Recurrence Relations for Moments and Estimation of Parameters of Extended Exponential Distribution Based on Progressive Type-II Right-Censored Order Statistics

Devendra Kumar; Mansoor Rashid Malik; Sanku Dey; Muhammad Qaiser Shahbaz

doi:10.2991/jsta.d.190514.003

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Volume 18, Issue 2, June 2019, Pages 171 - 181

Recurrence Relations for Moments and Estimation of Parameters of Extended Exponential Distribution Based on Progressive Type-II Right-Censored Order Statistics

Authors

Devendra Kumar¹, Mansoor Rashid Malik², Sanku Dey³, Muhammad Qaiser Shahbaz⁴^{, *}

¹Department of Statistics, Central University of Haryana, Mahendergarh, India

²Department of Statistics, Amity University, Noida, India

³Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India

⁴Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia

^*Corresponding author. Email: qshahbaz@gmail.com

Corresponding Author

Muhammad Qaiser Shahbaz

Received 19 July 2018, Accepted 13 April 2019, Available Online 30 May 2019.

DOI: 10.2991/jsta.d.190514.003 How to use a DOI?
Keywords: Progressive Type-II right-censored order statistics; Single moments; Product moments; Recurrence relations; Extended exponential distribution
Abstract: In this article we derive the recurrence relations for the single and product moments based on progressively Type-II right-censored order statistics for the extended exponential (EE) distribution. The estimation of the model parameters under progressively Type-II right-censored order statistics are obtained by maximum likelihood method. Furthermore, Monte Carlo simulation study has been carried out to compare the performances of the proposed method. Finally, a real data set has been analyed for illustrative purposes.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The most commonly censoring schemes found in statistics literature are Type-I and Type-II censoring, but the conventional Type-I and Type-II censoring schemes do not have the suppleness of allowing removal of units at points other than the terminal point of the experiment. For this reason; we consider here a more general censoring scheme called the progressive Type-II censoring scheme. Several authors have studied progressive Type-II censoring and properties of order statistics arising from such a progressively censored life test. Some key references are Cohen [1] and Thomas and Wilson [2]. We refer readers to Balakrishnan and Aggarwala [3] for an excellent review of the progressive censoring.

Consider an experiment in which n units are placed on life test. In progressive censoring schemes, the experimenter decides before hand the quantity m, the number of failures to be observed. When the first failure is observed, R1 of the n−1 surviving units are randomly selected and removed. At the second observed failure, R2 of the n−2−R1 surviving units are randomly selected and removed. The experiment finally terminates at the time of the mth failure when all remaining Rm=n−m−R1−R2−…−Rm−1 surviving units are removed. The censoring numbers Ri;i=1…m−1 are prefixed. We will denote the m ordered failure times thus observed by X1:m:n,…,Xm:m:n. It is evident that n=m+∑k=1mRk. The resulting m ordered values which are obtained from this type of censoring are referred to as progressively Type-II right-censored order statistics.

A random variable X is said to follow the extended exponential (EE) distribution with parameters α and λ. if its probability density function (pdf) and cumulative distribution function (cdf) are given as

(1)fx;α,λ=αλ1+λxα−1e1−1+λxα, x>0,α,λ>0

and the corresponding cumulative density function (cdf) is

(2)Fx;α,λ=1−e1−1+λxα, x>0,α,λ>0.

Here α and λ are the shape and scale parameters, respectively. Hereafter, a random variable X that follows the distribution in (1) is denoted by X∼EEα,λ. An important characteristic of this distribution is that the density function (1) has a decreasing probability function like an exponential distribution but its mode is at zero which has been dealt in detail by Nadarajah and Haghighi [4]. Faster decay of the upper tail. The shape of the hazard rate of this distribution shows increasing, decreasing, and constant like a Weibull or generalized exponential distribution. For α=1, the distribution reduces to standard exponential distribution. Further, this distribution is a particular member of the three-parameter power generalized Weibull distribution, introduced by Nikulin and Haghighi [5]. Lemonte [6] introduced exponentiated Nadarajah and Haghighi (ENH) distribution in the lines of Gupta and Kundu [7] where he provided detailed mathematical properties of the ENH distributions. He also discussed the estimation of the unknown parameters of the distribution by the method of maximum likelihood for complete samples as well as for censored samples. Mirmostafaee et al. [8] studied recurrence relations for the single and product moments of record values and associated inference for EE distribution. Kumar et al. [9] obtained recurrence relations for the single and product moments of order statistics from the EE distribution. They also obtained the best linear unbiased estimators (BLUEs) for the location and scale parameters.

One can observe from (1) and (2) that

(3)fx=α∑u=0α−1α−1uλu+1xu1−Fx.

provided that α≥1 is an integer. This equation will be exploited in order to derive some recurrence relations for the single and product moments of progressive Type-II right-censored order statistics for the EE distribution. This relation will be exploited in this section to derive recurrence relations for the single moments of progressively Type-II right-censored order statistics from the generalized half-logistic distribution.

