Estimating the parameter of an exponential distribution under multiply type II hybrid censoring^†

2018, 29 ( 3), 807–814

Estimating the parameter of an exponential distribution under multiply type II hybrid censoring ^†

Wonhee Lee ¹ · Kyeongjun Lee ²

1 Department of Statistics, Daegu University

2 Division of Mathematics and Big Data Science, Daegu University

Received 3 April 2018, revised 10 May 2018, accepted 12 May 2018

Abstract

In this paper, we propose a multiply type II hybrid censoring scheme. And, we derive statistical inference on the scale parameter for the exponential distribution under proposed multiply type II hybrid censoring scheme. The scale parameter is estimated by maximum likelihood estimator (MLE) and approximate MLE using Taylor series expansion. We also obtain the asymptotic confidence interval (CI) for scale parameter under multiply type II hybrid censoring scheme. We compare the MLE and approximate MLE in the terms of the mean square error (MSE) and bias. The simulation procedure is repeated 10,000 times for the sample size n = 20, 40 and various multiply type II hybrid censoring. Consequently, approximate MLE is generally more efficient than the MLE. The average CL of approximate MLE is longer than the corresponding average CL of MLE.

Keywords: Approximate maximum likelihood estimator, confidence interval, exponen- tial distribution, multiply type II hybrid censoring, Taylor series expansion.

1. Introduction

For a random variable X with probability density function (pdf) f (x) and cumulative density function (cdf) F (x), its exponential distribution has the cdf

F (x; θ) = 1 − exp 

− x θ



, x > 0, θ > 0, (1.1) and pdf

f (x; θ) = 1 θ exp 

− x θ



, x > 0, θ > 0. (1.2) In life-time test, there are many situation in which units are lost or removed from test before failure. The loss may occur unconsciously or carelessly. For example, unconsciously

† This research was supported by the Daegu University Research Grant, 2017.

1

Graduate Student, Department of Statistics, Daegu University, Gyeongsan 68453, Korea.

2

Corresponding author: Assistant professor, Division of Mathematics and Big Data Science, Daegu

University, Gyeongsan 68453, Korea. E-mail : indra [email protected]

II hybrid censoring scheme. Type II hybrid censoring scheme may arise in a situation when the tester determines that at least r failures must be observed, and has prepaid for the use of the testing facility for T units of time.

However, some units can be failed between two points of observation with exact times of failure of these units unobserved under type II hybrid censored sample. For example, loss may arise in life-testing experiments when the failure times of some units were not observed due to mechanical or experimental difficulties. Therefore, the new censoring method is needed.

This study has two aims. The first is to propose a multiply type II hybrid censoring scheme. The second aim is to consider the maximum likelihood estimator (MLE) of the θ when the data are multiply type II hybrid censored samples. However, MLE cannot be obtained in a closed form. We use the approximate MLE as an approximate estimator of θ.

The rest of the paper is organized as follows. In Section 2, we propose the multiply type II hybrid censoring scheme. Also, we derive MLE of the θ for exponential distribution under the multiply type II hybrid censored samples. Moreover, the asymptotic confidence interval (CI) for θ is presented. In Section 3, we derive some approximate MLE of the θ for the exponential distribution under the multiply type II hybrid censored samples. The θ is estimated by using Taylor series expansion method. Also, the asymptotic CI for θ is presented. In Section 4, the description of different estimators that are compared by performing the Monte Carlo simulation is presented. In Section 5, we conclude the paper.

2. Multiply type II hybrid censoring

Under the type II hybrid censoring scheme, suppose the tester fails to observe the middle observations. Then, for known and , we have the following two cases under multiply type II hybrid censoring scheme;

Case I: X _a

₁

_:n < X _a

₂

_:n < · · · < X _a

_d

_:n < T, if X _a

_d

_:n < T, (2.1) Case II: X a

₁

:n < X a

₂

:n < · · · < X a

_d

:n < T < X a

_d

+1:n < · · · < X a

_r

:n ,

if d < r and X _a

_d

_:n < T < X _a

_r

_:n , (2.2)

where X a

_i

:n denote a i -th observed failure time, and R i denote the number of censoring be-

tween X a

_i

:n and X a

_i−1

:n . A schematic representation of the multiply type II hybrid censoring

scheme is presented in Figure 2.1.

Figure 2.1 The multiply type II hybrid censoring scheme

Then, the likelihood functions based on (2.1) and (2.2) are as follows.

