070 0.003254 0.004739 0.003216 0.003229 0.003450

Table 5.4Risk values of all estimates of R(t) for different choices of n,m, t, α = 1, β = 0.5, a = 4, b = 2, c = 4, d = 4.

n m Scheme t = 1 t = 2R(t) RSB(t) RLB(t) REB(t) R(t) RSB(t) RLB(t) REB(t)

20 15

(5, 0∗14) 0.008995 0.003769 0.003708 0.004239 0.009083 0.004213 0.004116 0.004852(0∗14, 5) 0.009261 0.003917 0.003830 0.004791 0.009549 0.004425 0.004323 0.005237(0∗5, 1∗5, 0∗5) 0.009223 0.003970 0.003899 0.005252 0.009227 0.004327 0.004239 0.005110

20 (0∗20) 0.007786 0.003726 0.003710 0.004142 0.007545 0.003931 0.003896 0.004518

30

15

(15, 0∗14) 0.009684 0.003859 0.003764 0.005856 0.009986 0.004431 0.004315 0.005180(0∗14, 15) 0.009196 0.003029 0.003122 0.004126 0.009552 0.003855 0.003610 0.004613(1∗15) 0.009383 0.003705 0.003554 0.004338 0.009901 0.004143 0.004001 0.005023(0∗6, 5∗3, 0∗6) 0.009635 0.003769 0.003642 0.004775 0.009929 0.004298 0.004140 0.005721

20

(10, 0∗19) 0.007173 0.003695 0.003672 0.004082 0.007269 0.003966 0.003922 0.004503(0∗19, 10) 0.007199 0.003736 0.003716 0.004234 0.007699 0.004099 0.004074 0.005585(0∗9, 5∗2, 0∗9) 0.007171 0.003794 0.003777 0.004209 0.007608 0.004175 0.004139 0.004898(0∗5, 1∗10, 0∗5) 0.007221 0.003807 0.003791 0.004284 0.007334 0.004031 0.004003 0.004674

25

(5, 0∗24) 0.006239 0.003656 0.003647 0.003978 0.005991 0.003687 0.003665 0.004110(0∗24, 5) 0.005902 0.003553 0.003550 0.003865 0.006003 0.003740 0.003729 0.004221(0∗10, 1∗5, 0∗10) 0.005646 0.003443 0.003436 0.003729 0.005517 0.003449 0.003434 0.003876(0∗12, 5, 0∗12) 0.005865 0.003554 0.003552 0.003866 0.005750 0.003596 0.003583 0.004043

30 (0∗30) 0.005283 0.003374 0.003371 0.003625 0.004875 0.003234 0.003218 0.003524

choices of t . In all cases, Bayesian estimates against the loss function L2 is evaluated for h = 1 whilefor the loss function L3 we take q = 1. Also, compact notations have been used to represent differentcensoring schemes in respective tables, for example, censoring scheme (0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0,0, 0) is denoted as (0∗5, 1∗5, 0∗5). Based on tabulated MSE values, following conclusions can be drawnfrom these tables.1. Tables 5.1 and 5.2 exhibit that Bayesian estimates of α and β derived against loss functions L1, L2

and L3 perform better than the MLE’s α and β respectively for various sample sizem.2. For estimating α, the Bayesian estimate αLB (against the linex loss function L2) is a better choice

among all its competitors. Performance of the estimate αSB is also good as can be seen fromTables 5.1 and 5.2.

3. In the case of β we found that the performance of the estimate βEB is better than all its competitorsfor all tabulated values of m. The Bayesian estimate βLB also performs quite well in all suchsituations.

390 M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391

4. It is observed that performance of Bayesian estimates RSB(t) (against squared error loss functionL1) and RLB(t) (against the linex loss function L2) of the reliability function R(t) is better than otherestimates for various choices ofm and t (see Tables 5.3 and 5.4).We also notice that performance ofRSB(t) and RLB(t) for these choices ofm and t cannot be compared as for some choices one dominatethe other while for other cases opposite is true.

5. As the sample proportion mn increases performance of all estimates improves in terms of theirmean

square error values. We observed similar behavior for other censoring schemes as well.