The joint pdf of the progressively Type-II censored samples X_1:m:n, X_2:m:n, ⋯, X_m:m:n, is given by

(4)fX1:m:n,X2:m:n,⋯,Xm:m:nx1,x2,⋯,xm=Cn,m−1∏i=0mfxi1−FxiRi−∞<x1<x2<⋯<xm<∞,

where

(5)Cn,m−1=nn−R1−1⋯(n−R1−R2−⋯−Rm−1−m+1).

And fx and Fx are given by (1) and (2).

Let the progressively Type-II right-censored sample X1:m:nR1,R2,⋯,Rm, X2:m:nR1,R2,⋯,Rm,⋯, Xm:m:nR1,R2,⋯,Rm with censoring scheme R1,R2,⋯,Rm, m≤n arise from EE distribution with pdf and cdf given in (1) and (2), respectively.

The single moments of the progressive Type-II censored order statistics for the EE distribution can be written as

(6)μi:m:nR1,R2,⋯,Rmk=Exi:m:nR1,R2,⋯,Rmk =Cn,m−1∫∫⋯∫0<x1<x2<⋯<xm<∞xikfx11−Fx1R1fx21−Fx2R2 × fx31−Fx3R3⋯fxm1−FxmRmdx2dx3⋯dxm,

where f. and F. are given respectively in (1), (2), and Cn,m−1 as defined in (5).

Means and variances of a distribution can be computed by using recurrence relations for single and product moments for any continuous distribution. Several papers have been published on recurrence relation for progressively Type-II right-censored order statistics for different distributions. Recent works in this area are those of Balakrishnan et al. [10], Balakrishnan and Saleh [11], Dey et al. [12], and Malik and Kumar [13], and the references cited therein.

The motivation of the paper is two fold: first, we derive recurrence relations for the single and product moments of progressive Type-II right-censored order statistics. These recurrence relations will allow one for the recursive computation of these moments w.r.to. the given censoring scheme, and second is to obtain the maximum likelihood estimators and confidence intervals (CIs) of the unknown parameters of the model. The uniqueness of this study comes from the fact that we provide explicit expressions for single and product moments using progressive Type-II right-censored order statistics along with parameter estimation using Maximum Likelihood Estimates (MLE).

This article unfolds as follows: In Sections 2 and 3, we provide the recurrence relations for single and product moments of progressive Type-II right-censored samples from EE distribution. In Section 4, we discuss the maximum likelihood estimation method of the unknown parameters along with approximate CI. A Monte Carlo simulation study is presented in Section 5 to evaluate the performances of the estimation method discussed in Section 5. Then, in Section 6, we illustrate the methodology developed in this manuscript and the usefulness of the EE based on progressive Type-II right-censored order statistics using a real data example. Finally, Section 7 concludes the paper.

2. RECURRENCE RELATION FOR SINGLE MOMENTS

In this section, we derive several new recurrence relations for the single moments of progressive Type-II censored order statistics for all sample sizes n and all censoring schemes (R1,R2,⋯,Rm), m≤n from EE distribution.

Theorem 2.1.

For 2≤m ≤n and k≥0,

(7)μ1:m:nR1,R2,⋯,Rm(u+k+1)=11+R1α∑u=0α−1α−1uλu+1u+k+1−1μ1:m:nR1,R2,⋯,Rm(k) − n−R1−11+R1μ1:m−1:nR1+1+R2,⋯,Rm(u+k+1).

Proof:

From equations (5) and (6), we have

(8)μ1:m:nR1,R2,⋯,Rm(k)=Cn,m−1∫∫⋯∫0<x1<x2<⋯<xm<∞ × Lx2fx21−Fx2R2fx31−Fx3R3⋯fxm ×1−FxmRmdx2dx3⋯dxm,

where

(9)Lx2=∫0x2x1kfx11−Fx1R1dx1.

Using (3) in (9), we get

(10)Lx2=∫0x2x1kα∑u=0α−1α−1uλu+1x1u1−Fx11−Fx1R1dx1 =α∑u=0α−1α−1uλu+1∫0x2x1u+k1−Fx1R1+1dx1.

Integrating (10) by parts, we get after simplification

(11)=∑u=0α−1(α−1 u)αλu+1u+k+1[[1−F(x2)]R1+1x2u+k+1+(R1+1)∫0x2x1u+k+1×∫2x[1−F(x1)]].

Substituting the value of Lx2 from (11) in (8) and using (6), we simply have

μ1:m:n(R1,R2,⋯,Rm)(k)=∑u=0α−1(α−1 u)αλu+1(u+k+1)[∫∫⋯∫x2u+k+1(1−F(x2))R1+1f(x2)×(1−F(x2))R2⋯f(xm)(1−F(xm))Rm+(1+R1)μ1:m:n(R1,R2,⋯,Rm)(u+k+1)∫]=∑u=0α−1(α−1 u)αλu+1(u+k+1)[(n−R1−1)μ1:m−1:n(R1+1+R2,⋯,Rm)(u+k+1)+(1+R1)μ1:m:n(R1,R2,⋯,Rm)(u+k+1)],

upon rearrangement the above equations, yields the relation in (7).

