Approximate maximum product spacing estimation of exponential distribution under multiply progressive censoring^†

Approximate maximum product spacing estimation of exponential distribution under multiply progressive

censoring ^†

Hyejin Shin ¹ · Kyeongjun Lee ²

12 Division of Mathematics and Big Data Science, Daegu University

Received 29 August 2019, revised 11 September 2019, accepted 11 September 2019

Abstract

There are many situation in life testing experiments in which units are lost or removed from experimentation before failure. In this paper, we propose the estimators of the parameter and reliability function of the ED under MPC scheme. First, we derive the MLE and MPSE for the parameter and reliability function of ED. And we derive the approximate MPSE for the parameter and reliability function of ED using Talyor series expansion. We also compare the proposed estimators in the sense of the root mean squared error (RMSE) and bias under MPC scheme. Finally, the validity of the proposed methods are demonstrated by a real data.

Keywords: Approximate MPSE, exponential distribuiton, multiply progressive censor- ing, Taylor series expansion.

1. Introduction

Consider an exponential distribution (ED) with pdf

f (x; σ) = 1 σ exp 

− x σ



(1.1) and

F (x; σ) = 1 − exp 

− x σ



, x ≥ 0, σ > 0. (1.2)

Singh and Kumar (2007) consider a Bayes estimation of the ED under multiply censoring scheme. Wang (2008) proposed a goodness of fit (GOF) test statistics for the ED under progressive censoring (PC) samples. Recently, Lee and Cho (2017) consider exact inference

† This work was supported by Daegu University Undergraduate Research Program, 2019.

1

Undergraduate student, Division of Mathematics and Big Data Science, Daegu University, Gyeongsan 38453, Korea.

2

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

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

for competing risks data under generalized progressive hybrid censored ED. Lee and Lee (2018) consider an estimation of the ED under multiply type II hybrid censoring scheme.

There are many situation in life testing experiments in which units are lost or removed from experimentation before failure. Lee et al. (2012) introduced multiply progressive censoring (MPC) scheme. Under the PC scheme, suppose the experimenter fails to observe the middle observations. Now, let

X a

₁

:m:n ≤ X a

₂

:m:n ≤ · · · ≤ X a

_s−1

:m:n ≤ X a

_s

:m:n , r i−1 = a i − a i−1 − 1, i = 2, 3, · · · , s (1.3) be the available MPC sample. Lee et al. (2012) consider the approximate maximum likeli- hood estimators (MLEs) for ED under MPC scheme. Yun and Lee (2018) proposed a GOF test statistics for the ED under MPC samples.

The maximum product spacings (MPS) method was introduced by Cheng and Amin (1983). Under classical estimation set up, the MPS method is quite effective and several authors advocated the use of this method as an alternative to MLE, and found that this estimation method provides better estimates than MLE. Also, it is observed that maximum product spacings estimators (MPSE) possesses almost all properties being possessed by MLE.

The main aim of this paper is to propose the estimators of the parameter and reliabil- ity function of the ED under MPC scheme. First, we derive the MLE and MPSE for the parameter and reliability function of ED. And we derive the approximate MPSE for the parameter and reliability function of ED using Talyor series expansion. We also compare the proposed estimators in the sense of the root mean squared error (RMSE) and bias under MPC scheme.

The rest of the paper is organized as follows. In Section 2, different estimation proce- dures are discussed and estimators of parameter and reliability function using MLE, MPSE and approximate MPSE is proposed. In Section 3, comparison of proposed estimators is conducted using Monte Carlo simulation. Also, we analyzed two real data sets. Finally, we conclude the paper in Section 4.

