Journal of Statistical Theory and Applications

Volume 20, Issue 2, June 2021, Pages 267 - 278

Progressively Censored N-H Exponential Distribution

Authors
M. R. Mahmoud, R. M. Mandouh*
Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt
*Corresponding author. Email: rshmndoh@yahoo.com; rshmndoh@gmail.com
Corresponding Author
R. M. Mandouh
Received 3 May 2019, Accepted 1 December 2020, Available Online 29 March 2021.
DOI
https://doi.org/10.2991/jsta.d.210322.002How to use a DOI?
Keywords
Progressive type II censoring samples, Maximum likelihood estimation, Bayesian estimation, Lindely's approximation, Gibbs sampling
Abstract

An extended version of exponential distribution is considered in this paper. This lifetime distribution has increasing, decreasing and constant hazard rates. So it can be considered as another useful two-parameter extension/generalization of the exponential distribution. It can be used as an alternative to the gamma, Weibull and exponentiated exponential distributions. Maximum likelihood and Bayes estimates for two parameter of N-H exponential distribution are obtained based on a progressive type II censored samples. Bayesian estimates are obtained using squared error loss function. These Bayesian estimates are evaluated by applying the Lindely's approximation method.

Open Access

1. INTRODUCTION

The exponential distribution is a popular choices for analyzing lifetime data, but the constant hazard rate makes it not suited in several practical situations. In such situations the Weibull and gamma models are frequently employed. The either increasing or decreasing hazard rate of these distributions marked them unfit for the analysis of many lifetime data sets where the hazard rate is U-shaped. This led some authors to introduce extensions or generalization of the exponential distribution as alternative models. In [1] Gupta and Kundu introduced the exponentiated (or generalized) exponential distribution and in [2] Nadarajah and Kotz introduced a generalization of exponential distribution referred to as the beta exponential distribution which is generated from the logit of a beta random variable. Also, Gupta and Kundu [3] introduced a shape parameter to an exponential model using the idea of Azzalini and called it weighted exponential distribution. Nadarajah and Haghighi [4] introduced another extension of the exponential distribution which was later called the N-H exponential distribution. Here, we will be concerned with the N-H exponential distribution. This distribution has survival function given by

S(t)=e1(1+λt)α(1)

The corresponding cdf, pdf and hazard function are given by

F(x)=1e1(1+λt)α(2)
f(x)=αλ(1+λt)α1e1(1+λt)α(3)
and
h(x)=αλ(1+λt)α1(4)
some properties of this distribution derived by [4] and they discussed the estimation by method of moments and maximum likelihood in the case of complete samples. They also provided formulas for the associated Fisher information matrix. They discussed the maximum likelihood estimation (mle) in the general case of multicensored data which include type I and type II censoring as particular cases.

The conventional type I and type II censoring allow for units to be removed or lost from the test at prefixed time and prefixed no of failures, respectively. These types of censoring are common censoring schemes but they do not have the facility of allowing removal of units at points other than the terminal points of the test. In this article a progressive sample scheme as popularized by [5], in particular progressive type II right censoring scheme will be considered.

Under progressive type II right censoring scheme, n units are placed on a test at time 0. At the first failure, R1 surviving units are randomly removed from the remaining n1 surviving units. At the second failure, R2 surviving units are randomly removed from the remaining n2R1 units etc. This process continues until the mth failure recorded and at this time all remaining units (Rm=nmR1R2Rm1) are removed. In this censoring scheme, Ri and m are previously fixed. Note that if R1=R2==Rm1=0 i.e., Rm=nm, this scheme reduces to conventional type II one stage right censoring scheme. Also, one can note that if R1=R2==Rm1=Rm=0 i.e., m=n, this scheme reduces to the case of complete sample(i.e., the case of noncensoring) [5]

In this paper, we discuss the mle of the two unknown parameters of N-H exponential with approximate confidence interval under progressive type II right censoring.

2. MAXIMUM LIKELIHOOD ESTIMATION

Suppose we have X1:m:n,X2:m:n,,Xm:m:n a progressive type II censored sample taken from N-H exponential distribution given by (2) and let R1,R2,,Rm be the progressive censoring scheme. In this case the likelihood function take the form

L(α,λ)=Ci=1mf(xi:m:n;α,λ)[1F(xi:m:n;α,λ)]Ri(5)
where C=n(n1R1)(n2R2)(nmR1Rm1); and f(x) and F(x) take the form given by formulas (3) and (2), respectively. Substituting (2) and (3) into (5), the likelihood function becomes
L(α,λ|x)αmλmi=1m(1+λxi)α1(e1(1+λxi)α)Ri+1(6)
and the natural logarithm of likelihood function is
logLmlogα+mlogλ+(α1)i=1mlog(1+λxi)+i=1m(Ri+1)(1(1+λxi)α)(7)

Differentiating Equation (7) with respect to α and λ and equating each to zero, we get

logLαmα+i=1mlog(1+λxi)i=1m(Ri+1)(1(1+λxi)α)log(1+λxi)=0(8)
and
logLλmλ+(α1)i=1mxi(1+λxi)αi=1m(Ri+1)(1(1+λxi)α1xi=0(9)

Solving Equations (8) and (9), we obtain α^ and λ^. These equations will be normally solved numerically and some software packages utilize for this purpose.