Theorem 2.2.

For m=1,n=1,2,⋯ and k≥0,

(12)μ1:1:nn−1(u+k+1)=1nα∑u=0α−1α−1uλu+1u+k+1−1μ1:1:nn−1(k).

Proof:

Similar to the Proof of Theorem 2.1.

Theorem 2.3.

For 2≤i ≤m−1, m≤n and k≥0,

(13)μi:m:n(R1,R2,⋯,Rm)(u+k+1)=11+Ri[1α∑u=0α−1(α−1 u)λu+1(u+k+1)−1μi:m:n(R1,R2,⋯,Rm)(k)−(n−R1−R2−⋯−Ri−i)μi:m−1:n(R1,R2,⋯,Ri−1,Ri+Ri+1+1,Ri+2,⋯,Rm)(u+k+1)+(n−R1−R2−⋯−Ri−1−i+1)×μi−1:m−1:n(R1,R2,⋯,Ri−2,Ri−1+Ri+1,Ri+1,⋯,Rm)(u+k+1)1(a−1u)].

Proof:

Similar to the Proof of Theorem 2.1.

Theorem 2.4.

For 2≤m≤n, and k≥0,

(14)μm:m:nR1,R2,⋯,Rm(u+k+1)=11+Rmα∑u=0α−1α−1uλu+1u+k+1−1μm:m:nR1,R2,⋯,Rm(k) + μm−1:m−1:nR1,R2,⋯,Rm−2,Rm−1+Rm+1,Ri+1,⋯,Rm(u+k+1).

Proof:

Similar to the Proof of theorem 2.1.

Corollary 2.1.

By letting α=λ=1 in (7), we can deduce relation for the single moments of progressively Type-II censored order statistics for the standard exponential distribution

(15)μ1:m:nR1,R2,⋯,Rm(k+1)=11+R1k+1μ1:m:nR1,R2,⋯,Rm(k)−n−R1−1μ1:m−1:nR1+1+R2,⋯,Rm(k+1),

Corollary 2.2.

For α=λ=1 in (12), we get

(16)μ1:1:nn−1(k+1)=k+1nμ1:1:nn−1(k),

Corollary 2.3.

For α=λ=1 in (13), we get

(17)μi:m:n(R1,R2,⋯,Rm)(k+1)=11+Ri[μi−1(R1,R2)(k+1)(k+1)μi:m:n(R1,R2,⋯,Rm)(k)−(n−R1−R2−⋯−Ri−i)× μi:m−1:n(R1,R2,⋯,Ri−1,Ri+Ri+1+1,Ri+2,⋯,Rm)(k+1)+(n−R1−R2−⋯−Ri−1−i+1)×μi−1:m−1:n(R1,R2,⋯,Ri−2,Ri−1+Ri+1,Ri+1,⋯,Rm)(k+1)],

Corollary 2.4.

For α=λ=1 in (14), we get

(18)μm:m:nR1,R2,⋯,Rm(k+1)=k+11+Rmμm:m:nR1,R2,⋯,Rm(k)+μm−1:m−1:nR1,R2,⋯,Rm−2,Rm−1+Rm+1,Ri+1,⋯,Rm(k+1),

Deductions: When R1=R2=⋯=Rm=0 so that m=n, in which the case of progressive Type-II censored order statistics become the usual order statistics X1:n,X2:n,⋯,Xn:n, then

From (7): For k≥0, we get
(19)μ1:nu+k+1=1α∑u=0α−1α−1uλu+1u+k+1−1μ1:nk−n−1μ1:n−1:n1,0,0,⋯,0(u+k+1)
From (13): For k≥0, we get
(20)μi:nu+k+1=1α∑u=0α−1α−1uλu+1u+k+1−1μi:nk−n−iμi:nu+k+1+n−i+1μi−1:nu+k+1

3. RECURRENCE RELATION FOR PRODUCT MOMENTS

For EE distribution, we can write the (r,s)th product moment of the progressively Type-II right-censored order statistics as

(21)μr,s:m:nR1,R2,⋯,Rm=Exr:m:nR1,R2,⋯,Rmxs:m:nR1,R2,⋯,Rm =Cn,m−1∫∫...∫0<x1<x2<⋯<xm<∞xrxs fx11−Fx1R1fx2 ×1−Fx2R2⋯fxm1−FxmRmdx1dx2dx3⋯dxm,

where f. and F. are defined in (1) and (2) and Cn,m−1 is defined in (5).

Theorem 3.1.