Case I: L ∝

d

Y

i=1

f (x _a

_i

_:n )

d−1

Y

i=1

F x a

i+1

:n
− F (x a

i

:n )  ^R

i+1

F (x _a

₁

_:n ) ^R

¹

[1 − F (x _a

_d

_:n )] ^n−a

^d

,

Case II: L ∝

r

Y

i=1

f (x _a

_i

_:n )

r−1

Y

i=1

F x a

i+1

:n
− F (x a

i

:n )  ^R

i+1

F (x _a

₁

_:n ) ^R

¹

[1 − F (x _a

_r

_:n )] ^n−a

^r

.

Therefore, case I and case II can be combined, and can be written as

L ∝

m

Y

i=1

f (x a

_i

:n )

m+1

Y

i=1

F (x a

_i

:n ) − F x a

_i−1

:n
 ^R

i

, (2.3)

where m denotes the number of failures, F (x a

₀

:n ) = 0, F x a

_m+1

:n
= 1, R i = a i − a i−1 − 1, a 0 = 0.

On differentiating the log-likelihood functions with respect to θ of (2.3) and equation to zero, we obtain the estimating equation as

∂lnL

∂θ = − 1 θ ²

"

mθ −

m

X

i=1

x a

_i

:n −

m+1

X

i=1

R i

x _a

_i−1

_:n e ⁻

^xai−1θ

− x _a

_i

_:n e ⁻

^xaiθ

e ⁻

^xai−1θ

− e ⁻

^xaiθ

#

= 0. (2.4)

From (2.4), we get the MLE of as

θ = h (θ) , where

h (θ) = 1 m

" _m X

i=1

x a

_i

:n +

m+1

X

i=1

R i

x _a

_i−1

_:n e ⁻

^xai−1θ

− x _a

_i

_:n e ⁻

^xaiθ

e ⁻

^xai−1θ

− e ⁻

^xaiθ

#

. (2.5)

From (2.5), we propose a simple iterative scheme to solve for θ. This has been proposed

in the literature by Lee and Cho (2017). Start with an initial guess of θ ⁽⁰⁾ , then obtain

θ ⁽¹⁾ = h θ ⁽⁰⁾
and proceed in this way iteratively to obtain θ ⁽ⁿ⁾ = h θ ⁽ⁿ⁻¹⁾
. Stop the

∂θ ² = − θ ³

i=1

R _i

H ₀ (θ) + θ ⁴

i=1

R _i

H ₀ (θ) ² , (2.6)

where H j (θ) = x ^j _a

_i

_:n e ⁻

^xai:nθ

− x ^j _a

_i−1

_:n e ⁻

^xai−1:nθ

.

Let I (θ) denotes the Fisher information matrix of the θ. Then, the Fisher information matrix is obtained by taking expectations of minus Equation (2.6). Under some mild regu- larity conditions, ˆ θ is approximately normal with mean θ and variance I ⁻¹ (θ). In practice, we usually estimate I ⁻¹ (θ) by I ⁻¹  ˆ θ 

. A simpler and equally valid procedure is to use the approximation ˆ θ ∼ N 

θ, I ⁻¹  ˆ θ 

. Therefore, 100(1 − α)% CI for θ is

θ ± Z ˆ _α/2 r

I ⁻¹  ˆ θ  ,

where Z _α/2 is the percentile of the standard normal distribution with right-tail probability α/2.

3. Approximate maximum likelihood estimators

Let Z _a

_i:n

= X _a

_i:n

/θ and then the variables Z _a

_i

_:n have a standard exponential distribution with pdf f (z _a

_i

_:n ) = e ^−z

^ai:n

, and cdf F (z _a

_i

_:n ) = 1 − e ^−z

^ai:n

, z _a

_i

_:n > 0, we may rewrite the (2.3) as

L ∝ θ ^−m

m

Y

i=1

f (z _a

_i

_:n )

m+1

Y

i=1

F (z a

i

:n ) − F z _a

_i−1

_:n
 ^R

i

. (3.1)

On differentiating the log-likelihood functions with respect to and equation to zero, we obtain the estimating equation as

∂lnL

∂θ = − 1 θ [m −

m

X

i=1

z a

_i

:n + R 1

f (z a

₁

:n )

F (z a

₁

:n ) z a

₁

:n − (n − a m )z a

_m

:n

+

m−1

X

i=1

R i+1

f (z a

_i+1

:n )z a

_i+1

:n − f (z a

_i

:n )z a

_i

:n

F (z a

_i+1

:n ) − F (z a

_i

:n ) ]. (3.2)

Since (3.2) cannot be solved explicitly, it will be desirable to consider an approximation

to the likelihood equation which provide us with explicit estimators for θ. See for example

the work of Lee et al. (2011), Lee et al. (2013) and Gwag and Lee (2018). Let ξ _a