3. BAYESIAN ESTIMATION

Now we will discuss parameter estimation via Bayesian viewpoint. Assuming that the two parameters α and λ are unknown, we propose to use independent gamma priors for both α and λ with pdfs

g1(α)αa1ebα,         α>0,     a,b>0g2(λ)λc1edλ,        λ>0,      c,d>0

respectively. Two reasons for choosing gamma priors; one is its mathematical tractability and the other is due to the fact that if we known from experience some information about the parameter of interest, like its mean and its variance, it would be fairly easy to calculate the hyper-parameters of the gamma priors. The joint prior distribution for α and λ is

g(α,λ)αa1λc1e(bα+dλ)(10)
and

The joint posterior distribution of α and λ is obtained as

π(α,λx)αm+a1λm+c1em(bα+dλ)i=1m(1+λxi)α1ei=1m(Ri1)(1(1+λxi)α)
i.e.,
π(α,λx)=Kαm+a1λm+c1em(bα+dλ)i=1m(1+λxi)α1ei=1m(Ri1)(1(1+λxi)α)
where
K=[00αm+a1λm+c1em(bα+dλ)i=1m(1+λxi)α1et=1m(Ri1)(1(1+λxi)α)dαdλ]1

Therefore, the Bayes estimate of any function of α and λ say θ(α,λ) squared error loss function would be

θ˜Bayes=K00[θ(α,λ)αm+a1λm+c1em(bα+dλ)i=1m(1+λxi)α1ei=1m(Ri1)(1(1+λxi)α)dαdλ.

It is not easy to compute this integral, so one can use one of the approximation methods such as Lindely's approximation. Using this method we have the approximate Bayes estimate of α and λ under squared error loss function as follows, respectively

α~Lindely=α^+1/2[2ρασαα+2ρλσαλ+σαα2Lααα+2σαασλαLλαλ+σλασααLααλ+2σλα2Lλαλ+σαασλλLλλα+σαλσλλLλλλ](11)
λ~Lindely=λ^+1/2[2ρλσλλ+2ρασλα+σλλ2Lλλλ+2σαλ2Lαλα+σαλσααLααα+σλλσααLααλ+2σλλσαλLααλ+2σλλσαλLαλλ](12)
where α^ and λ^ are the mle of α and λ while ρα and ρλ are the derivatives of the logarithm of the joint prior distribution and σ, L obtained from differentiating log L. (see details in Appendix)

4. A NUMERICAL STUDY

Here, we present some numerical results to show the behavior of the proposed method for various sample sizes (n = 20, 25, 30, 50, 70, 100); different effective sample sizes(m = 5, 10, 15, 20, 25, 30, 50); two different priors (noninformative prior and informative prior); two different censoring schemes (details of the schemes are given in Tables 1 and 2).

n m R1,…,Rm Scheme Number
20 5 (3 * 5)a [1]
(5, 0, 5, 0, 5) [2]
(0, 5, 2, 5, 3) [3]
(0 * 4, 15) [4]
20 10 (1 * 2, 0, 2 * 2, 0, 2, 1 * 2) [5]
(10, 0 * 9) [6]
(0 * 9, 10) [7]
(2 * 2, 0 * 2, 0 * 3, 2 * 2) [8]
25 10 (2 * 2, 0 * 2, 1 * 4, 5, 2) [9]
(0 * 5, 5 * 3, 0 * 2) [10]
(1 * 4, 2 * 3, 0 * 2, 5) [11]
(0 * 4, 15, 0 * 5) [12]
25 15 (1, 2, 0 * 2, 1, 2, 0 * 3, 1, 0 * 3, 2) [13]
(0 * 10, 2 * 5) [14]
(1 * 10, 0 * 5) [15]
(1 * 5, 0 * 5, 1 * 5) [16]
30 15 (0 * 14, 15) [17]
(1 * 10, 0 * 4, 5) [18]
(0 * 5, 1 * 5, 2 * 5) [19]
(2 * 5, 0 * 9, 5) [20]
30 20 (3, 0 * 4, 3, 0 * 4, 1, 0 * 4) [21]
(1 * 10, 0 * 10) [22]
(1 * 5, 0 * 5, 1 * 5, 0 * 5) [23]
(2 * 5, 0 * 15) [24]
50 20 (1 * 5, 0 * 14, 25) [25]
(10, 0 * 4, 1 * 10, 2 * 5) [26]
(0 * 5, 2 * 10, 0 * 3, 5 * 2) [27]
(1 * 10, 0 * 5, 5 * 4) [28]
50 25 (0 * 5, 1 * 5, 0 * 5, 2 * 3, 3 * 2, 4 * 2, 0 * 3) [29]
(1 * 25) [30]
(0 * 24, 25) [31]
(1 * 5, 0 * 16, 4 * 5) [32]
100 30 (2 * 5, 0 * 5, 5 * 5, 0 * 5, 6 * 5, 1 * 5) [33]
(0 * 10, 2 * 10, 0 * 5, 10 * 5) [34]
(5 * 5, 0 * 20, 9 * 5) [35]
(2 * 5, 0 * 5, 10 * 5, 0 * 5, 2 * 5) [36]
100 50 (0 * 15, 1 * 5, 0 * 5, 5 * 5, 0 * 15, 4 * 5) [37]
(2 * 5, 1 * 10, 0 * 10, 5 * 5, 0 * 15, 1 * 5) [38]
(0 * 35, 5 * 5, 0 * 5, 5 * 5) [39]
(2 * 10, 0 * 20, 1 * 10, 2 * 20) [40]

