Estimating the parameters of the Weibull distribution under generalized type II hybrid censoring

Estimating the parameters of the Weibull distribution under generalized type II hybrid censoring

Kyeongjun Lee ¹

1 Division of Mathematics and Big Data Science, Daegu University

Received 30 June 2021, revised 12 July 2021, accepted 14 July 2021

Abstract

In hybrid censoring both the time and the number of failures are considered for the life testing of the product. The combination of type I and type II censoring is called the hybrid censoring. Though the type II hybrid censored scheme guarantees a pre-fixed number of failures, it might take a long time to complete the test. In order to provide a guarantee in terms of the time to complete the test, generalized type II hybrid censoring scheme was introduced. In this paper, we consider the MLEs of the parameters and reliability when the data are generalized type II hybrid censored Weibull data. However, the MLEs cannot be obtained in a closed form. We use the approximate MLEs using Taylor series expansion. Also, we consider the Bayes estimation for the parameters and reliability when the data are generalized type II hybrid censored Weibull data. In Bayes estimation, Lindley’s approximation is used to obtain the Bayes estimators. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, a real data set has been analysed for illustrative purposes.

Keywords: Bayes estimation, generalized type II hybrid censoring, Linex loss, squared error loss, Weibull distribution.

1. Introduction

Consider a life testing experiment in which n units are put on test. Assume that the life times of n units are independent and identically distributed (i.i.d) as Weibull distribution with the cumulative density function (cdf)

F (x; α, λ) = 1 − exp (−λx ^α ) , x > 0, α > 0, λ > 0, (1.1) the probability density function (pdf)

f (x; α, λ) = αλx ^α−1 exp (−λx ^α ) , x > 0, α > 0, λ > 0 (1.2) and the reliability function

R(t; α, λ) = exp (−λt ^α ) . (1.3)

1

Assistant professor, Division of Mathematics and Big Data Science, Daegu University, Gyeongsan

38453, Korea. E-mail: indra [email protected]

The estimation for the parameters of Weibull distribution based on the censored sample has been investigated by many authors such as Almetwally et al. (2020), Lee et al. (2020a, 2020b), Okasha and Mustafa (2020), Wang (2018), Wang et al. (2020). Almetwally et al.

(2020) considered a maximum product spacing estimation of Weibull distribution under adaptive type-II progressive censoring. Lee et al. (2020a, 2020b) considered Bayes estima- tion of Weibull distribution based on generalized adaptive progressive hybrid censoring.

Okasha and Mustafa (2020) considered E-Bayesian estimation for the Weibull distribution under adaptive type I progressive hybrid censored competing risks data. Wang (2018) and Wang et al. (2020) considered inference for Weibull competing risks data under generalized progressive hybrid censoring.

Hybrid censoring scheme is commonly used in the life testing situations which was intro- duced by Epstein (1954) for the exponential distribution as the life time distribution (Aslam et al., 2017). In hybrid censoring both the time and the number of failures are considered for the life testing of the product (Epstein, 1954). There are situations in which only the time is fixed for the life testing, which is known as the Type-I censoring. When the number of observed failures is fixed in the life testing, it is called the Type-II censoring. The combina- tion of these two censoring schemes is called the hybrid censoring (Balakrishnan and Kundu, 2013). Following Childs et al. (2003) introduced a type I hybrid censoring scheme and type II hybrid censoring scheme. In type II hybrid censoring scheme, the termination point is max {X r:n , T}. Here, the ith ordered life times of items be denoted by X i:n , r ∈ {1, 2, · · · , n}

and T ∈ (0, ∞) are pre-fixed. Though the type II hybrid censored scheme guarantees a pre- fixed number of failures, it might take a long time to observe r failures. In order to provide a guarantee in terms of the time to complete the test, Chandrasekar et al. (2004) introduced generalized type II hybrid censoring scheme. The detail description of the generalized type II hybrid censoring scheme is described as follows. If the rth failure occurs before time T 1 , terminate the experiment at T 1 ; if the rth failure occurs between T 1 and T 2 , terminate at X _r:n ; if the rth failures occur after T ₂ , terminate at T ₂ (see Chandrasekar et al., 2004). Let d ₁ and d ₂ denote the number of observed failures up to time T ₁ and T ₂ , respectively.

This study is to consider the MLEs of the parameters and reliability when the data are generalized type II hybrid censored data. However, the MLEs cannot be obtained in a closed form. We use the approximate MLEs using Taylor series expansion. Also, we consider the Bayes estimation for the parameters and reliability when the data are generalized type II hybrid censored data.

2. Maximum likelihood estimation

Assume that the failure times of the units are the Weibull distribution with cdf (1.1) and pdf (1.2). The likelihood functions for three different cases are as follows.

Case 1 : L ₁ (α, λ) = n!

(n − d 1 )! α ^d

λ ^d

d

Y

i=1

x ^α−1 _i:n exp

"

−λ

d

X

i=1

x ^α _i:n + (n − d ₁ )T ₁ ^α

!#

,

Case 2 : L 2 (α, λ) = n!

(n − r)! α ^r λ ^r

r

Y

i=1

x ^α−1 _i:n exp

"

−λ

r

X

i=1

x ^α _i:n + (n − r)x ^α _r:n

!#

,

Case 3 : L ₃ (α, λ) = n!

(n − d 2 )! α ^d

λ ^d

d

Y

i=1

x ^α−1 _i:n exp

"

−λ

d

X

i=1

x ^α _i:n + (n − d ₂ )T ₂ ^α

!#

.