_i

_:n = F ⁻¹ (p _a

_i

_:n ) = −ln (1 − p _a

_i

_:n ), where p _a

_i

_:n = a _i /n + 1, q _a

_i

_:n = 1 − p _a

_i

_:n , i = 1, · · · , m. Here, we expand the function _{F (z} ^{f (z}

^a1:n

⁾

a1:n

) z a

₁

:n and ^{f (z}

^ai:n

_{F (z} ^)z

^ai:n

^{−f (z}

^ai−1:n

^)z

^ai−1:n

ai:n

)−F (z

_ai−1:n

) in Taylor series around the points ξ _a

_i

_:n . Then, we can approximate the function by;

f (z a

1

:n )

F (z a

₁

:n ) z a

₁

:n ' α 1 + β 1 z a

₁

:n , (3.3) f (z a

_i

:n )z a

_i

:n − f (z z

_i−1

:n )z a

_i−1

:n

F (z a

_i

:n ) − F (z a

_i−1

:n ) ' α 2i + β 2i z a

_i

:n + γ 2i z a

_i−1

:n , (3.4) where

α ₁ = q _a

₁

_:n ξ _a ²

1

:n

p a

₁

:n



1 + q _a

₁

_:n p a

₁

:n



, β ₁ = q _a

₁

_:n p a

₁

:n



1 − ξ _a

₁

_:n − q _a

₁

_:n ξ _a

₁

_:n p a

₁

:n

 ,

α _2i = q a

_i

:n ξ _a ²

_i

_:n − q a

_i−1

:n ξ _a ²

_i−1

_:n p _a

_i

_:n − p a

i−1

:n

+  q a

i

:n ξ _a

_i

_:n − q a

i−1

:n ξ _a

_i−1

_:n p _a

_i

_:n − p a

i−1

:n

 ² ,

β 2i = q a

_i

:n

p _a

_i

_:n − p a

i−1

:n



1 − ξ a

_i

:n − q _a

_i

_:n ξ _a

_i

_:n − q _a

_i−1

_:n ξ _a

_i−1

_:n p _a

_i

_:n − p a

i−1

:n

 ,

γ 2i = − q _a

_i−1

_:n p _a

_i

_:n − p _a

_i−1

_:n



1 − ξ a

_i−1

:n − q _a

_i

_:n ξ _a

_i

_:n − q _a

_i−1

_:n xi _a

_i−1

_:n p _a

_i

_:n − p _a

_i−1

_:n

 .

By substituting (3.3) and (3.4) into (3.2), we may approximate the equation in (2.5) by

∂lnL

∂θ ' − 1 θ [m −

m

X

i=1

z a

_i

:n + R 1 (α 1 + β 1 z a

₁

:n ) − (n − a m )z a

_m

:n

+

m−1

X

i=1

R i+1 (α 2i + β 2i z a

_i+1

:n + γ 2i z a

_i

:n )] = 0. (3.5)

We can derive approximate MLE as follows;

θ ˆ _A = A

B , (3.6)

where

A =

m

X

i=1

x a

_i

:n − R 1 β 1 x a

₁

:n + (n − a m )x a

_m

:n −

m−1

X

i=1

R i+1 (β 2i x a

_i+1

:n + γ 2i x a

_i

:n ), B = m + R 1 α 1 + sum ^m−1 _i=1 R i+1 α 2i .

Next, the 100(1−α)% confidence interval (CI) for θ can be constructed from the asymptotic normality of the approximate MLE with V ar  ˆ θ 

estimated from the inverse of the observed

Fisher information matrix.

Therefore, 100(1 − α)% CI for θ is

θ ˆ A ± Z _α/2 r

I ⁻¹  ˆ θ A

 .

4. Numerical experiment

4.1. Data analysis

Nelson (1982) performed life tests and determined the times to breakdown of an insulating fluid in an accelerated test conducted at various test voltage. The observed times are as follows:

.1900, .7800, .9600, 1.3100, 2.7800, 3.1600, 4.1500, 4.6700, 4.8500, 6.5000, 7.3500, 8.0100, 8.2700, 12.0600, 31.7500, 32.5200, 33.9100, 36.7100, 72.8900.

For this data set, Lee et al. (2014) indicated that the exponential distribution provides a satisfactory fit. In this example, we assume that this data is the exponential distribution based on the multiply type II hybrid censoring scheme (i.e., T = 30, r = 12 and a _i = 1∼3, 6∼19). From (2.6) and (3.6), the MLE is ˆ θ = 19.477892 and approximate MLE is ˆ θ A

= 20.46149. Also, CI for MLE is (18.047529, 20.908255) and CI for approximate MLE is (17.487283, 2.435697).