(a) This 3 * 5 denotes that 3, 3, 3, 3, 3. This table is adapted from Dey and Dey [6].

Table 1

Different censoring schemes for the simulation study.

n m R1,…,Rm Scheme Number
20 5 (1, 4, 1 * 3a) [1]
(4, 1, 2 * 2, 3) [2]
(1, 0, 2, 5 * 2) [3]
(2, 1, 3 * 3) [4]
(2, 1, 4, 3 * 2) [5]
(2, 4, 9, 5, 3) [6]
50 5 (1, 4, 1 * 3) [7]
(4, 1, 2 * 2, 3) [8]
(1, 0, 2, 5 * 2) [9]
(2, 1, 3 * 3) [10]
(2, 1, 4, 3 * 2) [11]
(2, 4, 9, 5, 3) [12]
50 10 (1, 5, 1, 3, 2, 1, 3, 0, 3 * 2) [13]
(1, 8, 5, 1, 4 * 4, 2, 4) [14]
(3, 2, 5, 2, 3, 4, 2, 1, 3, 4) [15]
(1, 3, 6, 0 * 2, 6, 2, 3, 2, 0) [16]
(4, 3, 1, 2, 1, 2, 1 * 2, 2, 6) [17]
(4, 1, 3, 5, 4, 7, 2, 5, 3, 1) [18]
70 10 (1, 8, 5, 1, 4 * 4, 2, 4) [19]
(1, 8, 5, 1, 4 * 4, 2, 4) [20]
(3, 2, 5, 2, 3, 4, 2, 1, 3, 4) [21]
(1, 3, 6, 0 * 2, 6, 2, 3, 2, 0) [22]
(4, 3, 1, 2, 1, 2, 1 * 2, 2, 6) [23]
(4, 1, 3, 5, 4, 7, 2, 5, 3, 1) [24]
70 15 (1, 3, 0 * 2, 3, 2, 3 * 2, 2, 1, 2 * 2, 1, 3, 2) [25]
(2, 3 * 2, 4 * 3, 1, 3, 0, 2 * 2, 3, 1, 5, 1) [26]
(2, 3 * 2, 4 * 2, 1, 0, 2, 3 * 2, 2, 4 * 3, 5) [27]
(5, 3 * 2, 1, 2, 5, 4 * 2, 1, 3, 2, 3, 4, 2, 1) [28]
(4, 1, 4, 3, 6, 3, 2, 1, 3, 4, 2 * 2, 7, 1, 2) [29]
(1, 5, 2, 5, 1, 3, 6, 2, 4, 3, 5, 4, 1, 6, 3) [30]
100 15 (1, 3, 0 * 2, 3, 2, 3 * 2, 2, 1, 2 * 2, 1, 3, 2) [31]
(2, 3 * 2, 4 * 3, 1, 3, 0, 2 * 2, 3, 1, 5, 1) [32]
(2, 3 * 2, 4 * 2, 1, 0, 2, 3 * 2, 2, 4 * 3, 5) [33]
(5, 3 * 2, 1, 2, 5, 4 * 2, 1, 3, 2, 3, 4, 2, 1) [34]
(4, 1, 4, 3, 6, 3, 2, 1, 3, 4, 2 * 2, 7, 1, 2) [35]
(1, 5, 2, 5, 1, 3, 6, 2, 4, 3, 5, 4, 1, 6, 3) [36]
100 20 (4, 5, 2 * 2, 1, 4, 2, 1, 4, 5 * 2, 4, 3, 2, 5, 3, 1, 6, 1, 3) [37]
(5, 3, 1, 3, 4 * 2, 3 * 2, 0, 3 * 2, 1 * 2, 5, 4, 2, 3, 5, 6, 2) [38]
(1, 4, 2, 3, 5 * 2, 6, 3 * 2, 1, 3, 5, 3, 4, 3, 2 * 2, 4, 6, 3) [39]
(4, 2, 4, 2, 1, 9, 3, 5, 4, 2, 5, 1, 3, 5, 6, 3, 4 * 2, 1 * 2) [40]
(3, 4, 2, 1, 2, 5, 4, 6, 2, 4, 2 * 2, 0, 2 * 2, 6, 2, 1, 3 * 2) [41]
(1 * 2, 4, 3 * 2, 7, 2, 1, 2 * 3, 0, 3 * 2, 2 * 2, 4, 2, 3) [42]

(a) This 1 * 3 denotes that 1, 1, 1.