For 1≤i<j≤m−1 and m≤n,

(22)μi,j:m:n(R1,R2,⋯,Rm)(1,u+1)=1Rj+1[1∑u=0α−1(α−1 u)αλu+1(u+1)−1μi:m:n(R1,R2,⋯,Rm)−(n−R1−1−⋯−Rj−j)μi,j:m−1:n(R1,R2,⋯,Rj−1,Rj+Rj+1+1,⋯Rm)(1,u+1)+(n−R1−1−⋯−Rj−1−j+1)μi,j−1:m−1:n(R1,R2,⋯,Rj−1+Rj+1,⋯,Rm)(1,u+1)1a−1u].

Proof:

Using (3) and (6), we have

(23)μi:m:nR1,R2,⋯,Rm=An,m−1∫∫⋯∫0<x1<⋯<xj−1<xj+1<⋯<xm<∞ × ∫xj−1xj+1∑u=0α−1α−1uαλu+11−FxjRj+1dxjxi fx11−Fx1R1⋯fxj−1 × 1−Fxj−1Rj−1fxj+11−Fxj+1Rj+1⋯fxm × 1−FxmRmdx1dx2⋯dxj−1dxj+1⋯dxm.

By integrating the innermost integral by parts and then substituting into (23), we obtain

∑u=0α−1(α−1u)αλu+1∫xj−1xj+1[1−F(xj)]Rj+1dxj=∑u=0α−1(α−1u)αλu+1∫xj−1xj+1xj+1u+1[1−F(xj+1)]1+Rj−xj−1u+1[1−F(xj−1)]1+Rj+(1+Rj) ×∫xj−1xj+1[1−F(xj)]×∫xj−1xj+1[1−F(xj)]Rjf(xj)xju+1dxj],

which, when substituted into (23) and using (21), we have

μi:m:n(R1,R2,⋯,Rm)=∑u=0α−1(α−1 u)αλu+1(u+1)−1[(R+1)μi,j:m,n(R1,R2)(1,u+1)(n−R1−1−⋯−Rj−j)×μi,j:m−1:n(R1,R2,⋯,Rj−1,Rj+Rj+1+1,⋯Rm)(1,u+1)−(n−R1−1−⋯−Rj−1−j+1)×μi,j−1:m−1:n(R1,R2,⋯,Rj−1+Rj+1,⋯,Rm)(1,u+1)+(Rj+1)μi,j:m:n(R1,R2,⋯,Rm)(1,u+1)].

Upon rearrangement the above equations, yields the relation in (22).

Theorem 3.2.

For 1≤i≤m−1 and m≤n,

(24)μi,m:m:n(R1,R2,⋯,Rm)(1,u+1)=1Rm+1[1∑u=0α−1(α−1 u)αλu+1(u+1)−1μi:m:n(R1,R2,⋯,Rm)+ (n−R1−1−⋯−Rm−1−m+1)μi,m−1:m−1:n(R1,R2,⋯,Rm−1+Rm+1,⋯,Rm)(1,u+1)1(α−1μ)].

Proof:

Similar to the Proof of theorem 3.1.

Corollary 3.1.

By letting α=λ=1 in (22), we can obtain the recurrence relation for product moments of progressively Type-II censored order statistics for the standard exponential distribution.

μi,j:m:nR1,R2,⋯,Rm=1Rj+1μi:m:nR1,R2,⋯,Rm−n−R1−1−⋯−Rj−jμi,j:m−1:nR1,R2,⋯,Rj−1,Rj+Rj+1+1,⋯Rm +n−R1−1−⋯−Rj−1−j+1μi,j−1:m−1:nR1,R2,⋯,Rj−1+Rj+1,⋯,Rm,

Corollary 3.2.

For α=λ=1 in (24), we get

(25)μi,m:m:nR1,R2,⋯,Rm=1Rm+1μi:m:nR1,R2,⋯,Rm+n−R1−1−⋯−Rm−1−m+1 × μi,m−1:m−1:nR1,R2,⋯,Rm−1+Rm+1,⋯,Rm,

