Estimating the parameters of exponentiated logistic distribution under progressive censoring scheme ^†

Estimating the parameters of exponentiated logistic distribution under progressive censoring scheme ^†

Yeongjae Seong ¹ · Kyeongjun Lee ²

12 Dvision of Mathematics and Big Data Science, Daegu University

Received 21 October 2019, revised 13 November 2019, accepted 14 November 2019

Abstract

The exponentiated logistic distribution can be considered as a proportional reversed hazard family with the baseline distribution as the logistic distribution. The exponenti- ated logistic distribution has been used to model the data with a unimodal density. The main aim of this paper is to propose the estimators of the parameters (when shape pa- rameter is known) of the exponentiated logistic distribution under progressive censoring (PC) scheme. First, we derive the maximum product spacings estimators for parame- ters of exponentiated logistic distribution. And we derive the approximate maximum product spacings estimators for parameters of exponentiated logistic distribution using Talyor series expansions. We also compare the maximum product spacings estimators and approximate maximum product spacings estimators in the sense of the root mean squared error and bias for various PC schemes. In addition, real data example based on progressive censoring scheme have been also analysed for illustrative purposes.

Keywords: Approximate maximum product spacings estimation, exponentiated logis- tic distribution, maximum product spacings estimation, progressive censoring, Taylor series expansion.

1. Introduction

The exponentiated logistic distribution can be considered as a proportional reversed haz- ard family with the baseline distribution as the logistic distribution. The exponentiated logistic distribution has been used to model the data with a unimodal density. Ali et al.

(2007) derived the properties of exponentiated logistic distribution. The random variable X has a exponentiated logistic distribution if it has a probability density function (pdf) and cumulative distribution function (cdf) of the form:

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

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

f (x; θ, σ) = λ σ

 1 + exp



− x − θ σ

 −λ−1

exp



− x − θ σ

 ,

f (x; θ, σ) =

 1 + exp



− x − θ σ



σ > 0, λ > 0, − ∞ < θ < ∞.

where θ and σ are the location and scale parameters, respectively. Let the random variable Z = (X − θ)/σ, the pdf and cdf of the random variable Z with the standard exponentiated logistic distribution are given by

f (z) = λ [1 + exp(−z)] ^−λ−1 exp(−z),

F (z) = [1 + exp(−z)] ^−λ . (1.1)

From Eqs (1.3) and (1.4), we can be obtained as f (z)

F (z) = λ exp(−z) 1 + exp(−z) , f ⁰ (z)

f (z) = 1 + λ

1 + exp(−λ) exp(−z) − 1.

In most of the life testing and reliability experiments, the experimenter is often, unable to observe life time of all items put on test and the data available to the experimenter is censored data. In fact, in life testing and reliability experiments, censoring occurs in a natural way. Recently, PC scheme has become quite popular in a life-testing problem and reliability analysis. PC scheme can be described as follows. Promptly following the 1st observed failure time, R 1 surviving items are eliminated from the test at random. Similarly, following the 2nd observed failure time, R 2 surviving items are eliminated from the test at random. This process continues until, promptly following the mth observed failure time, all the remaining R _m = n − R ₁ − · · · − R m−1 − m items are eliminated from the test. In test, it is assumed that the removals of still operating units are carried out at observed failure times and that the progressive censoring scheme is known in advance. Consequently, the m ordered observed failure times, which we denote by X _1:m:n , X _2:m:n , · · · , X _m:m:n , are referred to as progressive censoring.