Table 2

Different censoring schemes from poisson with mean 3 for the simulation study.

For α=0.5, λ=1 and given n, m and a sampling scheme, we generate a samples for a given censoring scheme. we compute the mles for the two unknown parameters and also we compute approximate Bayes estimates by using two different priors. The first prior is noninformative prior (Prior0) i.e., we take the hyper-parameters value as a=b=c=d=0. The second prior is informative prior (Prior1) and we take the hyper-parameters as a=2,b=c=d=1 which chosen in such a way that the prior mean became the expected value of the corresponding parameter (see [7]). 1000 samples are generated and we compute the average bias and the corresponding mean squared error (MSE). Results are listed in Tables 36 where α^,λ^ represents the mle estimates and α˜,λ˜ represents the approximate Bayes estimates based on Lindely's approximation.

Prior 0
Prior 1
n m Scheme α^ α˜ α˜
20 5 [1] 0.4971915(0.528149) 0.4971415(0.528073) 0.4971915(0.528037)
[2] 0.4829925(0.514042) 0.4829210(0.513973) 0.4829030(0.513955)
[3] 0.5218350(0.539559) 0.5217500(0.539468) 0.5217300(0.539450)
[4] 0.5191600(0.548442) 0.5190850(0.548365) 0.5190650(0.548347)
20 10 [5] 0.3415230(0.319913) 0.3415040(0.319884) 0.3414970(0.319880)
[6] 0.2649890(0.242477) 0.2649600(0.242462) 0.2649545(0.242459)
[7] 0.2499280(0.204724) 0.2498940(0.204707) 0.2498885(9.204704)
[8] 0.3022270(0.282316) 0.3021885(0.282293) 0.3021795(0.282287)
25 10 [9] 0.3588315(0.324342) 0.3587875(0.324310) 0.3587775(0.324303)
[10] 0.3572780(0.335403) 0.3572250(0.335365) 0.3572175(0.335360)
[11] 0.3200430(0.293386) 0.3199970(0.293357) 0.3199845(0.293349)
[12] 0.3178110(0.324944) 0.3178705(0.324918) 0.3178645(0.324915)
25 15 [13] 0.1958475(0.150822) 0.1958060(0.150805) 0.1957970(0.150802)
[14] 0.2375360(0.188130) 0.2374980(0.188111) 0.2374890(0.188107)
[15] 0.2051600(0.158191) 0.2051200(0.158175) 0.2051120(0.158171)
[16] 0.2297045(0.180323) 0.2296675(0.180306) 0.2296595(0.180303)
30 15 [17] 0.6506570(0.556114) 0.6505650(0.555973) 0.6505600(0.555970)
[18] 0.6489950(0.565434) 0.648880(0.565299) 0.648880(0.565297)
[19] 0.2339490(0.191414) 0.2339100(0.191396) 0.2338995(0.191391)
[20] 0.2132375(0.167186) 0.2131960(0.167168) 0.2131865(0.167164)
30 20 [21] 0.1545180(0.103939) 0.1544675(0.103924) 0.1544590(0.103921)
[22] 0.1380175(0.092865) 0.1379555(0.0902693) 0.1379460(0.0902667)
[23] 0.1512940(0.101957) 0.1512420(0.101942) 0.1512330(0.101939)
[24] 0.1240075(0.0777336) 0.1239540(0.0777203) 0.1239440(0.0777178)
50 20 [25] 0.2211845(0.164537) 0.2211475(0.164521) 0.2211360(0.164516)
[26] 0.2100615(0.156362) 0.2100215(0.156345) 0.210012(0.156341)
[27] 0.1987010(0.155447) 0.1986630(0.155431) 0.1986510(0.155427)
[28] 0.2080380(0.157063) 0.208000(0.157047) 0.2079890(0.157042)
50 25 [29] 0.1816730(0.125538) 0.1816190(0.125518) 0.1816090(0.125514)
[30] 0.1649060(0.110223) 0.1648620(0.110209) 0.1648530(0.110206)
[31] 0.1656070(0.121050) 0.1655740(0.121040) 0.1655630(0.121036)
[32] 0.1638900(0.112809) 0.1638530(0.112797) 0.1638420(0.112793)
100 30 [33] 0.1778810(0.119835) 0.1778280(0.119816) 0.1778150(0.119812)
[34] 0.1483560(0.114650) 0.1483260(0.114641) 0.1483120(0.114637)
[35] 0.1623790(0.116393) 0.1623460(0.116382) 0.1623330(0.116378)
[36] 0.1595940(0.112488) 0.1595460(0.112472) 0.1595330(0.112468)
100 50 [37] 0.0976598(0.0564262) 0.0976067(0.0564159) 0.097594(0.0564135)
[38] 0.0736086(0.0384041) 0.0734818(0.0383854) 0.0734641(0.0383828)
[39] 0.0942280(0.0610751) 0.0941882(0.061076) 0.0941766(0.0610654)
[40] 0.0857108(0.04392267) 0.0856541(0.048214) 0.0856418(0.0439158)

mle, maximum likelihood estimation.