2. Estimation

2.1. Maximum likelihood estimation

This section deals with deriving MLE of the unknown parameter of an ED. Suppose that X a

₁

:m:n , X a

₂

:m:n , · · · , X a

_s

:m:n denote the observed values of such a MPC sample. The m, s and IR IR IR = (IR 1 , IR 2 , · · · , IR m ) are pre-fixed integers satisfying P m

i=1 IR i + m = n. Using Lee et al. (2012), Eq (1.1) and Eq (1.2), the likelihood function under MPC scheme is given by

L(σ) ∝

s

Y

i=2

F (x a

_i

:m:n ) − F (x a

_i−1

:m:n )  ^r

i−1

s

Y

i=1

f (x a

_i

:m:n ) [1 − F (x a

_i

:m:n )] ^IR

ⁱ

,

where r i = a i+1 − a i − 1.

By putting z _a

_i

_:m:n = x _a

_i

_:m:n /σ, the likelihood function can be written as L(σ) ∝ σ ^−s

s

Y

i=2

G(z a

_i

:m:n ) − G(z a

_i−1

:m:n )  ^r

i−1

s

Y

i=1

g(z a

_i

:m:n ) [1 − G(z a

_i

:m:n )] ^IR

ⁱ

, where g(z a

i

:m:n ) = exp(−z a

i

:m:n ) and G(z a

i

:m:n ) = 1 − exp(−z a

i

:m:n ).

Hence, the log-likelihood function becomes log L(σ) ∝ − s log σ +

s

X

i=2

r _i−1 log G(z a

i

:m:n ) − G(z _a

_i−1

_:m:n ) 

+

s

X

i=1

[log g(z a

_i

:m:n ) + IR i log {1 − G(z a

_i

:m:n )}] .

Differentiating the log-likelihood function partially with respect to σ and then equating to zero, we have

∂ log L(σ)

∂σ = − 1 σ

"

s +

s

X

i=2

r i−1

g(z _a

_i

_:m:n )z _a

_i

_:m:n − g(z _a

_i−1

_:m:n )z _a

_i−1

_:m:n G(z _a

_i

_:m:n ) − G(z _a

_i−1

_:m:n )

−

s

X

i=1

(1 + IR _i )z _a

_i

_:m:n

#

= 0. (2.1)

The MLE of σ is the solution of Eq (2.1). However, solutions for σ is not available.

Therefore, we propose to use the Newton-Raphson algorithm to solve it. See for example the work of Gwag and Lee (2018), Lee (2019a) and Lee (2019b). Using the MPSE of σ, say ˆ

σ M , the MLE of reliability function is obtained as

< ˆ M (t) = exp(−t/ˆ σ _M ).

2.2. Maximum product spacings estimation

This section deals with deriving MPSE of the unknown parameter of an ED. Using Ng et al. (2012), Lee (2019) and Eq (1.2), the product spacings under MPC scheme is given by

L(σ) ∝ F (x a

₁

:m:n )

s

Y

i=2

F (x a

_i

:m:n ) − F (x a

_i−1

:m:n )  1+r

i−1

s

Y

i=1

[1 − F (x a

_i

:m:n )] ^IR

ⁱ

. By putting z _a

_i

_:m:n = x _a

_i

_:m:n /σ, the likelihood function can be written as

L(σ) ∝ G(z a

₁

:m:n )

s

Y

i=2

[G(z a

_i

:m:n ) − G(z a

_i

−1:m:n )] ^1+r

ⁱ⁻¹

s

Y

i=1

[1 − G(z a

_i

:m:n )] ^IR

ⁱ

. Hence, the log-likelihood function becomes

log L(σ) ∝ log G(z a

1

:m:n ) +

s

X

i=2

(1 + r i−1 ) log G(z a

i

:m:n ) − G(z a

i−1

:m:n ) 

+

s

X

i=1

IR _i log [1 − G(z _a

_i

_:m:n )] ,

where g(z _a

_i

_:m:n ) = exp(−z _a

_i

_:m:n ) and G(z _a

_i

_:m:n ) = 1 − exp(−z _a

_i

_:m:n ).