Here, Cases 1, 2, and 3 can be combined, and be represented as

L (α, λ) = n!

(n − d)! α ^d λ ^d

d

Y

i=1

x ^α−1 _i:n exp

"

−λ

d

X

i=1

x ^α _i:n + (n − d)T ^α

!#

. (2.1)

Here, d means the number of failures. d = d 1 if x r:n < T 1 , D = r if T 1 < x r:n < T 2 , and d = d 2 if T 2 < x r:n . T means the terminated time. T = T 1 if x r:n < T 1 , T = x r:n if T 1 < x r:n < T 2 , and T = T 2 if T 2 < x r:n . Ignoring the constant, from (2.1), corresponding log likelihood functions are

lnL (α, λ) = dlnαλ + (α − 1)

d

X

i=1

lnx _i:n − λ

d

X

i=1

x ^α _i:n − λ (n − d) T ^α .

On differentiating the log-likelihood functions with respect to α and λ and equating to zero, we obtain the estimating equations:

∂lnL

∂α = d α +

d

X

i=1

lnx i:n − λ

d

X

i=1

x ^α _i:n lnx i:n − λ (n − d) T ^α lnT = 0, (2.2)

∂lnL

∂λ = d λ −

d

X

i=1

x ^α _i:n − (n − d) T ^α = 0.

Then, we get the MLE of λ, say ˆ λ(α), as a function of the MLE of λ as

λ(α) = ˆ d

P d

i=1 x ^α _i:n + (n − d) T ^α (2.3)

and putting the value of ˆ λ(α) into equation (2.2), we obtain d

α +

d

X

i=1

lnx _i:n − ˆ λ(α)

" _d X

i=1

x ^α _i:n lnx _i:n + (n − d) T ^α lnT

#

= 0. (2.4)

Then, we obtain

k(α) = α, (2.5)

where k(α) = d/{ˆ λ(α) h P d

i=1 x ^α _i:n lnx i:n + (n − d) T ^α lnT i

− P d

i=1 lnx i:n }.

From (2.5), we introduce a simple iterative scheme to solve for α. It has been proposed in the literature by Lee (2020a). Start with an initial guess of α, say α ⁽⁰⁾ , then obtain α ⁽¹⁾ = k(α ⁽⁰⁾ ) and proceed in this way iteratively to obtain α ⁽ⁿ⁺¹⁾ = k(α ⁽ⁿ⁾ ). Stop the iterative procedure, when |α ⁽ⁿ⁺¹⁾ − α ⁽ⁿ⁾ | < , some pre-assigned tolerance limit. Once we obtain the MLE of α, say ˆ α, then the MLE of λ can be obtain as ˆ λ = ˆ λ( ˆ α) from (2.3).

With α and λ replaced by the ˆ α and ˆ λ, in Eq (1.3), the reliability estimator of the Weibull distribution based on generalized type II hybrid censoring scheme are obtained as:

R(t) = exp ˆ 

−ˆ λt ^{α ˆ} 

. (2.6)

2.1. Approximate MLE

Because the log-likelihood equations cannot be solved explicitly, it will be desirable to consider an approximation to the likelihood equations that will provide explicit estimators of α and λ. In this section, therefore, we provide the approximate MLE which have explicit form. First of all, let y i:n = lnx i:n . Also, let z i:n = (y i:n − µ)/σ and T z = (T − µ)/σ where σ = 1/α and µ = −1/αlnλ. Then, the Eq (2.1) can be obtained as

L(µ, σ) = n!

(n − d)!

1 σ ^d

d

Y

i=1

f (z i:n )[1 − F (T z )] ^n−d , (2.7)

where F (z) = 1−exp(−e ^z ), f (z) = exp(z −e ^z ). On differentiating the log-likelihood function with respect to µ and σ, we obtain the two estimating equations

∂lnL

∂µ =

d

X

i=1

f ⁰ (z _i:n )

f (z i:n ) − (n − d) f (T z )

1 − F (T z ) = 0, (2.8)

∂lnL

∂σ = d +

d

X

i=1

f ⁰ (z i:n )

f (z i:n ) z i:n − (n − d) f (T z )

1 − F (T z ) T z = 0. (2.9) Because the Eqs (2.8) and (2.9) cannot be solved explicitly, we expand the function f ⁰ (z _i:n )/f (z _i:n ) and f (T z )/[1−F (T z )] in Taylor series expansion as follows. Let p _i = i/(n+1) and q _i = 1 − p _i for i = 1, 2, · · · , n. Also, let p _d

1

= (p _d

+ p _d

₊₁ )/2 and q _d

1

= 1 − p _d

1

for x _r:n < T ₁ . And, let p _d

2

= (p _d

+ p _d

₊₁ )/2 and q _d

2

= 1 − p _d

2

for T ₂ < x _r:n . Then, we expand the function f ⁰ (z _i:n )/f (z _i:n ) in Taylor series expansion around the points ξ _i = ln(−lnq _i ).