m	n	Scheme	λ=4, α=2		Mean
5	2	(0,3)	0.030068	0.067654
5	2	(3,0)	0.030068	0.180410
8	2	(6,0)	0.018792	0.169135
8	2	(0,6)	0.018792	0.040270
10	2	(8,0)	0.015034	0.165376
10	2	(0,8)	0.015034	0.031738
12	2	(10,0)	0.012528	0.162870
12	2	(0,10)	0.012528	0.026195
15	2	(13,0)	0.010022	0.160365
15	2	(0,13)	0.010022	0.020761
18	2	(16,0)	0.008352	0.158694
18	2	(0,16)	0.008352	0.017196
20	2	(18,0)	0.007517	0.157859
20	2	(0,18)	0.007517	0.015429
5	3	(2,0,0)	0.030068	0.105239	0.255581
5	3	(0,0,2)	0.030068	0.067654	0.117768
8	3	(5,0,0)	0.018792	0.093963	0.244306
8	3	(0,0,5)	0.018792	0.040270	0.065327
10	3	(7,0,0)	0.015034	0.090205	0.240547
10	3	(0,0,7)	0.015034	0.031738	0.050531
12	3	(9,0,0)	0.012528	0.087699	0.238041
12	3	(0,0,9)	0.012528	0.026195	0.041230
15	3	(12,0,0)	0.010022	0.085193	0.235536
15	3	(0,0,12)	0.010022	0.020761	0.032326
18	3	(15,0,0)	0.008352	0.083523	0.233865
18	3	(0,0,15)	0.008352	0.017196	0.026592
20	3	(17,0,0)	0.007517	0.082688	0.233030
20	3	(0,0,17)	0.007517	0.015429	0.023782
5	4	(1,0,0,0)	0.030068	0.080182	0.155353	0.305695
5	4	(0,0,0,1)	0.030068	0.067654	0.117768	0.192939
8	4	(4,0,0,0)	0.018792	0.068906	0.144077	0.294420
8	4	(0,0,0,4)	0.018792	0.040270	0.065327	0.095395
10	4	(6,0,0,0)	0.015034	0.065148	0.140319	0.290661
10	4	(0,0,0,6)	0.015034	0.031738	0.050531	0.072009
12	4	(8,0,0,0)	0.012528	0.062642	0.137813	0.288155
12	4	(0,0,0,8)	0.012528	0.026195	0.041230	0.057934
15	4	(11,0,0,0)	0.010022	0.060136	0.135308	0.285650
15	4	(0,0,0,11)	0.010022	0.020761	0.032326	0.044854
18	4	(14,0,0,0)	0.008352	0.058466	0.133637	0.283979
18	4	(0,0,0,14)	0.008352	0.017196	0.026592	0.036615
20	4	(16,0,0,0)	0.007517	0.057631	0.132802	0.283144
20	4	(0,0,0,16)	0.007517	0.015429	0.023782	0.032625
5	5	(0,0,0,0,0)	0.030068	0.067654	0.117768	0.192939	0.343281
8	5	(3,0,0,0,0)	0.018792	0.056378	0.106492	0.181663	0.332005
8	5	(0,0,0,0,3)	0.018792	0.040270	0.065327	0.095395	0.132981
10	5	(5,0,0,0,0)	0.015034	0.052619	0.102733	0.177904	0.328247
10	5	(0,0,0,0,5)	0.015034	0.031738	0.050531	0.072009	0.097066
12	5	(7,0,0,0,0)	0.012528	0.050114	0.100228	0.175399	0.325741
12	5	(0,0,0,0,7)	0.012528	0.026195	0.041230	0.057934	0.076727
15	5	(10,0,0,0,0)	0.010022	0.047608	0.097722	0.172893	0.323235
15	5	(0,0,0,0,10)	0.010022	0.020761	0.032326	0.044854	0.058522
18	5	(13,0,0,0,0)	0.008352	0.045937	0.096051	0.171223	0.321565
18	5	(0,0,0,0,13)	0.008352	0.017196	0.026592	0.036615	0.047353
20	5	(15,0,0,0,0)	0.007517	0.045102	0.095216	0.170387	0.320730
20	5	(0,0,0,0,15)	0.007517	0.015429	0.023782	0.032625	0.042022

Table 1

Means for selected progressive censoring schemes.