Under classical estimation set up, the maximum product spacings 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 in various situations. The

maximum product spacings method was introduced by Cheng and Amin (1983). Maximum

product spacings method is most suitable method, especially to those cases where one of

the parameter have an unknown shifted origin. It is also observed that maximum product

spacings estimators (MPSE) possesses almost all properties being possessed by MLE. Singh

et al. (2016) consider the maximum product spacings method for the estimation of parame-

ters of generalized inverted exponential distribution under PC scheme. Lee (2019a) consider

the MLE, MPSE and approximate MPSE of parameter and reliability function of half logis-

tic distribution under PC scheme. Lee (2019b) consider the MLE, MPSE and approximate

MPSE of parameter of Ryleigh distribution under doubly hybrid censoring scheme. Shin and

Lee (2019) consider the MLE, MPSE and approximate MPSE of parameter and reliability function of exponential distribution under multiply PC scheme.

The main aim of this paper is to propose the estimators of the parameters of the exponen- tiated logistic distribution under PC scheme. First, we derive the MPSEs for the parameters of exponentiated logistic distribution. And we derive the approximate MPSEs for the param- eters of exponentiated logistic distribution using Talyor series expansions. We also compare the MPSEs and approximate MPSEs in the sense of the root mean squared error (RMSE) and bias for various PC schemes.

The rest of the paper is organized as follows. In Section 2, MPSEs and approximate MPSEs of parameters using Taylor series expansion are discussed. In Section 3, we provides the real example and comparison of MPSEs and approximate MPSEs is conducted using Monte Carlo simulation. Finally concluding remark is presented in Section 4.

2. Estimation

2.1. Maximum product spacings estimation

In this section, we deal with deriving MPSE of the exponentiated logistic distribution parameters unver PC. Let X _1:m:n , X _2:m:n , · · · , X _m:m:n be the ordered observed values of such a PC sample. Here, the m and PC scheme R R R = (R 1 , R 2 , · · · , R m ) are pre-fixed integers.

Using Lee (2019a) and Eq (1.1), the product spacings under PC scheme is given by

L(θ, σ) ∝ F (x 1:m:n )

m

Y

i=2

[F (x i:m:n ) − F (x i−1:m:n )]

m

Y

i=1

[1 − F (x i:m:n )] ^R

ⁱ

.

By putting z i:m:n = (x i:m:n − θ)/σ, the likelihood function can be written as

L(θ, σ) ∝ F (z 1:m:n )

m

Y

i=2

[F (z i:m:n ) − F (z i−1:m:n )]

m

Y

i=1

[1 − F (z i:m:n )] ^R

ⁱ

.

Hence, the log-likelihood function becomes

log L(θ, σ) ∝ log F (z _1:m:n ) +

m

X

i=1

R _i log [1 − F (z _i:m:n )] +

m

X

i=2

log [F (z _i:m:n ) − F (z _i−1:m:n )] .

(2.1)

Differentiating the Eq (2.1) partially with respect to θ and σ, and then equating to zero,

respectively, we have

∂ log(θ, σ)

∂σ = − 1 σ

"

f (z 1:m:n )

F (z 1:m:n ) z 1:m:n −

m

X

i=1

R i

f (z i:m:n )

1 − F (z i:m:n ) z i:m:n

+

m

X

i=2

f (z i:m:n )z i:m:n − f (z i−1:m:n )z i−1:m:n

F (z _i:m:n ) − F (z _i−1:m:n )

#

= 0, (2.2)

∂ log(θ, σ)

∂θ = − 1 σ

"

f (z 1:m:n ) F (z _1:m:n ) −

m

X

i=1

R _i f (z i:m:n ) 1 − F (z _i:m:n )

+

m

X

i=2

f (z _i:m:n ) − f (z _i−1:m:n ) F (z i:m:n ) − F (z i−1:m:n )

#

= 0. (2.3)

The MPSEs of θ and σ are the solution of Eqs (2.2) and (2.3), respectively. However, solutions for θ and σ are 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 and Lee (2018), Kim and Lee (2018), Lee (2019a), Lee (2019b) and Shin and Lee (2019).

2.2. Approximate maximum product spacings estimation

Since the log-likelihood function is very complicated, the Eqs (2.2) and (2.3) do not admit an explicit solution for θ and σ. So, we need some approximate likelihood equations which give explicit solutions.

