# Minimum Variance Unbiased Estimation in the Gompertz Distribution under Progressive Type II Censored Data with Binomial Removals

Authors: Ashok Shanubhogue; N. R. Jain
Journal: ISRN Probability and Statistics
 (2013)

---

## Abstract

This paper deals with the problem of uniformly minimum variance unbiased estimation for the parameter of the Gompertz distribution based on progressively Type II censored data with binomial removals. We have obtained the uniformly minimum variance unbiased estimator (UMVUE) for powers of the shape parameter and its functions. The UMVUE of the variance of these estimators is also given. The UMVUE of (i) pdf, (ii) cdf, (iii) reliability function, and (iv) hazard function of the Gompertz distribution is derived. Further, an exact(1−α)100% confidence interval for the pth quantile is obtained. The UMVUE of pdf is utilized to obtain the UMVUE of P(X<Y). An illustrative numerical example is presented.


---

## Body

## 1. Introduction

A Type II censored sample is one for which onlym smallest observations in a sample of n items are observed. A generalization of Type II censoring is a progressive Type II censoring. Under this scheme, nunits of the same kind are placed on test at time zero, and m failures are observed. When the first failure is observed, a number r1 of surviving units are randomly withdrawn from the test; at the second failure time, r2 surviving units are selected randomly and taken out of the experiment, and so on. At the time of mth failure, the remaining rm=n-r1-r2-⋯-rm-1-m units are removed. Balakrishnan et al. [1] indicated that such scheme can arise in clinical trials where the drop out of patients may be caused by migration or by lack of interest. In such situations, the progressive censoring scheme with random removals is required. Many authors have discussed inference under progressive Type II censored samples using different life distributions including El-Din and Shafay [2], Kim et al. [3] Kim and Han [4], Ali Mousa and Jaheen [5], and Pérez-González and Fernández [6]. For a detailed discussion of progressive censoring, we refer to Balakrishnan and Aggarwala [7] and Balakrishnan [8]. Note that if  r1=r2=⋯=rm-1=0, this scheme reduces to the Type II censoring scheme. Also note that if r1=r2=⋯=rm=0, so that m=n, the progressively Type II censoring scheme reduces to the case of no censoring, that is, the case of a complete sample. In this paper, we use progressively Type II censoring scheme with binomial removals where the number of units removed at each failure time follows a binomial distribution.The Gompertz distribution was first introduced by Gompertz [9] to describe human mortality and establish actuarial tables. Since then, many investigators have used the Gompertz distribution or some related forms of it in a variety of  studies. There are many forms of the Gompertz distribution in the literature.The Gompertz distribution is applied in actuarial science, reliability and life testing studies, and epidemiological and biomedical studies. Several such situations have been discussed by Ananda et al. [10], Walker and Adham [11], Jaheen [12], and many others. For a review of literature on estimating parameters of the Gompertz distribution, one may refer to Gordon [13], Chen [14], Wu et al. [15], Garg et al. [16], Ismail [17], Al-Khedhairi and El-Gohary [18], and many others.Inference for The Gompertz distribution based on  progressively Type II censored data is discussed by many authors. Wu et al. [19] obtained the maximum likelihood estimators of the two-parameter Gompertz distribution under progressive Type II censoring with binomial removals. They had also given the expected test time to complete the censoring test. Wu et al. [20] discussed the problem of interval estimation for the two-parameter Gompertz distribution under progressive Type II censored data. Many authors have studied the problem of estimation of P=P(X<Y) for various distributions. This model involves two independent random variables X and Y. The term P=P(X<Y) is the reliability of a system of strength Y is subjected to a stress X. The system fails if the applied stress exceeds its strength. An extensive review of this topic is given in Kotz.et al. [21].  Saraçoğlu and kaya [22] obtained the maximum likelihood estimate of stress strength reliability for the Gompertz distribution. Saraçoğlu et al. [23] have obtained maximum likelihood estimate and UMVUE of stress strength reliability for the Gompertz distribution when X and Yare independent but not identically random variables belonging to the Gompertz distribution when complete sample is available.In this paper, we discuss the problem of UMVUE for shape parameter of  the  Gompertz distribution based on progressively Type II censored data with binomial removals. In Section2, the conditional  likelihood function is given. In Section 3, the UMVUE of  parameter of θ and its functions are derived. Also, the UMVUE of the (i) pdf, (ii) cdf, (iii) reliability function (iv) hazard function are obtained. In Section 4. the UMVUE of P(X<Y) is obtained by using the UMVUE of p.d.f. In Section 5 an exact (1-α)100% confidence interval for pth quantile is obtained. An illustrative numerical example is presented.