Table 3

The absolute value of bias and the mean squared error for the mle and Bayes estimates of α are listed for various sample sizes and sampling schemes.

Prior 0
Prior 1
n m Scheme λ^ u(α,λ) λ˜
20 5 [1] 0.1159850(2.440330) 0.1159210(2.440330) 0.1159860(2.440330)
[2] 0.1664680(2.877200) 0.1663370(2.877670) 0.1664760(2.877720)
[3] 0.0521421(2.208430) 0.0520099(2.208420) 0.0521356(2.208430)
[4] 0.1428650(2.925990) 0.1427380(2.925950) 0.1428660(2.925990)
20 10 [5] 0.0468047(1.475010) 0.0468103(1.475010) 0.4672770(1.475010)
[6] 0.1489790(1.847030) 0.1489070(1.847010) 0.1490000(1.847040)
[7] 0.0996390(1.687940) 0.0995607(1.687920) 0.0996578(1.687940)
[8] 0.1553390(1.709930) 0.1552530(1.709900) 0.1553590(1.709930)
25 10 [9] 0.1151180(1.894140) 0.1150350(1.894120) 0.1151330(1.894140)
[10] 0.00097442(1.27014) 0.0097176(1.27014) 0.00980259(1.27014)
[11] 0.1550760(1.930400) 0.1549820(1.930370) 0.1551000(1.930400)
[12] 0.0614795(1.129176) 0.0614042(1.29176) 0.0614893(1.29177)
25 15 [13] 0.1042340(1.083920) 0.1041470(1.083910) 0.1042640(1.083930)
[14] 0.0970686(1.266200) 0.0969918(1.266180) 0.0970923(1.266200)
[15] 0.0998006(1.173930) 0.0997249(1.173910) 0.0998328(1.173930)
[16] 0.0980328(1.323670) 0.0979575(1.323650) 0.0980556(1.323670)
30 15 [17] 0.2909370(0.592421) 0.2909970(0.592457) 0.2909670(0.592439)
[18] 0.2861400(0.628215) 0.2861990(0.628249) 0.2861690(0.628232)
[19] 0.1142530(1.281150) 0.1151730(1.281130) 0.1152810(1.281150)
[20] 0.1052180(1.131820) 0.1051320(1.131800) 0.1052460(1.131820)
30 20 [21] 0.0397746(0.722620) 0.0396972(0.722614) 0.0398096(0.722623)
[22] 0.0163531(0.645947) 0.0162683(0.645945) 0.0163926(0.645949)
[23] 0.0528775(0.718214) 0.0527984(0.718206) 0.0529175(0.718278)
[24] 0.0919885(0.879429) 0.0919003(0.879413) 0.0920422(0.879439)
50 20 [25] 0.1060660(1.094770) 0.1059960(1.094750) 0.1060980(1.094770)
[26] 0.0639464(0.918429) 0.0638751(0.918420) 0.0639725(0.918433)
[27] 0.1027290(1.003060) 0.1026540(1.003050) 0.1027620(1.003070)
[28] 0.0921558(0.996051) 0.0920827(0.996037) 0.092190(0.996057)
50 25 [29] 0.0226088(0.513429) 0.0226722(0.513432) 0.0225785(0.513428)
[30] 0.0412888(0.746001) 0.0412199(0.745995) 0.0413179(0.746004)
[31] 0.1573980(1.070030) 0.1573300(1.070010) 0.1574360(1.070050)
[32] 0.0939640(0.882445) 0.0938659(0.882432) 0.0939705(0.882451)
100 30 [33] 0.0221276(0.492459) 0.0221921(0.462462) 0.0220961(0.492458)
[34] 0.1468470(0.877965) 0.1467860(0.877947) 0.1468920(0.877978)
[35] 0.1084710(0.777371) 0.1084100(0.777358) 0.1085090(0.777379)
[36] 0.0209500(0.707031) 0.0208773(0.707029) 0.0209798(0.499895)
100 50 [37] 0.0221536(0.365813) 0.0220891(0.365810) 0.0221887(0.365815)
[38] 0.00189239(0.266265) 0.00180332(0.266265) 0.00195382(0.266265)
[39] 0.0798200(0.450539) 0.0797579(0.450530) 0.7985540(0.450545)
[40] 0.0351824(0.391901) 0.0351142(0.0482149) 0.0352210(0.391903)

mle, maximum likelihood estimation.

Table 4

The absolute value of bias and the mean squared error for the mle and Bayes estimates of λ are listed for various sample sizes and sampling schemes.