Let

ψ _i:m:n = F ⁻¹ (p _i:m:n ) = − log 

p ^−1/λ _i:m:n − 1  , where

p i:m:n = E(U i:m:n ) = 1 −

m

Y

j=m−i+1

j + R m−i+1 + · · · + R m

1 + j + R m−i+1 + · · · + R m

, q i:m:n = 1 − p i:m:n .

First, we can approximate the functions by Taylor series expansion as follows f (z _1:m:n )

F (z 1:m:n ) z _1:m:n ' κ 1 + δ ₁ z _1:m:n , (2.4) f (z i:m:n )

1 − F (z i:m:n ) z i:m:n ' κ 1i + δ 1i z i:m:n (2.5) and

f (z _i:m:n )z _i:m:n − f (z _i−1:m:n )z _i−1:m:n

F (z i:m:n ) − F (z i−1:m:n ) ' α _1i + β _1i z _i:m:n + γ _1i z _i−1:m:n , (2.6)

where

κ 1 = λp ^1/λ _1:m:n h

1 − p ^1/λ _1:m:n i ψ ² _1:m:n , δ 1 = λ h

1 − p ^1/λ _1:m:n ψ 1:m:n

i h

1 − p ^1/λ _1:m:n i , κ 1i = − ψ ² _i:m:n

q _i:m:n



f ⁰ (ψ i:m:n ) + f ² (ψ i:m:n ) q _i:m:n

 ,

δ _1i = 1 q i:m:n



f (ψ _i:m:n ) + f ⁰ (ψ _i:m:n )ψ _i:m:n + f ² (ψ _i:m:n ) q i:m:n

ψ _i:m:n

 ,

α 1i = K _i:m:n ² − f ⁰ (ψ i:m:n )ψ _i:m:n ² − f ⁰ (ψ i−1:m:n )ψ _i−1:m:n ² p i:m:n − p i−1:m:n

,

β 1i = (1 − K i:m:n )f (ψ i:m:n ) + f ⁰ (ψ i:m:n )ψ i:m:n

p i:m:n − p i−1:m:n

,

γ 1i = − (1 − K i:m:n )f (ψ i−1:m:n ) + f ⁰ (ψ i−1:m:n )ψ i−1:m:n

p i:m:n − p i−1:m:n

,

K i:m:n = f (ψ i:m:n )ψ i:m:n − f (ψ i−1:m:n )ψ i−1:m:n

p i:m:n − p i−1:m:n

.

By substituting the Eqs (2.4), (2.5) and (2.6) into the Eq (2.2), we can approximate the likelihood equation for σ as follows

∂ log(θ, σ)

∂σ ' − 1 σ

"

(κ 1 + δ 1 z 1:m:n ) −

m

X

i=1

R i (κ 1i + δ 1i z i:m:n )

+

m

X

i=1

(α 1i + β 1i z i:m:n + γ 1i z i−1:m:n )

#

= 0. (2.7)

Next, we can approximate the functions by Taylor series expansion as follows f (z _1:m:n )

F (z 1:m:n ) ' κ 2 + δ 2 z 1:m:n , (2.8)

f (z i:m:n )

1 − F (z i:m:n ) ' κ 2i + δ 2i z i:m:n (2.9) and

f (z _i:m:n ) − f (z _i−1:m:n )