Also, we expand the function f (T z )/[1 − F (T z )] in Taylor series expansion around the points ξ d

= ln(−lnq d

) for x r:n < T 1 , ξ r = ln(−lnq r ) for T 1 < x r:n < T 2 and ξ d

= ln(−lnq d

) for T 2 < x r:n . We can approximate the functions by

f ⁰ (z _i:n )

f (z i:n ) ' δ i − β i z i:n , f (T z )

1 − F (T z ) ' 1 − δ d

+ β d

T z , (2.10) where δ i = 1 + lnq i [1 − ln(−lnq i )], β i = −lnq i , d = d ^∗ ₁ for x r:n < T 1 , d = r for T 1 < x r:n < T 2

and d = d ^∗ ₂ for T 2 < x r:n . By substituting Eqs (2.10) into Eqs (2.8) and (2.9), we obtain

∂lnL

∂µ '

d

X

i=1

(δ _i − β i z _i:n ) − (n − d)(1 − δ _d

+ β _d

T z ) = 0, (2.11)

∂lnL

∂σ ' d +

d

X

i=1

(δ i − β i z _i:n ² ) − (n − d)(1 − δ d

+ β d

T z )T z = 0. (2.12)

From Eqs (2.11) and (2.12), we obtain as

ˆ µ =

P d

i=1 β _i y _i:n + β _d

(n − d)T z

P d

i=1 β i + β d

(n − d) − P d

i=1 δ i − (n − d)(1 − δ d

) P d

i=1 β i + β d

(n − d) σ, ˆ (2.13) ˆ

σ = −B + √

B ² + 4dC

2d , (2.14)

where B =

d

X

i=1

δ i y i:n − P d

i=1 β i y i:n + β d

(n − d)T z

P d

i=1 β _i + β _d

(n − d)

!

− (n − d)(1 − δ d

) T z − P d

i=1 β i y i:n + β d

(n − d)T z

P d

i=1 β _i + β _d

(n − d)

!

− 2 P d

i=1 δ i − (n − d)(1 − δ d

) P d

i=1 β i + β d

(n − d)

" _d X

i=1

β _i y _i:n − P d

i=1 β i y i:n + β d

(n − d)T z

P d

i=1 β i + β d

(n − d)

!

−β d

(n − d) T z − P d

i=1 β _i y _i:n + β _d

(n − d)T z

P d

i=1 β i + β d

(n − d)

!#

,

C =

d

X

i=1

β i y i:n − P d

i=1 β _i y _i:n + β _d

(n − d)T z

P d

i=1 β i + β d

(n − d)

! ²

+ (n − d)β d

T z − P d

i=1 β i y i:n + β d

(n − d)T z

P d

i=1 β i + β d

(n − d)

! 2

.

Since C > 0, the estimator ˆ σ is always positive root. Hence, get the approximate MLEs of the α and λ under generalized type II hybrid censoring scheme as follows

ˆ α _A = 1

ˆ

σ and ˆ λ _A = exp(− ˆ α _A µ). ˆ

With α and λ replaced by the ˆ α _A and ˆ λ _A , in Eq (1.3), the reliability estimator of the Weibull distribution based on generalized type II hybrid censoring scheme are obtained as:

R(t) ˆ A = exp 

−ˆ λ A t ^{α ˆ}



. (2.15)

3. Bayes estimation

In this section, we derive the Bayes estimators for parameters of a Weibull distribution under squared error, linex loss and entropy loss functions. Here, we assumed that the pa- rameters α and λ are independent and follow the γ(a ₁ , b ₁ ) and γ(a ₂ , b ₂ ) prior distributions with a ₁ , b ₁ , a ₂ , b ₂ > 0. Then, the joint prior distribution of α and λ is of the form

π(α, λ) ∝ α ^a

⁻¹ λ ^a

⁻¹ e ^(−b

^α−b

^λ) , α > 0, λ > 0. (3.1) Based on the Eq (3.1), the joint density of the α, λ and X X X, can be written as follows.

π(α, λ, X X X) ∝ α ^d+a

⁻¹ λ ^d+a

⁻¹

d

Y

i=1

x ^α−1 _i:n exp

"

−λ (

b ₂ +

d

X

i=1

x ^α _i:n + (n − d)T ^α )

− b ₁ α

# ,

where X X X = (x 1:n , x 2:n , · · · , x d:n ). Now, based on the above priors, we compute the Bayes

estimates of and using Lindley’s approximation. For the two parameter case, (θ 1 , θ 2 ), the

Lindley’s approximation can be written as

ˆ

g =g  ˆ θ ₁ , ˆ θ ₂  + 1

2 [u ₁₁ κ ₁₁ + u ₁₂ κ ₁₂ + u ₂₁ κ ₂₁ + u ₂₂ κ ₂₂ + l ₃₀ {(u ₁ κ ₁₁ + u ₂ κ ₁₂ )κ ₁₁ } (3.2) + l ₀₃ {(u 2 κ ₂₂ + u ₁ κ ₂₁ )κ ₂₂ } + l 21 {3u 1 κ ₁₁ κ ₁₂ + u ₂ (κ ₁₁ κ ₂₂ + 2κ ² ₁₂ )}

+l 12 {3u 2 κ 22 κ 21 + u 1 (κ 22 κ 11 + 2κ ² ₂₁ )} + p 1 (u 1 κ 11 + u 2 κ 21 ) + p 2 (u 2 κ 22 + u 1 κ 12 ), where

l ij = ∂ ^i+j l(θ 1 , θ 2 )

∂θ ⁱ ₁ ∂θ ₂ ^j , p i = ∂lnπ (θ 1 , θ 2 )

∂θ _i , u i = ∂g(θ 1 , θ 2 )

∂θ _i , u ij = ∂ ² g(θ 1 , θ 2 )