4.2. Simulation results

To compare the performance of the MLE and approximate MLE, we simulated the MSEs and biases through Monte Carlo simulation. The multiply type II hybrid censored samples from the exponential distribution are generated for sample size n = 20, 40 and various multiply type II hybrid censoring schemes. Using this data, the MSEs and biases of MLE and approximate MLE are simulated by the Monte Carlo method based on 10,000 times for sample size n = 20, 40 and various multiply type II hybrid censored samples with θ = 1. Also, we obtain the average confidence lengths and corresponding coverage percentages.

The simulation results are given in Table 4.1.

From Table 4.1, the following observations can be made. For all the method, the MSE of the

estimates decreases as the sample size n increases. As the time T increases, the MSE of the

estimates decreases. And, as the multiply censoring size decreases, the MSE of the estimates

decreases. The MLE of θ is compared with approximate MLE in terms of MSE and bias. The

estimator ˆ θ A is generally more efficient than the ˆ θ. The average CL of approximate MLE is

longer than the corresponding average CL of MLE. Coverage probabilities of approximate

MLE are mostly below the nominal level.

Table 4.1 Relative MSE, bias, CL and CP for the MLE and approximate MLE

n T R

i

a

r

θ ˆ θ ˆ

A

MSE (bias) CL (CP) MSE (bias) CL (CP)

20 1.5 R

4

= 3 16 .0543 .8581 .0445 .9222

R

i

= 0 for i 6= 4 (.0251) (96.0) (-.0659) (93.8)

13 .0665 .7234 .0463 .9926

(.0214) (98.2) (-.1439) (91.2)

10 .0734 .6690 .0455 1.0233

(.0085) (99.4) (-.1357) (91.2)

R

3

= 2, R

7

= 1 16 .0536 .8581 .0440 .9216

R

i

= 0 for i 6= 3, 7 (.0247) (96.6) (-.0666) (93.4)

13 .0657 .7236 .0461 .9917

(.0208) (.9840) (-.1447) (91.2)

10 .0725 .6705 .0452 1.0223

(.0079) (99.2) (-.1366) (91.0)

2 R

4

= 3 16 .0548 .8555 .0443 .9208

R

i

= 0 for i 6= 4 (.0241) (96.0) (-.0656) (93.4)

13 .0617 .7166 .0454 .9558

(.0083) (98.0) (-.1431) (92.2)

10 .0604 .6501 .0449 .9574

(-.0007) (99.0) (-.1474) (92.2)

R

3

= 2, R

7

= 1 16 .0531 .8555 .0438 .9201

R

i

= 0 for i 6= 3, 7 (.0236) (96.6) (-.0663) (93.0)

13 .0610 .7172 .0452 .9550

(.0078) (98.2) (-.1438) (92.0)

10 .0596 .6514 .0447 .9565

(-.0013) (99.2) (-.1482) (92.0)

40 1.5 R

10

= 3 36 .0253 .6138 .0238 .6344

R

i

= 0 for i 6= 10 (.0106) (95.6) (-.0494) (95.6)

32 .0360 .5583 .0301 .6631

(-.1466) (97.4) (.0078) (93.6)

28 .0379 .4886 .0296 .6684

(-.1613) (98.0) (-.0027) (93.6)

R

7

= 2, R

11

= 1 36 .0253 .6137 .0237 .6344

R

i

= 0 for i 6= 7, 11 (.0107) (95.4) (-.0495) (95.6)

32 .0360 .5585 .0301 .6707

(-.1467) (97.4) (.0079) (94.0)

28 .0379 .4885 .0329 .6989

(-.1614) (98.0) (.0053) (92.6)

2 R

10

= 3 36 .0253 .6138 .0238 .6344

R

i

= 0 for i 6= 10 (.0106) (95.6) (-.0494) (95.6)

32 .0350 .5527 .0293 .6707

(-.1445) (97.6) (.0125) (93.8)

28 .0369 .4831 .0329 .6989

(-.1601) (97.2) (.0052) (92.6)

R

7

= 2, R

11

= 1 36 .0253 .6135 .0237 .6344

R

i

= 0 for i 6= 7, 11 (.0107) (95.4) (-.0494) (95.6)

32 .0349 .5530 .0293 .6632

(-.1446) (97.6) (.0126) (93.6)

28 .0369 .4829 .0296 .6684

(-.1602) (97.4) (-.0026) (93.6)

this article, the proposed method can be extended for other distributions such as Weibull distribution.

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