F (z i:m:n ) − F (z i−1:m:n ) ' α 2i + β _2i z _i:m:n + γ _2i z _i−1:m:n , (2.10)

where

κ 2 = λ h

1 + p ^1/λ _1:m:n ψ 1:m:n

i h

1 − p ^1/λ _1:m:n i , δ 2 = −λp ^1/λ _1:m:n h

1 − p ^1/λ _1:m:n i , κ 2i = 1

q _i:m:n



f (ψ i:m:n ) − f ⁰ (ψ i:m:n )ψ i:m:n − f ² (ψ i:m:n ) q _i:m:n ψ i:m:n

 ,

δ _2i = 1 q i:m:n



f ⁰ (ψ _i:m:n ) + f ² (ψ _i:m:n ) q i:m:n

 ,

α 2i = (1 + K i:m:n )f (ψ i:m:n ) − f ⁰ (ψ i:m:n )ψ i:m:n

p _i:m:n − p _i−1:m:n ,

− (1 + K i:m:n )f (ψ i−1:m:n ) − f ⁰ (ψ i−1:m:n )ψ i−1:m:n

p _i:m:n − p _i−1:m:n ,

β 2i = 1

p _i:m:n − p _i−1:m:n



f ⁰ (ψ i:m:n ) − f ² (ψ i:m:n ) p _i:m:n − p _i−1:m:n



+ f (ψ i:m:n )f (ψ i−1:m:n ) (p _i:m:n − p _i−1:m:n ) ² , γ _2i = f (ψ _i:m:n )f (ψ _i−1:m:n )

(p i:m:n − p i−1:m:n ) ² − 1 p i:m:n − p i−1:m:n



f ⁰ (ψ _i−1:m:n ) + f ² (ψ _i−1:m:n ) p i:m:n − p i−1:m:n

 .

By substituting the Eqs (2.8), (2.9) and (2.10) into the Eq (2.3), we can approximate the likelihood equation for θ as follows

∂ log(θ, σ)

∂θ ' − 1 σ

"

(κ 2 + δ 2 z 1:m:n ) −

m

X

i=1

R i (κ 2i + δ 2i z i:m:n )

+

m

X

i=1

(α 2i + β 2i z i:m:n + γ 2i z i−1:m:n )

#

= 0. (2.11)

Upon solving the Eqs (2.7) and (2.11) for θ, we can obtain an approximate MPSE of θ as follows

θ = ˆ D ₁ F ₂ − D ₂ F ₁ D 1 E 2 − D 2 E 1

, (2.12)

where

D 1 =κ 2 −

m

X

i=1

R i κ 2i +

m

X

i=2

α 2i ,

E ₁ =δ ₂ −

m

X

i=1

R _i δ _2i +

m

X

i=2

(β _2i + γ _2i ),

F 1 =δ 2 x 1:m:n −

m

X

i=1

R i δ 2i x i:m:n +

m

X

i=2

(β 2i x i:m:n + γ 2i x i−1:m:n ),

D 2 =κ 1 −

m

X

i=1

R i κ 1i +

m

X

i=2

α 1i ,

E 2 =δ 1 −

m

X

i=1

R i δ 1i +

m

X

i=2

(β 1i + γ 1i ),

F ₂ =δ ₁ x _1:m:n −

m

X

i=1

R _i δ _1i x _i:m:n +

m

X

i=2

(β _1i x _i:m:n + γ _1i x _i−1:m:n ).

Also, solving the Eq (2.7) for σ using Eq (2.12), we can obtain an approximate MPSE of σ as follows

ˆ σ 1 = B 1

A ₁ , (2.13)

where

A 1 =κ 1 −

m

X

i=1

R i κ 1i +

m

X

i=2

α 1i ,

B 1 =δ 1 (x 1:m:n − ˆ θ 1 ) −

m

X

i=1

R i δ 1i (x i:m:n − ˆ θ 1 ) +

m

X

i=2

h

β 1i (x i:m:n − ˆ θ 1 ) + γ 1i (x i−1:m:n − ˆ θ 1 ) i .