Differentiating the log-likelihood function partially with respect to σ and then equating to zero, we have

∂ log L(σ)

∂σ = − 1 σ

"

g(z _a

₁

_:m:n )

G(z a

₁

:m:n ) z a

₁

:m:n +

s

X

i=2

(1 + r i−1 ) g(z a

_i

:m:n )z a

_i

:m:n − g(z a

_i−1

:m:n )z a

_i−1

:m:n

G(z a

_i

:m:n ) − G(z a

_i−1

:m:n )

−

s

X

i=1

IR i z a

_i

:m:n

#

= 0. (2.2)

The MPSE of σ is the solution of Eq (2.2). However, solutions for σ is not available.

Therefore, we propose to use the Newton-Raphson algorithm to solve it. See for example the work of Lee and Lee (2018), Lee (2019a) and Lee (2019b). Using the MPSE of σ, say ˆ

σ _A , the MPSE of reliability function is obtained as

< ˆ _A (t) = exp(−t/ˆ σ _A ).

2.3. Approximate maximum product spacings estimation

In this Section, we use the approximate MPSE as an approximate estimator of σ. Let ψ a

_i

:m:n = F ⁻¹ (p a

_i

:m:n ) = − log (1 − p a

_i

:m:n ) ,

where

p i:m:n = E(U i:m:n ) = 1 −

m

Y

j=m−i+1

j + IR _m−i+1 + · · · + IR _m 1 + j + IR m−i+1 + · · · + IR m

.

First, we can approximate the functions by Taylor series expansion as follows g(z a

₁

:m:n )

G(z _a

₁

_:m:n ) z _a

₁

_:m:n ' ν ₁ + κ ₁ z _a

₁

_:m:n (2.3) and

g(z a

_i

:m:n )z a

_i

:m:n − g(z a

_i−1

:m:n )z a

_i−1

:m:n

G(z a

_i

:m:n ) − G(z a

_i−1

:m:n ) ' α i + β _i z _a

_i

_:m:n + γ _i z _a

_i−1

_:m:n , (2.4)

where

ν 1 = q a

₁

:m:n

p a

₁

:m:n

ψ _a ²

₁

_:m:n +



1 + q a

₁

:m:n

p a

₁

:m:n

 , κ 1 = q a

₁

:m:n

p a

₁

:m:n



1 − ψ a

₁

:m:n − q a

₁

:m:n

p a

₁

:m:n

ψ a

₁

:m:n

 ,

α i = q _a

_i

_:m:n ψ _a ²

i

:m:n − q a

i−1

:m:n ψ ² _a

i−1

:m:n

p a

_i

:m:n − p a

_i−1

:m:n

+  q a

_i

:m:n ψ a

_i

:m:n − q a

_i−1

:m:n ψ a

_i−1

:m:n

p a

_i

:m:n − p a

_i−1

:m:n

 2

, β _i = q _a

_i

_:m:n

p a

_i

:m:n − p a

_i−1

:m:n



1 − ψ _a

_i

_:m:n − q a

_i

:m:n ψ a

_i

:m:n − q a

_i−1

:m:n ψ a

_i−1

:m:n

p a

_i

:m:n − p a

_i−1

:m:n

 ,

γ _i = − q a

i−1

:m:n

p a

_i

:m:n − p a

_i−1

:m:n



1 − ψ _a

_i−1

_:m:n − q a

i

:m:n ψ a

i

:m:n − q a

i−1

:m:n ψ a

i−1

:m:n

p a

_i

:m:n − p a

_i−1

:m:n

 . By substituting the Eqs (2.3) and (2.4) into the Eq (2.2), we can approximate the Eq (2.2) for σ as follows

∂ log L(σ)

∂σ = − 1 σ

"

(ν 1 + κ 1 z a

₁

:m:n ) +

s

X

i=2

(1 + r i−1 )(α i + β i z a

_i

:m:n + γ i z a

_i−1

:m:n )

−

s

X

i=1

IR i z a

_i

:m:n

#

. (2.5)

Upon solving the Eq (2.5) for σ, we can obtain an approximate MPSE of σ as follows