m	n	Scheme	λ=4, α=2		Variance
5	2	(0,3)	0.002140	0.005483
5	2	(3,0)	0.002140	0.055641
8	2	(6,0)	0.000835	0.054337
8	2	(0,6)	0.000835	0.001927
10	2	(8,0)	0.000535	0.054036
10	2	(0,8)	0.000535	0.001195
12	2	(10,0)	0.000371	0.053872
12	2	(0,10)	0.000371	0.000813
15	2	(13,0)	0.000237	0.053739
15	2	(0,13)	0.000237	0.000510
18	2	(16,0)	0.000165	0.053666
18	2	(0,16)	0.000165	0.000350
20	2	(18,0)	0.000133	0.053635
20	2	(0,18)	0.000133	0.000281
5	3	(2,0,0)	0.002140	0.015515	0.069016
5	3	(0,0,2)	0.002140	0.005483	0.011428
8	3	(5,0,0)	0.000835	0.014211	0.067712
8	3	(0,0,5)	0.000835	0.001927	0.003413
10	3	(7,0,0)	0.000535	0.013910	0.067411
10	3	(0,0,7)	0.000535	0.001195	0.002031
12	3	(9,0,0)	0.000371	0.013746	0.067248
12	3	(0,0,9)	0.000371	0.000813	0.001348
15	3	(12,0,0)	0.000237	0.013613	0.067114
15	3	(0,0,12)	0.000237	0.000510	0.000827
18	3	(15,0,0)	0.000165	0.013540	0.067041
18	3	(0,0,15)	0.000165	0.000350	0.000559
20	3	(17,0,0)	0.000133	0.013509	0.067010
20	3	(0,0,17)	0.000133	0.000281	0.000447
5	4	(1,0,0,0)	0.002140	0.008084	0.021459	0.074961
5	4	(0,0,0,1)	0.002140	0.005483	0.011428	0.024803
8	4	(4,0,0,0)	0.000835	0.006780	0.020155	0.073657
8	4	(0,0,0,4)	0.000835	0.001927	0.003413	0.005554
10	4	(6,0,0,0)	0.000535	0.006479	0.019854	0.073356
10	4	(0,0,0,6)	0.000535	0.001195	0.002031	0.003123
12	4	(8,0,0,0)	0.000371	0.006316	0.019691	0.073192
12	4	(0,0,0,8)	0.000371	0.000813	0.001348	0.002009
15	4	(11,0,0,0)	0.000237	0.006182	0.019557	0.073059
15	4	(0,0,0,11)	0.000237	0.000510	0.000827	0.001198
18	4	(14,0,0,0)	0.000165	0.006109	0.019485	0.072986
18	4	(0,0,0,14)	0.000165	0.000350	0.000559	0.000797
20	4	(16,0,0,0)	0.000133	0.006078	0.019453	0.072954
20	4	(0,0,0,16)	0.000133	0.000281	0.000447	0.000632
5	5	(0,0,0,0,0)	0.002140	0.005483	0.011428	0.024803	0.078305
8	5	(3,0,0,0,0)	0.000835	0.004179	0.010124	0.023499	0.077001
8	5	(0,0,0,0,3)	0.000835	0.001927	0.003413	0.005554	0.008897
10	5	(5,0,0,0,0)	0.000535	0.003878	0.009823	0.023198	0.076700
10	5	(0,0,0,0,5)	0.000535	0.001195	0.002031	0.003123	0.004609
12	5	(7,0,0,0,0)	0.000371	0.003715	0.009659	0.023035	0.076536
12	5	(0,0,0,0,7)	0.000371	0.000813	0.001348	0.002009	0.002845
15	5	(10,0,0,0,0)	0.000237	0.003581	0.009526	0.022901	0.076402
15	5	(0,0,0,0,10)	0.000237	0.000510	0.000827	0.001198	0.001641
18	5	(13,0,0,0,0)	0.000165	0.003508	0.009453	0.022828	0.076330
18	5	(0,0,0,0,13)	0.000165	0.000350	0.000559	0.000797	0.001069
20	5	(15,0,0,0,0)	0.000133	0.003477	0.009422	0.022797	0.076298
20	5	(0,0,0,0,15)	0.000133	0.000281	0.000447	0.000632	0.000841

Table 2

Variances for selected progressive censoring schemes.

4. PARAMETER ESTIMATION UNDER PROGRESSIVE TYPE-II CENSORED ORDER STATISTICS

Let X1:m:n,X2:m:n, . . . ,Xm:m:n be the ordered m observed failures under Type-II progressively censored sample from EEα,λ with censoring scheme R1,R2,…,Rm. For notational convenience, we will use Xi in place of Xi:m:n. Thus the likelihood function is given by

Lx|α,λ=Cn,m−1∏i=1mαλ1+λxiα−1exp1−1+λxiα ×exp1−1+λxiαRi.

The corresponding log-likelihood function is given by

(26)lnLx|α,λ=D+mlnα+mlnλ+α−1∑i=1mln1+λxi+m −∑i=1m1+λxiα+∑i=1mRi−∑i=1mRi1+λxiα,

where D=ln Cn,m−1.

By differentiating the log-likelihood function (26), we obtain the MLEs of α and λ, α^, λ^ by solving numerically the following nonlinear equations:

(27)∂ ln Lx|α,λ∂α=mα+∑i=1mln1+λxi−∑i=1mln 1+λxi1+λxiα −∑i=1mln1+λxiRi1+λxiα=0

(28)∂ ln Lx|α,λ∂λ=mλ+α−1∑i=1mxi1+λxi−α∑i=1mxi1+λxiα−1 −α∑i=1mxiRi1+λxiα−1=0

4.1. Approximate CIs

Once the ML estimates of α and λ are obtained, we can apply the asymptotic normality of the MLEs to compute the apoximate CIs for the parameters. The observed variance–covariance matrix for the MLEs of the unknown parameters θ=α,λ is

I−1θ=−∂2log L∂α2−∂2log L∂α∂λ−∂2log L∂λ∂α−∂2log L∂λ2|α,λ=α^,λ^−1 =varα^covα^,λ^covλ^,α^varλ^.