Next, by substituting the Eqs (2.5), (2.6) and (2.8) into the Eq (2.2), we can approximate the likelihood equation for σ as follows

∂ log(θ, σ)

∂σ ' − 1 σ

"

(κ ₂ + δ ₂ z _1:m:n )z _1:m:n −

m

X

i=1

R _i (κ _1i + δ _1i z _i:m:n )

+

m

X

i=2

(α _1i + β _1i z _i:m:n + γ _1i z _i−1:m:n )

#

= 0. (2.14)

Upon solving the Eq (2.14) for σ, we can obtain an approximate MPSE of σ as follows

ˆ

σ 2 = −B 2 + pB ₂ ² − 4A 2 C ₂ 2A 2

, (2.15)

where

A 2 =

m

X

i=2

α 1i −

m

X

=1

R i κ 1i ,

B 2 =κ 2 (x 1:m:n − ˆ θ 1 ) −

m

X

i=1

R i δ 1i (x i:m:n − ˆ θ 1 ) +

m

X

i=2

h

β 1i (x i:m:n − ˆ θ 1 ) + γ 1i (x i−1:m:n − ˆ θ 1 ) i , C 2 =δ 2 (x 1:m:n − ˆ θ 1 ) ² .

Also, by substituting the Eqs (2.6), (2.8) and (2.9) into the Eq (2.2), we can approximate the likelihood equation for σ as follows

∂ log(θ, σ)

∂σ ' − 1 σ

"

(κ ₂ + δ ₂ z _1:m:n )z _1:m:n −

m

X

i=1

R _i (κ _2i + δ _2i z _i:m:n )z _i:m:n

+

m

X

i=2

(α 1i + β 1i z i:m:n + γ 1i z i−1:m:n )

#

= 0. (2.16)

Upon solving the Eq (2.16) for σ, we can obtain an approximate MPSE of σ as follows

ˆ

σ 3 = −B 3 + pB ₃ ² − 4A 3 C 3

2A ₃ , (2.17)

where

A ₃ =

m

X

i=2

α _1i ,

B ₃ =κ ₂ (x _1:m:n − ˆ θ ₁ ) −

m

X

i=1

R _i κ _2i (x _i:m:n − ˆ θ ₁ ) +

m

X

i=2

h

β _1i (x _i:m:n − ˆ θ ₁ ) + γ _1i (x _i−1:m:n − ˆ θ ₁ ) i ,

C 3 =δ 2 (x 1:m:n − ˆ θ 1 ) ² −

m

X

=1

R i δ 2i (x i:m:n − ˆ θ 1 ) ² .

3. Simulation results and example

3.1. Simulation results

In this section, a Monte Carlo simulation is conducted to compare the performance of

MPSE and approximate MPSEs. We consider various PC schemes. We have used three

different PC schemes. First, R _m = n − m and R _i = 0 for i = 1, · · · , m − 1. Second, R ₁ = n − m and R _i = 0 for i = 2, · · · , m. Third, R ₁ = R _m = (n − m)/2 and R _i = 0 for i = 2, · · · , m − 1. Various PC schemes have been taken into consideration to calculate bias and RMSE of MPSEs and approximate MPSEs.

From Tables 3.1 and 3.2, MSEs and biases of MPSEs and approximate MPSEs of param- eters are presented for various PC scheme. In general, we observed that the MSEs decrease as the sample size n increases. For a fixed n, the MSEs decrease generally as the PC sam- ples size m decreases. In addition, we observed that the MSEs decrease as the value of λ increases. Also, first PC scheme is smaller than the other PC schemes in terms of MSE.

In estimation of location parameter, MPSE is better than the corresponding approximate MPSEs. However, in estimation of scale parameter, approximate MPSEs are better than the corresponding MPSE. In particular, approximate MPSE ˆ σ ₁ is better than the corresponding approximate MPSE ˆ σ ₂ and ˆ σ ₃ .