∂θ _i θ _j , κ 11 κ 12

κ 21 κ 22



= −l 11 −l 12

−l 21 −l 22

 −1

. To compute Bayesian estimators of α, λ and R(t), we have

l 30 = 2d ˆ α ³ − ˆ λ

" _d X

i=1

x ^α _i:n ^ˆ (lnx i:n ) ³ + (n − d)T ^{α ˆ} (lnT) ³

#

, l 03 = 2d

λ ˆ ³ , l 12 = 0

l ₂₁ = −

" _d X

i=1

x ^α _i:n ^ˆ (lnx _i:n ) ² + (n − d)T ^{α ˆ} (lnT) ²

#

, p ₁ = a 1 − 1 ˆ

α − b ₁ , p ₂ = a 2 − 1 λ ˆ − b ₂ . Also, we have

κ 11 = M 3

M 1 − M ₂ ² , κ 22 = M 4

M 1 − M ₂ ² , κ 12 = κ 21 = − M 2

M 1 − M ₂ ² , where M ₁ = 

d/ ˆ αˆ λ  ²

+ d/ˆ λ n P d

i=1 x ^α _i:n ^ˆ (lnx _i:n ) ² + (n − d)T ^{α ˆ} (lnT) ² o

, M ₂ = P d i=1 x ^α _i:n ^ˆ lnx i:n +(n−d)T ^{α ˆ} lnT, M 3 = d/ ˆ λ ² , M 4 = d/ ˆ α ² +ˆ λ h

P d

i=1 x ^α _i:n ^ˆ (lnx i:n ) ² + (n − d)T ^{α ˆ} (lnT) ² i . Now, we compute the Bayes estimate for α under squared error loss function using Lind- ley’s approximation. Then, we observe that

g(α, λ) = α, u 1 = 1, u 2 = u 11 = u 12 = u 21 = u 22 = 0.

Therefore, using equation (3.2), the Bayes estimator for α under squared error loss function is given by

ˆ

α S = ˆ α + 0.5[l 30 κ ² ₁₁ + l 03 κ 21 κ 22 + 3l 21 κ 11 κ 12 ] + p 1 κ 11 + p 2 κ 12 . (3.3) Similarly, the Bayes estimator for λ under squared error loss function is given by

λ ˆ _S = ˆ λ + 0.5[l ₃₀ κ ₁₂ κ ₁₁ + l ₀₃ κ ² ₂₂ + l ₂₁ (κ ₁₁ κ ₂₂ + 2κ ² ₁₂ )] + p ₁ κ ₂₁ + p ₂ κ ₂₂ . (3.4) Also, we observe that

g(α, λ) = exp(−λt ^α ), u ₁ = R ^(α) (t) = −λt ^α ln t exp(−λt ^α ), u ₂ = R ^(λ) (t) = −t ^α exp(−λt ^α ), u 11 = R ^(αα) (t) = −λt ^α (ln t) ² exp(−λt ^α )(1 − λt ^α ), u 22 = R ^(λλ) (t) = t ^2α exp(−λt ^α ), u 12 = u 21 = R ^(αλ) (t) = t ^α ln t exp(−λt ^α )(−1 + λt ^α ln t).

Then, the Bayes estimator for R(t) under squared error loss function is given by

R(t) ˆ S = ˆ R(t) + 1

2 [u 11 κ 11 + 2u 12 κ 12 + u 22 κ 22 + l 30 (u 1 κ 11 + u 2 κ 12 )κ 11

+l ₀₃ (u ₂ κ ₂₂ + u ₁ κ ₂₁ )κ ₂₂ + l ₂₁ {3u ₁ κ ₁₁ κ ₁₂ + u ₂ (κ ₁₁ κ ₂₂ + 2κ ² ₁₂ )} 

+ p 1 (u 1 κ 11 + u 2 κ 21 ) + p 2 (u 2 κ 22 + u 1 κ 12 ). (3.5) Next, we compute the Bayes estimate for α under linex loss function using Lindley’s approximation. Then, we observe that

g(α, λ) = e ^−hα , u ₁ = −he ^−cα , u ₁₁ = h ² e ^−hα , u ₂ = u ₁₂ = u ₂₁ = u ₂₂ = 0.

Therefore, using Eqs (3.2), the Bayes estimator for α under linex loss function is given by ˆ

α _L = − 1

h ln[e ^{−h ˆ α} + 0.5{u ₁₁ κ ₁₁ + l ₃₀ u ₁ κ ² ₁₁ + l ₀₃ u ₁ κ ₂₁ κ ₂₂ + 3l ₂₁ u ₁ κ ₁₁ κ ₁₂ }

+ p 1 u 1 κ 11 + p 2 u 1 κ 12 ]. (3.6)

Similarly, the Bayes estimator for λ under linex loss function is given by ˆ λ L = − 1

h ln[e ^{−hˆ λ} + 0.5{u 22 κ 22 + l 30 u 2 κ 12 κ 11 + l 03 u 2 κ ² ₂₂ + l 21 u 2 (κ 11 κ 22 + 2κ ² ₁₂ )}

+ p 1 u 2 κ 21 + p 2 u 2 κ 22 ]. (3.7)

Also, we observe that

g(α, λ) = e ^{−h exp(−λt}

⁾ = e ^−hR(t) , u ₁ = −hR ^(α) (t)e ^−hR(t) , u ₂ = −hR ^(λ) (t)e ^−hR(t) , u 11 = −h[R ^(αα) (t) − h(R ^(α) (t)) ² ]e ^−hR(t) , u 22 = −h[R ^(λλ) (t) − h(R ^(λ) (t)) ² ]e ^−hR(t) , u 12 = u 21 = −h[R ^(αλ) (t) − hR ^(α) (t)R ^(λ) (t)]e ^−hR(t) .