ˆ σ A1 =

P s

i=1 IR _i x _a

_i

_:m:n − P s

i=2 (1 + r _i−1 )(β _i x _a

_i

_:m:n + γ _i x _a

_i−1

_:m:n ) − κ ₁ x _a

₁

_:m:n ν 1 + P s

i=2 (1 + r i−1 )α i

. (2.6) Next, we can approximate the Eq (2.2) by the Eq (2.4) and

g(z _a

₁

_:m:n )

G(z a

₁

:m:n ) ' ν 2 + κ ₂ z _a

₁

_:m:n , (2.7) where

ν ₂ = q _a

₁

_:m:n p a

₁

:m:n

+



1 + ψ _a

₁

_:m:n + q _a

₁

_:m:n p a

₁

:m:n

ψ _a

₁

_:m:n



and κ ₂ = − q _a

₁

_:m:n p ² _a

₁

_:m:n .

By substituting the Eq (2.4) and Eq (2.7) into the Eq (2.2), we can approximate the Eq (2.2) for σ as follows

∂ log L(σ)

∂σ = − 1 σ

"

(ν 2 + κ 2 z a

₁

:m:n )z a

₁

:m:n +

s

X

i=2

(1 + r i−1 )(α i + β i z a

_i

:m:n + γ i z a

_i−1

:m:n )

−

s

X

i=1

IR _i z _a

_i

_:m:n

#

. (2.8)

Upon solving the Eq (2.8) for σ, we can obtain an approximate MPSE of σ as follows

ˆ

σ _A2 = −B + √

B ² − 4AC

2A , (2.9)

where

A =

s

X

i=2

(1 + r _i−1 )α _i ,

B =ν ₂ x _a

₁

_:m:n −

s

X

i=1

IR _i x _a

_i

_:m:n +

s

X

i=2

(1 + r _i−1 )(β _i x _a

_i

_:m:n + γ _i x _a

_i−1

_:m:n ), C =κ ₂ x ² _a

1

:m:n .

Using the approximate MPSEs of σ, say ˆ σ A1 and ˆ σ A2 , the approximate MPSEs of reliability function are obtained as

< ˆ A1 (t) = exp(−t/ˆ σ A1 ) and ˆ < A2 (t) = exp(−t/ˆ σ A2 ). (2.10)

3. Simulation studies and illustrative example

3.1. Simulation studies

In this Section, we performed a simulation study to evaluate the performance of different methods presented above section. The performance of proposed results can be evaluated in terms of RMSE and bias for point estimates. We chose the sample size n = 20 and n = 40, and various MPC schemes from the standard exponential distribution are generated by using the algorithm presented in Lee et al. (2012). Using this random data, the RMSEs and biases of proposed estimators are simulated by the Monte Carlo method based on 1,000 runs. The obtained results were presented in Table 3.1 and Table 3.2, respectively.

From Table 3.1, we can clearly see that there is little difference between the MLE (ˆ σ _M ), MPSE (ˆ σ _A ) and approximate MPSEs (ˆ σ _A1 , ˆ σ _A2 ) in term of RMSE. Also, it is clear that when sample size grows, MLE (ˆ σ M ), MPSE (ˆ σ A ) and approximate MPSEs (ˆ σ A1 , ˆ σ A2 ) of parameter are all closer to the true value and the corresponding RMSE decrease. The approximate MLE (ˆ σ A1 ) is generally more efficient than the MLE (ˆ σ M ) in term of RMSE.

From Table 3.2, the performances of the MLE ( ˆ < M ), MPSE ( ˆ < A ) and approximate MPSEs

( ˆ < A1 , ˆ < A2 ) are very similar in term of RMSE. Also, it is clear that when sample size grows,

MLE ( ˆ < M ), MPSE ( ˆ < A ) and approximate MPSEs ( ˆ < A1 , ˆ < A2 ) of parameter are all closer

to the true value and the corresponding RMSE decrease. The approximate MLE ( ˆ < _A1 ) is

generally more efficient than the MLE ( ˆ < M ) in term of RMSE.