Table 3.1 The relative RMSEs and biases of parameter estimators with MPSEs and approximate MPSEs (λ = 1.5)

RMSE (bias)

n m R R R θ σ

θ ˆ θ ˆ

1

σ ˆ σ ˆ

1

σ ˆ

2

σ ˆ

3

25 23 (0*22,2) .3343(-.1041) .3343(-.1011) .1773(.0237) .1751(.0164) .1760(.0186) .1762(.0197) (2,0*22) .3464(-.1092) .3470(-.1060) .1715(-.0067) .1702(-.0080) .1710(-.0056) .1717(-.0035) (1,0*21,1) .3406(-.1079) .3408(-.1047) .1743(.0141) .1727(.0082) .1735(.0105) .1741(.0126) 21 (0*20,4) .3345(-.1021) .3344(-.1017) .1957(.0392) .1927(.0308) .1937(.0331) .1942(.0347) (4,0*20) .3652(-.1209) .3614(-.1207) .1844(-.0080) .1814(-.0067) .1823(-.0038) .1843(.0010) (2,0*19,2) .3452(-.1123) .3455(-.1122) .1893(.0273) .1873(.0203) .1883(.0227) .1898(.0267) 19 (0*18,6) .3427(-.1002) .3404(-.1008) .2104(.0454) .2054(.0375) .2064(.0397) .2069(.0416) (6,0*18) .3867(-.1319) .3931(-.1386) .1951(-.0134) .1880(-.0106) .1889(-.0074) .1920(.0006) (3,0*17,3) .3624(-.1590) .3641(-.1188) .2027(.0325) .1970(.0254) .1980(.0279) .2001(.0336) 35 32 (0*31,3) .2710(-.0835) .2706(-.0826) .1554(.0210) .1540(.0164) .1545(.0179) .1547(.0187) (3,0*31) .2927(-.0873) .2926(-.0873) .1481(-.0050) .1477(-.0044) .1481(-.0026) .1486(-.0006) (1,0*30,2) .2810(-.0857) .2807(-.0849) .1538(.0166) .1527(.0126) .1532(.0142) .1536(.0156) 30 (0*29,5) .2739(-.0829) .2740(-.0833) .1565(.0313) .1552(.0260) .1559(.0276) .1562(.0286) (5,0*29) .2942(-.0933) .2956(-.0965) .1485(-.0034) .1485(-.0013) .1492(.0006) .1508(.0043) (2,0*28,3) .2820(-.0883) .2824(-.0894) .1539(.0251) .1528(.0208) .1536(.0224) .1545(.0249) 28 (0*27,7) .2749(-.0803) .2754(-.0815) .1666(.0359) .1645(.0302) .1651(.0316) .1654(.0327) (7,0*27) .3065(-.0996) .3091(-.1046) .1547(-.0048) .1541(-.0025) .1546(-.0006) .1563(.0046) (3,0*26,4) .2877(-.0892) .2887(-.0915) .1621(.0287) .1605(.0239) .1611(.0254) .1622(.0285) 45 41 (0*40,4) .2408(-.0624) .2410(-.0626) .1352(.0177) .1347(.0142) .1352(.0154) .1353(.0160) (4,0*40) .2511(-.0651) .2522(-.0671) .1327(-.0045) .1326(-.0033) .1332(-.0020) .1341(.0000) (2,0*39,2) .2462(-.0650) .2467(-.0659) .1338(.0119) .1336(.0095) .1342(.0107) .1347(.0122) 38 (0*37,7) .2487(-.0648) .2492(-.0655) .1407(.0231) .1398(.0189) .1402(.0200) .1403(.0207) (7,0*37) .2687(-.0736) .2712(-.0775) .1365(-.0070) .1361(-.0047) .1365(-.0034) .1378(.0000) (3,0*36,4) .2572(-.0698) .2583(-.0714) .1386(.0177) .1381(.0145) .1385(.0156) .1392(.0177) 35 (0*34,10) .2497(-.0656) .2493(-.0672) .1522(.0289) .1511(.0247) .1515(.0257) .1517(.0266) (10,0*34) .2829(-.0836) .2866(-.0906) .1399(-.0083) .1393(-.0045) .1407(-.0031) .1426(.0021) (5,0*33,5) .2644(-.0754) .2659(-.0787) .1467(.0215) .1462(.0185) .1466(.0197) .1478(.0228)