## 2. The Model

Let the failure time distribution be the Gompertz with probability density function,(1)f(x)={θecxexp{-θc(ecx-1)}θ>0,c>0,x>00otherwise,
where θ and c are the parameters. We assume that c is known.The cumulative distribution function is given by(2)F(x)=1-exp{-θc(ecx-1)},x>0.
The survival function is given by
(3)S(x)=exp{-θc(ecx-1)},x>0.
The density given in (1) can be written as
(4)f(x)=a(x)[h(θ)]d(x)[g(θ)],
where
(5)a(x)=ecx,h(θ)=exp(-θ),d(x)=1c[ecx-1],g(θ)=1θ,suchthata(x)>0,g(θ)=∫x>0a(x)[h(θ)]d(x)dx.Let(X1,R1),(X2,R2),…,(Xm,Rm) denote a progressively Type II censored sample, where Xi=Xi:m:n,  for i=1,2,…m, and X1<X2<⋯<Xm. The conditional likelihood  function  can be written as, see Cohen [24],
(6)L(θ;x_/R_=r_)=A∏i=1mf(xi)·[s(xi)]ri,
where A=n(n-r1-1)(n-r1-r2-2)⋯(n-r1-r2-⋯rm-1-m+1) and 0≤ri≤(n-m-r1-r2-⋯ri-1) for i=1,2,…m-1.Substituting (1) and (3) in (6) we get
(7)L(θ;x_/R_=r_)=Aθmec∑i=1mxiexp{-θc[∑i=1m(1+ri)ecxi-n]}.
Suppose that an individual unit being removed from the life test is independent of others but with the same probability p.Then the number of units removed at each failure time follows a binomial distribution, and, following Wu et al. [19], the joint probability mass function of r1,r2,…rm-1 is given by
(8)P(R_=r_)=(n-m)!p∑i=1m-1ri(1-p)(m-1)(n-m)-∑i=1m-1(m-i)ri(n-m-∑i=1m-1ri)!∏i=1m-1ri!,
that is,
(9)P(R_=r_)=(n-m)![p/(1-p)m]∑i=1m-1ri(1-p)∑i=1m-1iri∏i=1mri![(1-p)-(n-m)](m-1).
The unconditional likelihood function is
(10)L(θ,p;x_,r_)=L(θ;x_/R_=r_)P(R_=r_).
Using (7) and (9) in (10) we can write the full likelihood function as
(11)L(θ,p;x_,r_)=Aθmec∑i=1mxiexp{-θc[∑i=1m(1+ri)ecxi-n]}×(n-m)![p/(1-p)m]∑i=1m-1ri(1-p)∑i=1m-1iri∏i=1mri![(1-p)-(n-m)](m-1).