Table 3.1 The relative RMSEs and biases of proposed parameter estimators

n m IR s a

i

σ ˆ

M

ˆ σ

A

σ ˆ

A1

σ ˆ

A2

20 18 (0*17,2) 16 1∼4, 7∼18 .2399(.0073) .2321(-.0550) .2320(-.0532) .2415(-.0186)

14 1∼4, 9∼18 .2403(.0078) .2324(-.0550) .2322(-.0529) .2417(-.0182)

12 1∼4, 11∼18 .2415(.0082) .2334(-.0558) .2333(-.0533) .2428(-.0184)

16 (0*15,4) 14 1∼5, 8∼16 .2608(.0070) .2516(-.0430) .2517(-.0415) .2598(-.0106)

12 1∼5, 10∼16 .2605(.0065) .2513(-.0440) .2514(-.0420) .2593(-.0109)

10 1∼5, 12∼16 .2623(.0072) .2527(-.0448) .2530(-.0409) .2613(-.0092)

14 (0*13,6) 12 1∼3, 6∼14 .2742(.0094) .2647(-.0333) .2649(-.0317) .2793(.0064)

10 1∼3, 8∼14 .2741(.0091) .2646(-.0341) .2650(-.0314) .2796(.0069)

8 1∼3, 10∼14 .2753(.0098) .2654(-.0345) .2665(-.0278) .2819(.0112)

18 (2,0*17) 16 1∼4, 7∼18 .2399(.0073) .2429(-.1065) .2381(-.0956) .2457(-.0545)

14 1∼4, 9∼18 .2404(.0080) .2434(-.1107) .2384(-.0952) .2461(-.0540)

12 1∼4, 11∼18 .2421(.0084) .2450(-.1086) .2398(-.0955) .2476(-.0541)

16 (4,0*15) 14 1∼5, 8∼16 .2609(.0070) .2619(-.1175) .2566(-.1040) .2625(-.0636)

12 1∼5, 10∼16 .2607(.0064) .2624(-.1201) .2565(-.1045) .2621(-.0638)

10 1∼5, 12∼16 .2647(.0079) .2660(-.1245) .2592(-.1016) .2651(-.0605)

14 (6,0*13) 12 1∼3, 6∼14 .2743(.0095) .2753(-.1273) .2684(-.1106) .2921(-.0517)

10 1∼3, 8∼14 .2746(.0092) .2762(-.1298) .2687(-.1103) .2928(-.0515)

8 1∼3, 10∼14 .2788(.0115) .2800(-.1342) .2708(-.1048) .2955(-.0458)

40 36 (0*35,4) 32 1∼13, 18∼36 .1631(-.0034) .1618(-.0346) .1618(-.0339) .1646(-.0163)

30 1∼13, 20∼36 .1636(-.0033) .1622(-.0347) .1621(-.0339) .1649(-.0162)

28 1∼13, 22∼36 .1638(-.0036) .1626(-.0352) .1625(-.0343) .1654(-.0166)

32 (0*31,8) 8 1∼10, 15∼32 .1795(-.0023) .1771(-.0272) .1771(-.0264) .1800(-.0093)

26 1∼10, 17∼32 .1795(-.0024) .1770(-.0274) .1771(-.0264) .1799(-.0093)

24 1∼10, 19∼32 .1792(-.0025) .1768(-.0277) .1768(-.0264) .1798(-.0092)

28 (0*27,12) 24 1∼8, 13∼28 .1887(-.0001) .1859(-.0215) .1860(-.0205) .1898(-.0026)

22 1∼8, 15∼28 .1891(.0001) .1863(-.0214) .1864(-.0200) .1903(-.0020)

20 1∼8, 17∼28 .1891(.0003) .1863(-.0213) .1865(-.0190) .1905(-.0009)

36 (4,0*35) 32 1∼13, 18∼36 .1631(-.0033) .1686(-.0677) .1665(-.0623) .1676(-.0414)

30 1∼13, 20∼36 .1637(-.0032) .1691(-.0678) .1669(-.0622) .1680(-.0413)