Then, the Bayes estimator for R(t) under squared error loss function is given by R(t) ˆ L = − 1

h ln[e ^{−h ˆ R(t)} + 1

2 [u 11 κ 11 + 2u 12 κ 12 + u 22 κ 22 + l 30 (u 1 κ 11 + u 2 κ 12 )κ 11

+l 03 (u 2 κ 22 + u 1 κ 21 )κ 22 + l 21 {3u 1 κ 11 κ 12 + u 2 (κ 11 κ 22 + 2κ ² ₁₂ )} 

+ p ₁ (u ₁ κ ₁₁ + u ₂ κ ₂₁ ) + p ₂ (u ₂ κ ₂₂ + u ₁ κ ₁₂ ). (3.8)

4. Illustrative example and simulation results

4.1. Simulated results

From the above section, the mean squared errors (MSEs) of the estimators are simulated by

Monte Carlo method (based on 10,000 Monte Carlo runs) for sample size n = 20, 40, 60, and

different r, T 1 and T 2 values. Before progressing further, first we describe how we generate

generalized type II hybrid censored data for a given set n, r, T 1 and T 2 . If X r:n < T 1 ,

then we have Case I and we find d 1 such that X d

:n < T 1 < X d

+1:n . The corresponding

generalized type II hybrid censored sample is (X 1:n , X 2:n , · · · , X d

:n ). If T 1 < X r:n < T 2 ,

then we have Case II and the corresponding generalized type II hybrid censored sample

is (X _1:n , X _2:n , · · · , X _r:n ). If T ₂ < X _r:n , then we have Case III and we find d ₂ such that

X _d

_:n < T ₂ < X _d

_+1:n . The corresponding generalized type II hybrid censored sample is (X _1:n , X _2:n , · · · , X _d

_:n ). Without loss of generality we take α = λ = 1 in each case. All Bayes estimates are computed when hyperparameters take values of a 1 = a 2 = b 1 = b 2 = 0.00001.

Bayes estimates for α and λ are derived with respect to squared and Linex loss functions.

Under Linex loss function, Bayes estimates are obtained for h = −0.5, 0.5, 1.5. We mainly compare the performances of the proposed estimators for α, λ and R(t), in terms of their biases and MSEs for different generalized type II hybrid censoring schemes, and these values are tabulated in Tables 4.2 ∼ 4.4.

Table 4.1 The relative MSEs and biases of estimators for α

n T

T

r α ˆ α ˆ

α ˆ

α ˆ

h = −.5 h = .5 h = 1.5

20 1.0 2.0 18 .0566(.0625) .0548(.0531) .0543(.0536) .0577(.0653) .0512(.0419) .0464(.0198) 16 .0690(.0873) .0662(.0760) .0669(.0805) .0724(.0945) .0619(.0067) .0541(.0406) 14 .0804(.0981) .0771(.0861) .0788(.0933) .0864(.1099) .0719(.0769) .0613(.0461) 2.5 18 .0557(.0711) .0537(.0615) .0532(.0610) .0567(.0721) .0501(.0501) .0451(.0293) 16 .0681(.0911) .0653(.0797) .0659(.0841) .0715(.0980) .0609(.0704) .0530(.0446) 14 .0803(.0987) .0770(.0868) .0787(.0939) .0863(.1105) .0717(.0775) .0611(.0467) 1.5 2.0 18 .0532(.0586) .0517(.0496) .0508(.0496) .0539(.0612) .0480(.0381) .0437(.0162) 16 .0573(.0673) .0556(.0575) .0551(.0598) .0591(.0727) .0515(.0470) .0460(.0228) 14 .0580(.0606) .0564(.0511) .0560(.0542) .0601(.0678) .0523(.0406) .0468(.0151) 2.5 18 .0522(.0672) .0506(.0581) .0497(.0570) .0528(.0680) .0469(.0462) .0424(.0257) 16 .0564(.0711) .0547(.0612) .0541(.0634) .0582(.0762) .0505(.0507) .0449(.0267) 14 .0578(.0613) .0563(.0518) .0558(.0548) .0600(.0684) .0521(.0413) .0466(.0157) 30 1.0 2.0 28 .0328(.0367) .0323(.0309) .0317(.0304) .0329(.0377) .0306(.0231) .0289(.0092) 25 .0422(.0547) .0412(.0478) .0413(.0496) .0433(.0579) .0394(.0414) .0362(.0255) 22 .0491(.0688) .0477(.0612) .0483(.0655) .0514(.0754) .0455(.0556) .0409(.0365) 2.5 28 .0314(.0427) .0308(.0372) .0302(.0353) .0313(.0419) .0291(.0288) .0274(.0161) 25 .0420(.0593) .0409(.0524) .0409(.0540) .0431(.0621) .0390(.0459) .0357(.0303) 22 .0491(.0691) .0477(.0615) .0483(.0657) .0513(.0757) .0455(.0558) .0409(.0368) 1.5 2.0 28 .0327(.0365) .0322(.0308) .0316(.0302) .0328(.0374) .0305(.0229) .0288(.0090) 25 .0366(.0462) .0359(.0398) .0356(.0409) .0373(.0489) .0341(.0330) .0316(.0177) 22 .0380(.0440) .0373(.0376) .0370(.0397) .0389(.0484) .0354(.0311) .0327(.0144) 2.5 28 .0313(.0426) .0307(.0370) .0301(.0351) .0312(.0417) .0290(.0286) .0273(.0159) 25 .0364(.0509) .0356(.0444) .0353(.0453) .0370(.0532) .0337(.0375) .0311(.0224) 22 .0379(.0443) .0373(.0378) .0370(.0400) .0389(.0487) .0353(.0314) .0326(.0147) 40 1.0 2.0 38 .0216(.0247) .0213(.0203) .0210(.0199) .0216(.0252) .0205(.0147) .0197(.0044) 34 .0249(.0362) .0244(.0312) .0244(.0322) .0253(.0380) .0236(.0264) .0222(.0152) 30 .0333(.0531) .0325(.0474) .0329(.0506) .0345(.0576) .0315(.0436) .0290(.0299) 2.5 38 .0202(.0281) .0199(.0242) .0195(.0223) .0200(.0270) .0191(.0176) .0183(.0085) 34 .0246(.0407) .0241(.0356) .0241(.0364) .0250(.0421) .0232(.0308) .0218(.0198) 30 .0333(.0534) .0325(.0476) .0329(.0509) .0345(.0579) .0314(.0438) .0289(.0301) 1.5 2.0 38 .0216(.0247) .0213(.0203) .0210(.0199) .0216(.0252) .0205(.0147) .0197(.0044) 34 .0237(.0333) .0232(.0231) .0292(.0292) .0240(.0350) .0224(.0235) .0212(.0124) 30 .0253(.0328) .0249(.0278) .0249(.0296) .0258(.0360) .0240(.0233) .0226(.0110) 2.5 38 .0202(.0281) .0199(.0242) .0195(.0223) .0200(.0270) .0191(.0176) .0183(.0085) 34 .0234(.0378) .0229(.0329) .0228(.0335) .0237(.0391) .0220(.0279) .0207(.0170) 30 .0253(.0330) .0249(.0281) .0248(.0299) .0258(.0362) .0240(.0236) .0225(.0113)