The derivatives in Iθ are given as follows:

(29)I11=∂2ln Lx|α,λ∂α2=−mα2−∑i=1mln1+λxi21+λxiα −∑i=1mln1+λxi2Ri1+λxiα,

(30)I22=∂2ln Lx|α,λ∂λ2=−mλ2−α−1∑i=1mxi21+λxi −αα−1∑i=1mxi21+λxiα−2−αα−1∑i=1mxi2Ri1+λxiα−2

(31)I12=∂2ln Lx|α,λ∂α∂λ=∑i=1mxi1+λxi−∑i=1mxi1+λxiα−1 +α∑i=1mln1+λxixi1+λxiα−1−∑i=1mRixi1+λxiα−1 −α∑i=1mln1+λxiRixi1+λxiα−1.

Therefore, the above approach is used to derive the approximate 1001−τ%confidence intervalCIs of the parameters θ=α,λ as in the following forms:

α^±zτ2Varα^,λ^±zτ2Varλ^,

Here, Zτ2 is the upper τ2th percentile of the standard normal distribution.

We have generated progressive Type-II censored order statistics from EE distribution by using following algorithm given by Balakrishnan and Sandhu [14]:

Define γj=∑i=jmRi+1 and generate m independent Beta-distributed random variable B1,B2,…,Bm with Bj~Betaγj,1.
Let V0=1; calculate Vk=BkVk−1, k=1,2,…,m
Let Ur:m:n=1−Vr ; r=1,2,…,m.
Compute Xr:m:n=1λ1−1−Ur:m:n1/α−1 as random progressive Type-II censored ordered sample from EE distribution.

5. SIMULATION

In this section, a Monte Carlo simulation is performed to compute the average estimates of the MLEs to evaluate the performance of the proposed method for different censoring schemes. For simplicity of notation, we denote these censoring schemes, for instance, by 8,0*2,3,0*2,2 which represents the censoring scheme R=8,0,0,3,0,0,2 with n=20 and m=7. The results show that the average estimates of MLEs overestimates the true parameter values in most of the censoring schemes. The censoring schemes with zero removals in middle provide underestimates the true parameter values. We have also obtained 95% CI under various censoring schemes. It shows that all the censoring schemes cover true value of the parameters with 95% confidence. The results are listed in Table 3. In all cases we have used α = 1.5 and λ = 0.5 (for other combinations of α, λ are not reported).

n	m	Scheme	α	λ	CI for α		CI for λ
					Lower	Upper	Lower	Upper
10	2	(8,0)	2.617	0.365	2.392	2.842	0.185	0.545
10	2	(7,1)	2.156	0.417	1.919	2.393	0.235	0.599
10	2	(5,3)	1.445	0.416	1.161	1.729	0.253	0.579
10	2	(3,5)	1.462	0.397	1.056	1.868	0.326	0.468
10	2	(0,8)	1.481	0.419	0.938	2.024	0.364	0.474
10	4	(6,0*3)	1.931	0.496	1.743	2.119	0.284	0.708
10	4	(4,2,0*2)	2.118	0.475	1.991	2.245	0.434	0.516
10	4	(4,0*2,2)	1.681	0.457	1.589	1.773	0.296	0.618
10	4	(2,2,2,0)	1.581	0.483	1.399	1.763	0.279	0.687
10	4	(0*3,6)	1.618	0.436	1.4	1.836	0.336	0.536
10	5	(5,0*4)	1.689	0.517	1.456	1.922	0.329	0.705
10	5	(3,2,0*3)	1.462	0.524	1.235	1.689	0.353	0.695
10	5	(3,0*3,2)	1.499	0.514	1.279	1.719	0.383	0.645
10	5	(2,2,1,0*2)	1.494	0.507	1.257	1.731	0.368	0.646
10	5	(0*4,5)	1.531	0.517	1.3	1.762	0.386	0.648
15	4	(11,0*3)	1.552	0.531	1.307	1.797	0.419	0.643
15	4	(8,3,0*2)	1.576	0.538	1.329	1.823	0.418	0.658
15	4	(6,5,0*2)	1.591	0.562	1.332	1.85	0.421	0.703
15	4	(5,4,2,0)	1.586	0.573	1.335	1.837	0.412	0.734
15	4	(0*3,11)	1.571	0.523	1.314	1.828	0.309	0.737
15	6	(9,0*5)	1.579	0.502	1.295	1.863	0.31	0.694
15	6	(7,2,0*4)	1.572	0.521	1.323	1.821	0.321	0.721
15	6	(6,2,0*3,1)	1.493	0.584	1.268	1.718	0.421	0.747
15	6	(3,2,1,1,1,1)	1.517	0.561	1.288	1.746	0.379	0.743
15	6	(0*5,9)	1.598	0.572	1.371	1.825	0.431	0.713
15	8	(7,0*7)	1.603	0.538	1.37	1.836	0.35	0.726
15	8	(5,2,0*6)	1.573	0.531	1.175	1.971	0.323	0.739
15	8	(4,0,3,0*5)	1.568	0.537	1.339	1.797	0.337	0.737
	8	(3,03,2,02,2)	1.487	0.521	1.166	1.808	0.341	0.701
15	8	(0*6,1,6)	1.548	0.532	1.181	1.915	0.367	0.697
20	5	(15,0*4)	1.581	0.571	1.277	1.885	0.383	0.759
20	5	(12,3,0*3)	1.589	0.573	1.332	1.846	0.363	0.783
20	5	(10,3,2,0*2)	1.572	0.583	1.245	1.899	0.454	0.712
20	5	(5,5,5,0*2)	1.603	0.567	1.235	1.971	0.373	0.761
20	5	(0*3,1,14)	1.492	0.582	1.153	1.831	0.38	0.784
20	7	(10,3,0*5)	1.533	0.581	1.135	1.931	0.426	0.736
20	7	(8,5,0*5)	1.586	0.595	1.225	1.947	0.452	0.738
20	7	(8,02,3,02,2)	1.493	0.504	1.105	1.881	0.349	0.659
20	7	(0*5,5,8)	1.521	0.509	1.202	1.84	0.303	0.715
20	7	(0*6,13)	1.597	0.548	1.236	1.958	0.348	0.748
20	10	(10,0*9)	1.603	0.541	1.205	2.001	0.357	0.725
20	10	(6,4,0*8)	1.584	0.538	1.245	1.923	0.336	0.74
20	10	(4,4,0*7,2)	1.587	0.586	1.242	1.932	0.386	0.786
20	10	(0*6,2,2,3,3)	1.544	0.562	1.17	1.918	0.399	0.725
20	10	(0*9,10)	1.53	0.586	1.163	1.897	0.421	0.751