Prior 0
Prior 1
n m Scheme α^ α˜ α˜
20 5 [1] 0.3264670(0.174211) 0.3263265(0.174119) 0.326554(0.174268)
[2] 0.3125085(0.175525) 0.3109845(0.174575) 0.3107235(0.174413)
[3] 0.2932695(0.156102) 0.2928570(0.155861) 0.2927425(0.155793)
[4] 0.2964170(0.165958) 0.2951265(0.165194) 0.2948700(0.165043)
[5] 0.3033180(0.170888) 0.3027190(0.170527) 0.3025795(0.170442)
[6] 0.3056350(0.171403) 0.3053230(0.171213) 0.3052520(0.171169)
50 5 [7] 0.2326710(0.104817) 0.2326680(0.104816) 0.2328510(0.104901)
[8] 0.227333(0.0996646) 0.227330(0.0996633) 0.227517(0.0997482)
[9] 0.2312230(0.101513) 0.2312220(0.101512) 0.2314180(0.101603)
[10] 0.2448950(0.108718) 0.2448930(0.108717) 0.2450840(0.108810)
[11] 0.2361100(0.102681) 0.2361080(0.102681) 0.2362990(0.102771)
[12] 0.2281270(0.102811) 0.2281250(0.102811) 0.2283180(0.102899)
50 10 [13] 0.2023535(0.098225) 0.2023340(0.098217) 0.2024420(0.098261)
[14] 0.2194495(0.113864) 0.2189520(0.113646) 0.2188260(0.113591)
[15] 0.1995715(0.100056) 0.1993410(0.0999638) 0.1995460(0.100046)
[16] 0.188810(0.0991293) 0.1887925(0.0991227) 0.188901(0.0991636)
[17] 0.1891650(0.0951523) 0.1891255(0.0951374) 0.1892485(0.0951839)
[18] 0.2115465(0.107693) 0.164785(0.0900948) 0.162992(0.0895071)
70 10 [19] 0.196174(0.0991527) 0.196152(0.0991441) 0.196262(0.0991872)
[20] 0.2076240(0.109312) 0.2071970(0.109135) 0.2070780(0.109086)
[21] 0.1925230(0.0981574) 0.1923310(0.0980836) 0.1925220(0.0981572)
[22] 0.178040(0.0892947) 0.178025(0.0892892) 0.178025(0.0892892)
[23] 0.181694(0.0942027) 0.181657(0.0941896) 0.181779(0.0942337)
[24] 0.2236670(0.113502) 0.1999880(0.103470) 0.1987630(0.102982)
70 15 [25] 0.1525525(0.074663) 0.1525445(0.0746605) 0.1526105(0.0746808)
[26] 0.1610365(0.078941) 0.160040(0.0789305) 0.161088(0.0789577)
[27] 0.1604285(0.0792977) 0.1594515(0.0789853) 0.159724(0.0790723)
[28] 0.153904(0.0788255) 0.153777(0.0787864) 0.159724(0.0790723)
[29] 0.160811(0.0844711) 0.160308(0.0843096) 0.1605155(0.0843761)
[30] 0.173772(0.0840183) 0.1730975(0.0837843) 0.172953(0.0837344)
100 15 [31] 0.131363(0.0640767) 0.131362(0.0640763) 0.131421(0.0640919)
[32] 0.137894(0.0678622) 0.137892(0.0678616) 0.137954(0.0678788)
[33] 0.127235(0.0637183) 0.127232(0.0637175) 0.127301(0.245602)
[34] 0.140579(0.068427) 0.140577(0.0684264) 0.140641(0.0684443)
[35] 0.142327(0.0668081) 0.142325(0.0668075) 0.142389(0.0668256)
[36] 0.133757(0.0670613) 0.133751(0.0670596) 0.133827(0.0670801)
100 20 [37] 0.619528(0.061622) 0.115463(0.0616065) 0.1195515(0.0616277)
[38] 0.124516(0.0623036) 0.124466(0.0622911) 0.124598(0.0632512)
[39] 0.133008(0.0679955) 0.1322985(0.0678074) 0.132512(0.0678639)
[40] 0.137380(0.0704051) 0.1368745(0.0702666) 0.1370595(0.0703172)
[41] 0.139426(0.0687665) 0.1394075(0.0687612) 0.139470(0.0687788)
[42] 0.120810(0.0583184) 0.1207935(0.0583143) 0.120857(0.0583296)

mle, maximum likelihood estimation.

Table 5

The average bias and the mean squared error for the mle and Bayes estimates of α and λ are listed for various sample sizes and sampling schemes (random).