In Tables 4.2 ∼ 4.4, the MSE and bias values of all estimates for α, λ and R(t) are

presented for various choices of n, r, T 1 and T 2 . In general, we observed that the MSEs

decrease as sample size n increases. For fixed sample size n, the MSEs decrease generally as

r decreases. Also, we observed that Bayes estimates are superior to the respective MLE and

approximate MLE in terms of MSEs. In particular, respective Bayes estimates under Linex

loss functions for α, λ and R(t) are better than the corresponding MLE and approximate

MLE. For estimating the α, λ and R(t), the choice h = 1.5 seem to be a reasonable choice

under Linex loss functions.

Table 4.2 The relative MSEs and biases of estimators for λ

n T

T

r ˆ λ λ ˆ

λ ˆ

ˆ λ

h = −.5 h = .5 h = 1.5

20 1.0 2.0 18 .0910(.0436) .0954(.0616) .0902(.0397) .0906(.0407) .0898(.0388) .0890(.0368) 16 .1001(.0673) .1043(.0820) .0985(.0628) .0992(.0640) .0979(.0616) .0967(.0592) 14 .1011(.0766) .1050(.0887) .0997(.0710) .1003(.0725) .0991(.0696) .0980(.0667) 2.5 18 .0900(.0488) .0942(.0677) .0891(.0452) .0895(.0461) .0887(.0443) .0879(.0424) 16 .0999(.0696) .1040(.0844) .0983(.0652) .0990(.0664) .0977(.0640) .0965(.0617) 14 .1000(.0769) .1041(.0890) .0987(.0714) .0993(.0728) .0981(.0699) .0970(.0670) 1.5 2.0 18 .0769(.0374) .0824(.0561) .0763(.0336) .0766(.0345) .0760(.0326) .0754(.0307) 16 .0771(.0419) .0825(.0589) .0763(.0378) .0767(.0388) .0761(.0367) .0755(.0347) 14 .0791(.0358) .0845(.0521) .0784(.0314) .0787(.0325) .0781(.0303) .0775(.0281) 2.5 18 .0761(.0426) .0815(.0622) .0753(.0390) .0757(.0399) .0750(.0381) .0744(.0363) 16 .0767(.0443) .0822(.0613) .0761(.0402) .0764(.0412) .0758(.0392) .0753(.0371) 14 .0781(.0361) .0833(.0524) .0774(.0317) .0777(.0328) .0770(.0306) .0764(.0285) 30 1.0 2.0 28 .0453(.0176) .0471(.0297) .0451(.0160) .0452(.0164) .0451(.0156) .0449(.0148) 25 .0547(.0350) .0558(.0451) .0543(.0332) .0545(.0336) .0542(.0327) .0540(.0318) 22 .0580(.0500) .0592(.0577) .0576(.0477) .0578(.0483) .0574(.0471) .0571(.0459) 2.5 28 .0449(.0218) .0467(.0352) .0447(.0203) .0448(.0207) .0447(.0200) .0445(.0193) 25 .0539(.0378) .0551(.0478) .0535(.0359) .0537(.0364) .0534(.0355) .0532(.0346) 22 .0580(.0501) .0591(.0578) .0575(.0478) .0577(.0484) .0573(.0472) .0570(.0461) 1.5 2.0 28 .0448(.0173) .0466(.0294) .0446(.0156) .0447(.0160) .0446(.0153) .0445(.0145) 25 .0456(.0250) .0472(.0355) .0454(.0232) .0455(.0236) .0454(.0227) .0452(.0219) 22 .0464(.0021) .0479(.0318) .0462(.0201) .0463(.0206) .0461(.0197) .0459(.0187) 2.5 28 .0444(.0215) .0462(.0349) .0442(.0200) .0443(.0204) .0442(.0197) .0440(.0190) 25 .0455(.0277) .0471(.0383) .0454(.0259) .0454(.0263) .0453(.0255) .0451(.0246) 22 .0456(.0222) .0471(.0319) .0454(.0203) .0455(.0207) .0453(.0198) .0451(.0189) 40 1.0 2.0 38 .0338(.0130) .0348(.0218) .0337(.0121) .0338(.0123) .0337(.0118) .0337(.0114) 34 .0384(.0241) .0391(.0318) .0383(.0231) .0383(.0233) .0383(.0229) .0382(.0224) 30 .0440(.0414) .0446(.0471) .0438(.0401) .0439(.0404) .0437(.0398) .0436(.0392) 2.5 38 .0332(.0153) .0343(.0254) .0331(.0144) .0331(.0146) .0331(.0142) .0330(.0138) 34 .0378(.0268) .0384(.0345) .0376(.0258) .0377(.0260) .0376(.0256) .0375(.0251) 30 .0439(.0415) .0445(.0473) .0437(.0403) .0438(.0406) .0437(.0399) .0435(.0393) 1.5 2.0 38 .0338(.0130) .0348(.0218) .0337(.0120) .0338(.0123) .0337(.0118) .0337(.0114) 34 .0351(.0207) .0359(.0285) .0350(.0197) .0350(.0199) .0350(.0195) .0349(.0190) 30 .0361(.0190) .0369(.0259) .0360(.0179) .0360(.0181) .0360(.0176) .0359(.0171) 2.5 38 .0332(.0152) .0343(.0253) .0331(.0144) .0331(.0146) .0331(.0142) .0330(.0138) 34 .0350(.0234) .0358(.0313) .0349(.0224) .0349(.0226) .0349(.0222) .0348(.0217) 30 .0354(.0191) .0362(.0260) .0353(.0180) .0354(.0183) .0353(.0177) .0352(.0172)