Table 3

The average estimates of the parameters for the MLE and Confidence Intervals (CI) are presented for different sample sizes and different sampling schemes.

6. ILLUSTRATIVE EXAMPLE

To illustrate the use of the method proposed in this article, we consider a data set from Lee and Wang [15] which corresponds to remission times (in months) of a random sample of 128 bladder cancer patients.

For the purposes of illustrating the method discussed in this article, a progressively Type-II censoring scheme R=0*48,80 has been used. The results are listed in Table 4. We fitted EE distribution to the data set by using the method of maximum likelihood and the results are compared with the other competitive models namely, exponentiated exponential, Kumaraswamy exponential, and exponential distribution. The statistics -ln(L) where -ln(L) denotes the log-likelihood function evaluated at the maximum likelihood estimates and AICs are listed in Table 4 for the data set. Based on the results displayed in Table 4, we can see that the EE distribution has the lowest Akike information criterion (AIC) value among all other competitive models, therefore, the EE distribution is a suitable model for the proposed data set and can chosen as the best model.

Distribution	Scheme	α	β	λ	−lnL	AIC
Extended exponentail	Complete	0.924	-	0.123	−409.574	823.147
		−0.149		−0.034
	Censored	0.802	-	0.182	−106.502	217.004
Exponentiated exponential		(−0.239)		(−0.099)
	Complete	1.235	-	0.124	−410.754	825.508
		(−0.151)		(−0.014)
	Censored	1.035	-	0.128	−107.743	219.486
Kumaraswamy exponential		(−0.235)		(−0.029)
	Complete	1.546	0.262	0.453	−409.965	825.931
		(−0.276)	(−0.026)	(−0.003)
	Censored	1.037	0.157	0.795	−107.659	221.318
		(−0.217)	(−0.027)	(−0.076)
Exponential	Complete			0.109	−412.188	826.377
				−0.01
	Censored			0.125	−107.754	217.508
				(−0.021)

Table 4

The MLEs and the -ln(L), AIC values of different models based on bladder cancer data.

7. CONCLUSION

In this paper, we have established several recurrence relations for single and product moments of progressive Type-II censored order statistics from EE distribution. Since recurrence relations reduce the amount of direct computation and hence reduce the time and labor, therefore the relations under consideration can be useful in computing the moments of higher order from the EE lifetime distribution. ML method of estimation is used for estimation of the parameters of the EE distribution based on progressively Type-II right-censored order statistics. The simulation results provide us some idea to choose the censoring schemes though it is not exhaustive. Finally, the work of this paper can be extended for Bayesian analysis of record values under different loss functions.

ACKNOWLEDGMENTS

The authors would like to thank the referees and the editors for careful reading and for fruitful comments which greatly improved the paper.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 18 - 2
Pages: 171 - 181
Publication Date: 2019/05/30
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.190514.003 How to use a DOI?
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TY  - JOUR
AU  - Devendra Kumar
AU  - Mansoor Rashid Malik
AU  - Sanku Dey
AU  - Muhammad Qaiser Shahbaz
PY  - 2019
DA  - 2019/05/30
TI  - Recurrence Relations for Moments and Estimation of Parameters of Extended Exponential Distribution Based on Progressive Type-II Right-Censored Order Statistics
JO  - Journal of Statistical Theory and Applications
SP  - 171
EP  - 181
VL  - 18
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190514.003
DO  - 10.2991/jsta.d.190514.003
ID  - Kumar2019
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