6. Data analysis

In this section we perform analysis on two different data set to illustrate our proposed methods ofestimation.

Example 1 (Simulated Data). In this example analysis on a simulated data is presented.We consideredthe case when n = 25 and m = 20. Then we simulated a progressively type II censored sample fromthe Burr(α, β) distribution when α = 1, β = 1 and (0∗3, 1, 0∗3, 1, 0∗3, 1, 0∗3, 1, 0∗3, 1) being thecensoring scheme. Samples from this data are given as:0.026512,0.063909,0.085911,0.135388,0.207235,0.226529,0.398520,0.462585,0.491429,0.530789,0.692413,0.924024,1.695117,1.722038,2.052806,2.200573,2.342361,2.652641,3.621829,6.831203.

We evaluated Bayesian estimates under the assumption that hyperparameters take values asa = b = c = d = 4. All estimates of α and β are presented in Table 6.1. We observed thatBayesian estimates are less biased than the corresponding maximum likelihood estimates of α andβ . In that regard, among different Bayesian estimates, αEB and βEB are better than their respectivecompetitors. The maximum likelihood estimates and Bayesian Estimates of the reliability functionR(t) are presented in Table 6.2 for two different choices of t . It is easy to observe that values ofreliability estimates are marginally higher in case of Bayesian estimates (under squared error andlinex) than the correspondingMLE of the reliability function. However, opposite is true for the entropyloss function.

Example 2 (Real Data). In this example a real data set is analyzed. This data is reported in [14] andwas also used by Soliman [16] which describes times to breakdown of an insulating fluid betweenelectrodes recorded at the voltage of 34 kV (minutes). The complete data set contains 19 observationsand is given as0.19,0.78,0.96,1.31,2.78,3.16,4.15,4.67,4.85,6.50,7.35,8.01,8.27,12.06,31.75,32.52,33.91,36.71,72.89.

Zimmer et al. [21] reported that the Burr(α, β) distribution fits the data quite well. A progressivelytype II censored sample of size m = 8 simulated randomly from the sample of size n = 19 withcensoring scheme (0∗2, 3, 0, 3, 0∗2, 5) is obtained as (see [16]),0.19,0.78,0.96,1.31,2.78,4.85,6.50,7.35.

Table 6.1Estimates of α and β for Example 1.

α αSB αLB αEB β βSB βLB βEB

1.0854 1.06557 1.03552 1.01303 1.05218 1.04906 1.02995 1.01373

Table 6.2Estimates of R(t) for different choices of t for Example 1.

t = 1 t = 2R(t) RSB(t) RLB(t) REB(t) R(t) RSB(t) RLB(t) REB(t)

0.471261 0.48483 0.481541 0.470648 0.295595 0.314774 0.311614 0.292564

M.K. Rastogi, Y.M. Tripathi / Statistical Methodology 9 (2012) 381–391 391

Table 6.3Estimates of α and β for Example 2.

α αSB αLB αEB β βSB βLB βEB

0.224834 0.232486 0.223907 0.219452 1.55166 1.65159 1.46685 1.42076

Table 6.4Estimates of R(t) for different choices of t for Example 2.

t = 1 t = 2R(t) RSB(t) RLB(t) REB(t) R(t) RSB(t) RLB(t) REB(t)

0.855694 0.855449 0.854495 0.848915 0.735059 0.736779 0.734426 0.727726

We have tabulated respective MLE’s and Bayesian estimates of α and β in Table 6.3. Estimates of R(t)are presented in Table 6.4 for two choices of t . For comparison purpose (with corresponding MLE’s),Bayesian estimates are evaluated against noninformative prior distribution. This corresponds to thecase when a = b = c = d = 0. We observed that Bayesian estimates αLB and αEB are marginallysmaller than the corresponding MLE of α while opposite is true for the Bayesian estimate obtainedunder the squared error loss function. Similar behavior is observed for the parameter β . It can bealso seen from the Table 6.4 that the MLE and Bayesian procedures have similar values for reliabilityestimates.

Acknowledgments

Authors would like to thank the Associate Editor and two referees for their valuable suggestions.

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