28 1∼13, 22∼36 .1640(-.0036) .1698(-.0687) .1675(-.0627) .1686(-.0417)

32 (8,0*31) 8 1∼10, 15∼32 .1795(-.0023) .1833(-.0731) .1810(-.0665) .1829(-.0432)

26 1∼10, 17∼32 .1795(-.0023) .1833(-.0731) .1810(-.0665) .1829(-.0432)

24 1∼10, 19∼32 .1792(-.0026) .1833(-.0743) .1807(-.0665) .1827(-.0432)

28 (12,0*27) 24 1∼8, 13∼28 .1887(.0001) .1935(-.0790) .1903(-.0707) .1945(-.0440)

22 1∼8, 15∼28 .1895(.0004) .1943(-.0791) .1909(-.0701) .1952(-.0433)

20 1∼8, 17∼28 .1898(.0009) .1944(-.0795) .1907(-.0687) .1951(-.0419)

Table 3.2 The relative RMSEs and biases of proposed reliability estimators

n m IR s a

i

< ˆ

_M

(t = 2) < ˆ

_A

(t = 2) < ˆ

_A1

(t = 2) < ˆ

_A2

(t = 2) 20 18 (0*17,2) 16 1∼4, 7∼18 .0615(.0021) .0593(-.0139) .0593(-.0134) .0615(-.0046)

14 1∼4, 9∼18 .0616(.0023) .0594(-.0138) .0594(-.0133) .0616(-.0045) 12 1∼4, 11∼18 .0620(.0024) .0597(-.0140) .0596(-.0134) .0619(-.0045) 16 (0*15,4) 14 1∼5, 8∼16 .0665(.0021) .0640(-.0106) .0641(-.0102) .0661(-.0023) 12 1∼5, 10∼16 .0665(.0020) .0640(-.0109) .0640(-.0103) .0660(-.0024) 10 1∼5, 12∼16 .0669(.0022) .0643(-.0110) .0644(-.0100) .0666(-.0019) 14 (0*13,6) 12 1∼3, 6∼14 .0696(.0027) .0671(-.0081) .0672(-.0077) .0706(.0019) 10 1∼3, 8∼14 .0696(.0026) .0671(-.0083) .0672(-.0076) .0707(.0020) 8 1∼3, 10∼14 .0698(.0028) .0673(-.0084) .0676(-.0067) .0712(.0031) 18 (2,0*17) 16 1∼4, 7∼18 .0615(.0021) .0615(-.0269) .0605(-.0242) .0622(-.0138) 14 1∼4, 9∼18 .0617(.0023) .0617(-.0270) .0606(-.0241) .0623(-.0137) 12 1∼4, 11∼18 .0621(.0024) .0620(-.0274) .0609(-.0241) .0627(-.0137) 16 (4,0*15) 14 1∼5, 8∼16 .0666(.0021) .0659(-.0294) .0647(-.0261) .0663(-.0158) 12 1∼5, 10∼16 .0665(.0020) .0660(-.0300) .0647(-.0262) .0662(-.0159) 10 1∼5, 12∼16 .0675(.0024) .0668(-.0311) .0654(-.0254) .0670(-.0150) 14 (6,0*13) 12 1∼3, 6∼14 .0696(.0027) .0686(-.0316) .0673(-.0275) .0726(-.0129) 10 1∼3, 8∼14 .0696(.0026) .0688(-.0322) .0674(-.0274) .0727(-.0128) 8 1∼3, 10∼14 .0706(.0032) .0696(-.0332) .0679(-.0260) .0734(-.0114) 40 36 (0*35,4) 32 1∼13, 18∼36 .0430(-.0008) .0426(-.0091) .0426(-.0089) .0433(-.0042) 30 1∼13, 20∼36 .0431(-.0008) .0427(-.0091) .0427(-.0089) .0434(-.0042) 28 1∼13, 22∼36 .0432(-.0009) .0428(-.0092) .0428(-.0090) .0435(-.0043) 32 (0*31,8) 8 1∼10, 15∼32 .0472(-.0006) .0466(-.0071) .0466(-.0069) .0473(-.0024) 26 1∼10, 17∼32 .0472(-.0006) .0465(-.0071) .0466(-.0069) .0473(-.0024) 24 1∼10, 19∼32 .0472(-.0006) .0465(-.0072) .0465(-.0069) .0473(-.0024) 28 (0*27,12) 24 1∼8, 13∼28 .0492(.0001) .0484(-.0055) .0485(-.0052) .0495(-.0005) 22 1∼8, 15∼28 .0493(.0002) .0485(-.0055) .0486(-.0051) .0496(-.0004) 20 1∼8, 17∼28 .0493(.0002) .0485(-.0054) .0486(-.0048) .0496(-.0001) 36 (4,0*35) 32 1∼13, 18∼36 .0430(-.0008) .0442(-.0177) .0437(-.0163) .0440(-.0108) 30 1∼13, 20∼36 .0431(-.0008) .0443(-.0178) .0438(-.0163) .0441(-.0108) 28 1∼13, 22∼36 .0432(-.0009) .0445(-.0180) .0440(-.0164) .0443(-.0109) 32 (8,0*31) 8 1∼10, 15∼32 .0472(-.0006) .0479(-.0191) .0474(-.0174) .0479(-.0113) 26 1∼10, 17∼32 .0472(-.0006) .0479(-.0191) .0474(-.0174) .0479(-.0113) 24 1∼10, 19∼32 .0472(-.0006) .0479(-.0194) .0473(-.0174) .0478(-.0113) 28 (12,0*27) 24 1∼8, 13∼28 .0492(.0002) .0501(-.0205) .0494(-.0183) .0504(-.0114) 22 1∼8, 15∼28 .0494(.0002) .0503(-.0205) .0495(-.0182) .0506(-.0112) 20 1∼8, 17∼28 .0495(.0004) .0503(-.0206) .0494(-.0178) .0506(-.0108)