Prior 0
Prior 1
n m Scheme λ^ λ˜ λ˜
20 5 [1] 0.1450170(0.414535) 0.1539360(0.414510) 0.1540930(0.414558)
[2] 0.1222220(0.595943) 0.1225950(0.596034) 0.1221080(0.595915)
[3] 0.0959207(0.553236) 0.0961172(0.553274) 0.0958474(0.553222)
[4] 0.0977636(0.500359) 0.0981262(0.500431) 0.9762980(0.500333)
[5] 0.1127710(0.486695) 0.1130020(0.486747) 0.1126970(0.486678)
[6] 0.1295870(0.516272) 0.1297450(0.516313) 0.1295510(0.516263)
50 5 [7] 0.0638504(0.339343) 0.3393430(0.582532) 0.3393460(0.582534)
[8] 0.0764455(0.305708) 0.0764358(0.305706) 0.0764867(0.305714)
[9] 0.0891549(0.300518) 0.0891353(0.300515) 0.0892121(0.300528)
[10] 0.0753222(0.338732) 0.0753097(0.338730) 0.0753691(0.338739)
[11] 0.0953258(0.298049) 0.0953122(0.298046) 0.0953706(0.298057)
[12] 0.0628087(0.316734) 0.0627925(0.316732) 0.0628660(0.316741)
50 10 [13] 0.0846792(0.362508) 0.0846487(0.362503) 0.0847311(0.362517)
[14] 0.0865706(0.407572) 0.0867438(0.407602) 0.0864742(0.407556)
[15] 0.0345237(0.423932) 0.0344032(0.423924) 0.0346810(0.423943)
[16] 0.0276860(0.386191) 0.0276545(0.386189) 0.0277470(0.386194)
[17] 0.0441424(0.396225) 0.0440963(0.396221) 0.0442209(0.396232)
[18] 0.0894087(0.389970) 0.0906514(0.390193) 0.0876544(0.389659)
70 10 [19] 0.0660680(0.390865) 0.0665740(0.390660) 0.0666644(0.390872)
[20] 0.0429737(0.465414) 0.0431430(0.465429) 0.0428696(0.465405)
[21] 0.0507068(0.369911) 0.0505973(0.369900) 0.050848(0.369926)
[22] 0.0436478(0.369652) 0.0436186(0.369650) 0.0436186(0.369650)
[23] 0.0342054(0.389776) 0.0341602(0.389773) 0.0342852(0.389782)
[24] 0.0115685(0.376568) 0.116635(0.376789) 0.114665(0.376333)
70 15 [25] 0.0297971(0.305182) 0.0297808(0.305181) 0.0298339(0.305185)
[26] 0.0550084(0.316372) 0.0549709(0.312447) 0.0550635(0.316378)
[27] 0.0591451(0.335305) 0.0589468(0.335281) 0.0593916(0.335334)
[28] 0.0193960(0.368465) 0.0193132(0.368462) 0.0194974(0.368469)
[29] 0.0209317(0.381710) 0.0207688(0.381703) 0.0211117(0.381718)
[30] 0.0646878(0.324553) 0.0648706(0.324577) 0.0645641(0.324537)
100 15 [31] 0.0003732(0.322194) 0.0003785(0.322194) 0.0003493(0.322194)
[32] 0.0004736(0.349041) 0.0004647(0.349041) 0.0005050(0.349041)
[33] 0.0138877(0.300106) 0.0139039(0.300106) 0.0138432(0.300105)
[34] 0.0322061(0.284619) 0.0321953(0.284618) 0.0322364(0.284621)
[35] 0.0390078(0.288319) 0.0389973(0.288319) 0.0390370(0.288322)
[36] 0.0073213(0.341275) 0.00734502(0.341275) 0.0072675(0.341274)
100 20 [37] 0.0123209(0.296028) 0.0122654(0.296027) 0.0123992(0.296030)
[38] 0.0273074(0.271238) 0.0272604(0.271235) 0.0273765(0.271241)
[39] 0.0186184(0.311621) 0.0184501(0.311615) 0.0188311(0.311629)
[40] 0.0037491(0.355722) 0.0035876(0.355721) 0.0039160(0.355723)
[41] 0.0386721(0.294793) 0.0386447(0.294791) 0.0387159(0.294796)
[42] 0.0168828(0.269872) 0.0168550(0.269871) 0.0169318(0.269874)

mle, maximum likelihood estimation.

Table 6

The average bias and the mean squared error for the mle and Bayes estimates of λ are listed for various sample sizes and sampling schemes (random).

From Tables 46, one can note that for fixed sample size n, as effective sample size m increases, the implementations improve in terms of bias and MSEs in both procedures. Bayes estimates under Lindely's method is similar to the mle estimates as expected. In case of Bayes estimates, the results of Prior0 is similar to Prior1.

5. A CASE OF REAL DATA

The data set in Table 7 represents the remission times (in months) of a sample of size 128 bladder cancer patients reported in [8]. The plot in Figure 1 depicts that N-H exponential distribution gives a reasonable fit to the data set. Using this data, the mle of the unknown parameters is α^=0.846, λ^=0.128 and the popular Kolmogorov-Smirnov goodness of fit test was carried out at level of significance. The result is the N-H exponential distribution is fit to this data with p-value 0.310952. we generate progressively type II censored data from the above data set and compute the estimates of the two unknown parameters and listed in Table 8. Approximate confidence intervals are computed in the case of mle.