3.2. Example

In order to analyze the example data, we use the MPSEs and approximate MPSEs in

Section 2. The example data given represent the strength measured on GPA for single

Table 3.2 The relative RMSEs and biases of parameter estimators with MPSEs and approximate MPSEs (λ = 2.0)

RMSE (bias)

n m R R R θ σ

θ ˆ θ ˆ

1

σ ˆ σ ˆ

1

σ ˆ

2

σ ˆ

3

25 23 (0*22,2) .3236(-.1041) .3220(-.0970) .1744(.0216) .1718(.0130) .1727(.0155) .1729(.0165) (2,0*22) .3306(-.1011) .3302(-.0952) .1693(-.0081) .1677(-.0113) .1684(-.0084) .1690(-.0068) (1,0*21,1) .3278(-.1050) .3266(-.0980) .1717(.0122) .1697(.0048) .1706(.0075) .1710(.0092) 21 (0*20,4) .3243(-.1066) .3223(-.1012) .1915(.0368) .1881(.0268) .1892(.0294) .1896(.0308) (4,0*20) .3453(-.1112) .3436(-.1081) .1819(-.0095) .1783(-.0107) .1793(-.0074) .1806(-.0036) (2,0*19,2) .3351(-.1126) .3335(-.1076) .1858(.0251) .1833(.0162) .1844(.0190) .1854(.0223) 19 (0*18,6) .3352(-.1050) .3327(-.1017) .2057(.0423) .1996(.0327) .2006(.0353) .2010(.0368) (6,0*18) .3711(-.1198) .3716(-.1217) .1921(-.0155) .1842(-.0157) .1850(-.0121) .1870(-.0056) (3,0*17,3) .3501(-.1179) .3514(-.1140) .1936(.0308) .1919(.0205) .1929(.0233) .1944(.0280) 35 32 (0*31,3) .2623(-.0841) .2609(-.0804) .1523(.0196) .1508(.0138) .1513(.0157) .1515(.0164) (3,0*31) .2781(-.0812) .2793(-.0796) .1475(-.0058) .1470(-.0067) .1474(-.0046) .1479(-.0030) (1,0*30,2) .2703(-.0850) .2700(-.0816) .1508(.0152) .1496(.0101) .1502(.0120) .1505(.0132) 30 (0*29,5) .2646(-.0872) .2637(-.0842) .1539(.0299) .1522(.0234) .1530(.0253) .1533(.0261) (5,0*29) .2783(-.0880) .2794(-.0892) .1478(-.0040) .1474(-.0039) .1482(-.0016) .1484(.0014) (2,0*28,3) .2708(-.0904) .2704(-.0883) .1514(.0239) .1501(.0181) .1510(.0201) .1517(.0221) 28 (0*27,7) .2676(-.0854) .2667(-.0830) .1632(.0338) .1607(.0269) .1613(.0286) .1616(.0295) (7,0*27) .2907(-.0923) .2923(-.0950) .1525(-.0060) .1516(-.0059) .1521(-.0036) .1533(.0007) (3,0*26,4) .2780(-.0914) .2776(-.0901) .1591(.0267) .1571(.0205) .1577(.0223) .1586(.0249) 45 41 (0*40,4) .2332(-.0640) .2328(-.0620) .1335(.0170) .1326(.0125) .1333(.0140) .1333(.0145) (4,0*40) .2402(-.0612) .2412(-.0619) .1313(-.0046) .1310(-.0049) .1316(-.0031) .1323(-.0015) (2,0*39,2) .2373(-.0650) .2374(-.0638) .1323(.0114) .1317(.0078) .1324(.0094) .1328(.0106) 38 (0*37,7) .2402(-.0673) .2399(-.0654) .1385(.0219) .1373(.0167) .1378(.0181) .1379(.0187) (7,0*37) .2549(-.0679) .2570(-.0704) .1350(-.0074) .1343(-.0068) .1348(-.0051) .1357(-.0024) (3,0*36,4) .2470(-.0704) .2474(-.0695) .1367(.0167) .1358(.0123) .1364(.0137) .1368(.0155) 35 (0*34,10) .2448(-.0692) .2447(-.0680) .1480(.0270) .1467(.0218) .1471(.0231) .1473(.0238) (10,0*34) .2710(-.0760) .2744(-.0814) .1372(-.0091) .1372(-.0074) .1376(-.0057) .1390(-.0013) (5,0*33,5) .2584(-.0759) .2592(-.0766) .1431(.0198) .1422(.0156) .1427(.0170) .1436(.0196)