3.2. Illustrative example

In order to illustrate the method proposed in the preceding section, here we consider a MPC samples. Nelson (1982) gives data on times to breakdown of an insulating fluid in an accelerated test conducted at various test voltage. In this data, Lee et al. (2012) concluded that the data follow an exponential distribution. Also, Lee et al. (2012) gives data and MPC scheme. The data and MPC schemes are as follows:

Table 3.3 The illustrative example under MPC scheme

a

i

1 2 3 4 5 6 7 10 11 12 13

x

a_i:m:n

0.19 0.78 0.96 1.31 2.78 4.15 4.85 31.75 32.52 33.91 36.71

IR

i

0 0 0 0 1 1 7 0 0 0 1

r

i

0 0 0 0 0 0 2 0 0 0 1

In this example, we obtain that MLE (ˆ σ _M = 14.4867, ˆ < _M (t = 30) = 0.1261), MPSE (ˆ σ A = 12.6480, ˆ < A (t = 30) = 0.0933) and approximate MPSE (ˆ σ A1 = 13.3028, ˆ < A1 (t = 30) = 0.1049, ˆ σ _A2 = 14.0264, ˆ < _A2 (t = 30) = 0.1178) using Eq (2.1), (2.2), (2.6) and (2.9).

4. Conclusions

In this paper, we propose the estimators of the parameter and reliability function of the ED under MPC scheme. First, we derive the MLE and MPSE for the parameter and reliability function of ED under MPC. Moreover, we have derived the approximate MPSE for parameter and reliability function under multiply progressive censored exponential data by using the Taylor series expansion. Finally, a simulation study has been used to test the performances of proposed methods. The results show that when sample size grows, MLE (ˆ σ M ), MPSE (ˆ σ A ) and approximate MPSEs (ˆ σ A1 , ˆ σ A2 ) of parameter are all closer to the true value and the corresponding RMSE decrease. The approximate MLE (ˆ σ A1 ) is generally more efficient than the MLE (ˆ σ M ) in term of RMSE.

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