## 3. Unbiased Estimation

Let(12)Yi=1c[ecXi-1],i=1,2,…,m,
then Yi have exponential distribution with mean 1/θ. We can show that Y1<Y2<⋯<Ym is a progressive Type II censored sample from an exponential distribution with mean 1/θ. Let us consider the following transformation:
(13)Z1=nY1,Zi=(n-i+1-r1-r2-⋯-ri-1)(Yi-Yi-1),i=2,3,…,m.
In order to derive the distribution of Zi, i=1,2,…,m consider the inverse transformation Y1=Z1/n and Yi=∑i=2m(Zi/(n-ri-1-i+1)),i=2,3,…m. The variables Z1,Z2,…,Zm defined in (13) are all independent and identically distributed with exponential distribution with mean 1/θ, see Thomas and Wilson [25]. The joint density of Z1,Z2,…,Zm, is
(14)f(z_,θ/R_=r_)=θmexp{-θ∑i=1mzi}.
It can be seen that
(15)∑i=1mZi=∑i=1m(1+ri)Yi.
Using (12) in (15), we have
(16)∑i=1mZi=1c∑i=1m(1+ri)[ecXi-1].
Let
(17)T=∑i=1mZi.=1c[∑i=1m(1+ri)ecXi-n].
Since (14) is a member of exponential family of distributions, Tis a complete sufficient statistic for θ. The distribution of T is gamma with parameters θ and m, which is again a member of exponential family of distributions. The pdf of T is given by
(18)f(t,θ)=B(t,m)[h(θ)]t[g(θ)]m,
where
(19)B(t,m)=tm-1|m-,h(θ)=exp(-θ),g(θ)=1θ.
Jani and Dave [26] have studied the problem of minimum variance unbiased estimation in a class of exponential family of distributions. They have shown that if  X1,X2,…Xn, be a random sample from density of the type given in (4) and the p.d.f. of its complete sufficient statistics can be written as the one given in (18), then the UMVUE of  [h(θ)]k is given by
(20)Hk,n=B(t-k,n)B(t,n),t>k
and the UMVUE of [g(θ)]k is
(21)Gk,n=B(t,n+k)B(t,n).
Following the results derived in Jani and Dave [26], we get the UMVUE of some important parametric functions as given below.(i) Using (20), the UMVUE of exp[-θk] is
(22)Hk,m=[1-k(1/c)[∑i=1m(1+ri)ecxi-n]]m-1,1c[∑i=1m(1+ri)ecxi-n]>k.(ii) Using (22), the UMVUE of the variance of Hk,m, is given by,
(23)V~ar[Hk,m]=[1-k(1/c)[∑i=1m(1+ri)ecxi-n]]2m-2-[1-2k(1/c)[∑i=1m(1+ri)ecxi-n]]m-1.(iii) Using (21), the UMVUE of (1/θ)k, is given by
(24)Gk,m=|m-|m+k-[1c[∑i=1m(1+ri)ecxi-n]]k.(iv) Using (24), the UMVUE of the variance of Gk,m is given by
(25)V~ar[Gk,m]=[1c[∑i=1m(1+ri)ecxi-n]]2k[(|m-|m+k-)2-(|m-|m+2k-)].(v) The UMVUE of densityf(x) given in (1), for fixed x, is given by,
(26)ϕx,m=((m-1)ecxt)(1-[ecx-1]ct)m-2,1c[ecx-1]<t<∞,m>2,
where
(27)t=1c[∑i=1m(1+ri)ecxi-n].(vi) The UMVUE of variance ofϕx,m,m>2 is given by
(28)V~ar[ϕx,m]={((m-1)ecxt)2[1-[ecx-1]ct]2m-4-((m-1)ecxt)×[1-[ecx-1]ct]m-2×(c(m-2)ecxct-[ecx-1])[1-[ecx-1][ct-[ecx-1]]]m-3,t>2c[ecx-1]((m-1)ecxt)2×[1-[ecx-1]ct]2m-4,1c[ecx-1]<t≤2c[ecx-1]0otherwise,wheret is given by (27).(vii) Consideringx as fixed, the UMVUE of  reliability  function R(x)=P(X>x), x≥0 is obtained as follows. Since R(x)=[h(θ)][ecx-1]/c, where h(θ) is given in (5) and using (22) with k=(1/c)[ecx-1], the UMVUE R~(x) of R(x) is given by
(29)R~(x)=[1-[ecx-1][∑i=1m(1+ri)ecxi-n]]m-1.(viii) Using (29), the UMVUE of the variance of R~(x) is given by
(30)V~ar[R~(x)]={[1-[ecx-1]ct]2m-2-[1-2[ecx-1]ct]m-10<x<12clog[ct+2][1-[ecx-1]ct]2m-212clog[ct+2]<x<12clog[ct+1]0otherwise.(ix) The UMVUE of cumulative distribution function given in (2) is
(31)F~(x)={0,x<01-(1-[ecx-1][∑i=1m(1+ri)ecxi-n])m-1,0<x<1clog[∑i=1m(1+ri)ecxi-n+1]1otherwise.Special Cases
(a)
Substitutingk=1 in (22), we get the UMVUE of exp[-θ] as,
(32)H1,m=[1-1(1/c)[∑i=1m(1+ri)ecxi-n]]m-1,1c[∑i=1m(1+ri)ecxi-n]>1.
(b)
Substitutingk=-1 in (22), we get UMVUE of exp[θ] as
(33)H-1,m=[1+1(1/c)[∑i=1m(1+ri)ecxi-n]]m-1,1c[∑i=1m(1+ri)ecxi-n]>0.
(c)
Substitutingk=1 in (24), we get the UMVUE of (1/θ) as
(34)G1,m=[∑i=1m(1+ri)ecxi-n]cm.
(d)
Substitutingk=-1 in (24), we get the UMVUE of θ as
(35)G-1,m=c(m-1)[∑i=1m(1+ri)ecxi-n].
(e)
The hazard function for the Gompertz distribution ish(x)=θecx. Using (35), the UMVUE of hazard function, for fixed x, can be given as
(36)h~(x)=c(m-1)ecx[∑i=1m(1+ri)ecxi-n].
Shanubhogue  and Jain [27] have studied the problem of minimum variance unbiased estimation in exponential distribution under progressive Type II censored data with binomial removals. They have given the UMVUE for parameter p and various functions of p. Since the joint density P(R_=r_) given in (9) is independent of θ, one gets the same estimators of p, and its various functions as given in Shanubhogue  and Jain [27].