 0.08 0.2 0.4 0.5 0.51 0.81 0.9 1.05 1.19 1.26 1.35 1.4 1.46 1.76 2.02 2.02 2.07 2.09 2.23 2.26 2.46 2.54 2.62 2.64 2.69 2.69 2.75 2.83 2.87 3.02 3.25 3.31 3.36 3.36 3.48 3.52 3.57 3.64 3.7 3.82 3.88 4.18 4.23 4.26 4.33 4.34 4.4 4.5 4.51 4.87 4.98 5.06 5.09 5.17 5.32 5.32 5.34 5.41 5.41 5.49 5.62 5.71 5.85 6.25 6.54 6.76 6.93 6.94 6.97 7.09 7.26 7.28 7.32 7.39 7.59 7.62 7.63 7.66 7.87 7.93 8.26 8.37 8.53 8.65 8.66 9.02 9.22 9.47 9.74 10.06 10.34 10.66 10.75 11.25 11.64 11.79 11.98 12.02 12.03 12.07 12.63 13.11 13.29 13.8 14.24 14.76 14.77 14.83 15.96 16.62 17.12 17.14 17.36 18.1 19.13 20.28 21.73 22.69 23.63 25.74 25.82 26.31 32.15 34.26 36.66 43.01 46.12 79.05
Table 7

Remission times of bladder cancer patients data.

n m Scheme α^mle λ^mle α^Lindely λ^Lindely
128 128 (0 * 128)a 0.846349 0.127828 0.873219 0.132073
(0.648884, 1.04381) (0.0774782, 0.178178)
128 53 (0*53, 75) 0.473106 0.244122 0.347462 0.291060
(0.210417, 0.735742) (0.0521221, 0.436122)
128 59 (0*59, 1*69) 1.771420 0.157970 5.94787 0.171523
(0.99333, 2.54951) (0.0706334, 0.245307)

The mle (CI in parenthesis) and Bayes estimates using different censoring schemes for a This 0*3(say) denotes that 0, 0, 0.

Table 8

The mle (CI in parenthesis) and Bayes estimates using different censoring schemes for the data set.

On the other hand, one can compute an approximate Bayes estimates for the two unknown parameters using the Gibbs sampling procedure which generates samples from the posterior distribution. Here, we obtain the approximate Bayes estimates under the assumptions of noninformative prior. A set of 10000 Gibbs samples was generated after a “burn-in-sample” of size 1000. Using these generated samples posterior summaries of interest can be derived and all the calculations are performed using the WinBUGS software. Table 9 lists the posterior descriptive summaries of interest. This table consists of MC error which considered as One way to assess the accuracy of the posterior estimates is by calculating the Monte Carlo error (MC error) for each parameter which estimates of the difference between the mean of sampled values and the true posterior mean. The simulation should be run until the MC error for each parameter of interest is less than about 5% of the sample standard deviation and this achieved in our example.

n m Scheme Posterior Summaries α˜Bayes λ˜Bayes
128 128 (0 * 128)a Mean (sd) 2.856(0.1253) 0.02215(0.001777)
credible interval (2.722, 3.180) (0.0188, 0.0256)
MC error 0.001682 2.163E-5
128 53 (0*53,75) mean (sd) 2.692(0.3328) 0.04565(0.003972)
credible interval (2.220, 3.454) (0.0353, 0.04967)
MC error 0.006645 7.798E-5

(a) This 0*3 (say) denotes that 0, 0, 0.

Table 9

The approximate Bayes estimates (sd in parenthesis and credible interval under different censoring schemes for the data set.

CONFLICTS OF INTEREST

The authors declare that there is no conflict of interest.

ACKNOWLEDGMENTS

The authors are grateful to the editor and referees for their comments that helped to improve the paper.

APPENDIX

Lindley [9] developed an asymptotic expansion to evaluate the ratio of the following integral

u=E[u(α,λ)]=00u(α,λ)exp[l(α,λx)+ρ(α,λ)]dαdλ00exp[l(α,λx)+ρ(α,λ)]dαdλ
where u(α,λ) is a function of α and λ only, l(α,λx) is the log-likelihood function (given by Equation (7)) and ρ(α,λ) is the logarithm of prior distribution (defined in Equation (10)).

Using Lindely's method, u can be approximated as

u=u+12[(uαα+2uαρα)σαα+(uαλ+2uαρλ)σαλ+(uλα+2uλρα)σλα+(uλλ+2uλρλ)(σλλ)+(uασαα+uλσαλ)(Lααασαα+Lαλασαλ+Lλαασλα+Lλλασλλ)+(uασλα+uλσλλ)(Lααασαα+Lαλλσαλ+Lλλλσλλ)](A1)

The right-hand side of above equation are evaluated at the mle u^. We have the joint prior distribution g(α,λ)αa1ebαλc1edλ and ρ(α,λ)=ln[g(α,λ)]. Also, ρα=ρα and ρλ=ρλ. When u=α, we have uα=uα=1 and uαα=0=uλ=uλλ=uλα=uαλ When u=λ, we have uλ=uλ=1 and uα=0=uαα=uλλ=uλα=uαλ

Journal
Journal of Statistical Theory and Applications
Volume-Issue
20 - 2
Pages
267 - 278
Publication Date
2021/03
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.d.210322.002How to use a DOI?
Open Access

TY  - JOUR
AU  - M. R. Mahmoud
AU  - R. M. Mandouh
PY  - 2021
DA  - 2021/03
TI  - Progressively Censored N-H Exponential Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 267
EP  - 278
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210322.002
DO  - https://doi.org/10.2991/jsta.d.210322.002
ID  - Mahmoud2021
ER  -