carbon fibers of 10 mm in gauge lengths (Badar and Priest, 1982). Lagos-Alvares et al

(2011) and Vasudeva Rao and Renuka (2018) have examined the goodness-of-fit of the

data to exponentiated logistic distribution and they found that the exponentiated logistic

distribution fits the data. In this example, we consider the case when the data are PC with

the following schemes: n = 63, m = 30 and R 1 = 18, R 30 = 15, R 2 = · · · = R 29 = 0. The

observations and PC scheme are given in Table 3.3. From the Eqs (2.2), (2.3), (2.12), (2.13),

(2.15) and (2.17), the MPSEs ˆ θ = -0.3905 and ˆ σ = 0.9794, the approximate MPSEs ˆ θ 1 =

-0.3588, ˆ σ 1 = 0.9706, ˆ σ 2 = 0.9707 and ˆ σ 3 = 0.9709 are obtained.

Table 3.3 The observations and PC scheme for example

i 1 2 3 4 5 6 7 8 9 10

R

i

18 0 0 0 0 0 0 0 0 0

x

i:m:n

1.901 2.2203 2.228 2.350 2.396 2.445 2.474 2.522 2.532 2.614

i 11 12 13 14 15 16 17 18 19 20

R

i

0 0 0 0 0 0 0 0 0 0

x

i:m:n

2.618 2.659 2.738 2.856 2.928 2.937 2.996 3.030 3.125 3.145

i 21 22 23 24 25 26 27 28 29 30

R

i

0 0 0 0 0 0 0 0 0 15

x

i:m:n

3.223 3.243 3.272 3.294 3.346 3.408 3.493 3.537 3.562 3.852

4. Conclusions

The exponentiated logistic distribution can be considered as a proportional reversed haz- ard family with the baseline distribution as the logistic distribution. The exponentiated logistic distribution has been used to model the data with a unimodal density. As the PC and exponentiated logistic distribution are widely employed in reliability engineering, it is necessary to infer on exponentiated logistic distribution parameters under PC. Considering that the PC is the general form of type II censoring scheme, it is useful and significant to infer on exponentiated logistic distribution parameters under PC.

In this paper, we propose the estimators of the parameters of the exponentiated logistic distribution under PC scheme. First, we derive the maximum product spacings estimators for parameters of exponentiated logistic distribution. And we derive the approximate maxi- mum product spacings estimators for parameters of exponentiated logistic distribution using Talyor series expansions. We also compare the maximum product spacings estimators and approximate maximum product spacings estimators in the sense of the root mean squared error and bias for various PC schemes. Consequently, we observed that the MSEs decrease as the value of λ increases. Also, last censoring scheme is smaller than the other PC schemes in terms of MSE. In estimation of location parameter, MPSE is better than the corresponding approximate MPSEs. However, in estimation of scale parameter, approximate MPSEs are better than the corresponding MPSE. In particular, approximate MPSE ˆ σ 1 is better than the corresponding approximate MPSE ˆ σ 2 and ˆ σ 3 .

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