4.2. Illustrative example

For illustrative purposes, we present here a real data analysis using the proposed methods.

The following data set are failure times of the air conditioning system of an airplane (Linhart and Zucchini, 1986; Gupta and Kundu, 2001). To check for goodness fo fit we compute the Anderson-Darling statistic, it is 0.552 and the associated p value is 0.159. Since the p value is quite high, we cannot reject the null hypothesis that the data are coming from the Weibull distribution. The ordered data are as follows: 1, 3, 5, 7, 11, 11, 11, 12, 14, 14, 14, 16, 16, 20, 21, 23, 42, 47, 52, 62, 71, 71, 87, 90, 95, 120, 120, 225, 246, 261.

From the above sample, we created an artificial data by generalized type II hybrid censored

sample. We took n = 30, r = 24, and Case 1 (T 1 = 100 and T 2 = 200), Case 2 (T 1 = 80

and T 2 = 200) and Case 3 (T 1 = 60 and T 2 = 80). For the Bayesian estimation, the prior

parameters are chosen a 1 = a 2 = b 1 = b 2 = 0.0001. The Bayes estimator based on the Linex

loss function with h = −0.5, 0.5 and 1.5 is obtained. Table 4.1 presents estimation for α, λ

and R(t)of generalized type II hybrid censored sample.

Table 4.3 The relative MSEs and biases of estimators for R(t = 1.5)

n T

T

r R(t) ˆ R(t) ˆ

R(t) ˆ

R(t) ˆ

h = −.5 h = .5 h = 1.5

20 1.0 2.0 18 .0103(-.0035) .0103(-.0061) .0103(.0004) .0103(.0007) .0103(.0001) .0103(-.0004) 16 .0114(-.0146) .0113(-.0158) .0113(-.0106) .0113(-.0103) .0113(-.0109) .0113(-.0115) 14 .0119(-.0191) .0119(-.0190) .0117(-.0146) .0117(-.0142) .0117(-.0150) .0117(-.0157) 2.5 18 .0093(-.0088) .0092(-.0118) .0092(-.0050) .0092(-.0047) .0092(-.0052) .0092(-.0057) 16 .0105(-.0174) .0104(-.0185) .0103(-.0113) .0103(-.0130) .0103(-.0136) .0103(-.0142) 14 .0116(-.0195) .0117(-.0195) .0114(-.0151) .0114(-.0147) .0114(-.0154) .0114(-.0162) 1.5 2.0 18 .0100(-.0027) .0100(-.0054) .0099(.0012) .0099(.0015) .0099(.0009) .0099(.0004)

16 .0101(-.0070) .0100(-.0088) .0101(-.0030) .0101(-.0027) .0101(-.0033) .0101(-.0039) 14 .0103(-.0024) .0105(-.0040) .0103(.0016) .0103(.0020) .0103(.0013) .0103(.0006) 2.5 18 .0090(-.0080) .0090(-.0111) .0090(-.0041) .0090(-.0039) .0090(-.0044) .0090(-.0049)

16 .0090(-.0097) .0091(-.0115) .0090(-.0058) .0090(-.0055) .0090(-.0061) .0090(-.0067) 14 .0101(-.0029) .0102(-.0045) .0101(.0012) .0101(.0015) .0101(.0008) .0101(.0002) 30 1.0 2.0 28 .0065(.0000) .0066(-.0019) .0065(.0022) .0065(.0023) .0065(.0020) .0065(.0017) 25 .0075(-.0073) .0075(-.0084) .0075(-.0051) .0075(-.0050) .0075(-.0053) .0075(-.0057) 22 .0082(-.0144) .0082(-.0145) .0081(-.0122) .0081(-.0120) .0081(-.0124) .0081(-.0128) 2.5 28 .0059(-.0040) .0059(-.0064) .0059(-.0018) .0059(-.0017) .0059(-.0020) .0059(-.0022) 25 .0069(-.0102) .0069(-.0113) .0069(-.0081) .0069(-.0079) .0069(-.0082) .0069(-.0086) 22 .0081(-.0146) .0081(-.0147) .0081(-.0124) .0081(-.0121) .0081(-.0126) .0081(-.0130) 1.5 2.0 28 .0065(.0000) .0065(-.0018) .0065(.0022) .0065(.0024) .0065(.0020) .0065(.0017)

25 .0068(-.0045) .0069(-.0057) .0068(-.0023) .0068(-.0021) .0068(-.0025) .0068(-.0028) 22 .0069(-.0028) .0069(-.0036) .0069(-.0006) .0069(-.0004) .0069(-.0008) .0069(-.0012) 2.5 28 .0059(-.0040) .0059(-.0064) .0059(-.0018) .0059(-.0016) .0059(-.0019) .0059(-.0022) 25 .0063(-.0074) .0063(-.0086) .0063(-.0052) .0063(-.0051) .0063(-.0054) .0063(-.0058) 22 .0067(-.0029) .0068(-.0038) .0067(-.0008) .0067(-.0006) .0067(-.0010) .0067(-.0014) 40 1.0 2.0 38 .0049(.0004) .0050(-.0009) .0049(.0019) .0049(.0020) .0049(.0018) .0049(.0015)

34 .0056(-.0050) .0056(-.0059) .0056(-.0035) .0056(-.0034) .0056(-.0037) .0056(-.0039) 30 .0066(-.0127) .0066(-.0128) .0066(-.0113) .0066(-.0111) .0066(-.0114) .0066(-.0117) 2.5 38 .0044(-.0021) .0044(-.0041) .0044(-.0006) .0044(-.0005) .0044(-.0007) .0044(-.0009) 34 .0051(-.0078) .0051(-.0087) .0051(-.0063) .0051(-.0062) .0051(-.0065) .0051(-.0067) 30 .0066(-.0129) .0066(-.0130) .0065(-.0115) .0065(-.0113) .0065(-.0116) .0065(-.0119) 1.5 2.0 38 .0049(.0004) .0050(-.0009) .0049(.0019) .0049(.0020) .0049(.0018) .0049(.0015)

34 .0053(-.0038) .0053(-.0048) .0053(-.0024) .0053(-.0023) .0053(-.0025) .0053(-.0028) 30 .0054(-.0031) .0053(-.0036) .0053(-.0017) .0053(-.0015) .0053(-.0018) .0053(-.0021) 2.5 38 .0044(-.0021) .0044(-.0041) .0044(-.0006) .0044(-.0005) .0044(-.0007) .0044(-.0009) 34 .0049(-.0066) .0049(-.0076) .0049(-.0052) .0049(-.0051) .0049(-.0053) .0049(-.0056) 30 .0053(-.0032) .0053(-.0038) .0052(-.0018) .0052(-.0017) .0052(-.0020) .0052(-.0023)

Table 4.4 Estimation for α, λ and R(t) for example

Case α ˆ α ˆ

α ˆ

α ˆ

h = −0.5 h = 0.5 h = 1.5

1 .8992 .8935 .8931 .8937 .8926 .8915

2 .8953 .8896 .8889 .8895 .8883 .8871

3 .8630 .8577 .8559 .8565 .8552 .8539

Case ˆ λ λ ˆ

λ ˆ

ˆ λ

h = −0.5 h = 0.5 h = 1.5

1 .0288 .0299 .0288 .0288 .0288 .0288

2 .0291 .0302 .0291 .0291 .0291 .0291

3 .0320 .0332 .0320 .0320 .0320 .0320

Case R(t) ˆ R(t) ˆ

R(t) ˆ

R(t) ˆ

h = −0.5 h = 0.5 h = 1.5

1 .0751 .0691 .0863 .0872 .0751 .0833

2 .0737 .0678 .0854 .0865 .0844 .0822

3 .0631 .0578 .0760 .0773 .0747 .0721

5. Conclusion

Hybrid censoring scheme is commonly used in the life testing situations which was intro- duced by Epstein (1954) for the exponential distribution as the life time distribution (Aslam et al., 2017). Childs et al. (2003) introduced a type I hybrid censoring scheme and type II hybrid censoring scheme. Though the type II hybrid censored scheme guarantees a pre-fixed number of failures, it might take a long time to complete the test. In order to provide a guarantee in terms of the time to complete the test, Chandrasekar et al. (2004) introduced generalized type II hybrid censoring scheme.

In this paper, we discussed estimators for parameters and reliability of the Weibull distri- bution based on generalized type II hybrid censored sample. The paper derived estimators for parameters and reliability by using the MLE, approximate MLE and Bayes estimators of the Weibull distribution based on generalized type II hybrid censored sample and compared them in terms of their MSE and bias. Bayes estimates using the non-informative prior are obtained under squared error and Linex loss functions, and it is observed that the Bayes estimates under the Linex loss functions work quite well in this case.

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