## 4. UMVU Estimator ofP(X<Y)

In the following theorem, we derive the UMVUE ofP(X<Y). Let m1 units (out of n1) on X and m2 units (out of n2) on Y are recorded which follow the Gompertz distributions, given in (1) with parameters θ1 and θ2, respectively. Let r1,r2,…rm1 and s1,s2,…sm2 be corresponding removals. We denote
(37)t1=1c[∑i=1m1(1+ri)ecxi-n1],(38)t2=1c[∑i=1m2(1+si)ecyi-n2].Theorem 1.
Under progressive Type II censored data, the UMVU estimator ofP=P(X<Y) for the density given in (1) is given by
(39)P~={∑j=0m2-1(-1)j(m1-1)!(m2-1)!(m1+j-1)!(m2-j-1)!(t1t2)jt1<t21-∑i=0m1-1(-1)i(m1-1)!(m2-1)!(m1-i-1)!(m2+i-1)!(t2t1)it1≥t2,
where t1 and t2 are given by (37) and (38), respectively.Proof.
We have(40)P~=∬Gϕx,m1ϕy,m2dxdy,
where
(41)G={(x,y):0<x<1clog[1+ct1],0<y<1clog[1+ct2],x<y}.
Using (26) in (40) and let t1<t2, we have
(42)P~=∫0(1/c)log[1+ct1]∫x(1/c)log[1+ct2][ecx(m1-1)t1]×[1-(ecx-1)ct1]m1-2[ecy(m2-1)t2]×[1-(ecy-1)ct2]m2-2dydx.
Now
(43)∫x(1/c)log[1+ct2][ecy(m2-1)t2][1-(ecy-1)ct2]m2-2dy=[1-(ecx-1)ct2]m2-1
After substituting (43) into (42), we get
(44)=∫0(1/c)log[1+ct1][ecx(m1-1)t1][1-(ecx-1)ct1]m1-2×[1-(ecx-1)ct2]m2-1dx.
Further simplification of (44) and applying the result (1-w)n=∑j=0n(nj)(-1)jwj, we get
(45)=(m1-1)∑j=0m2-1(m2-1j)(-1)j(t1t2)j∫01wj(1-w)m1-2dw.
Further simplification of (45) gives
(46)P~=∑j=0m2-1(-1)j(m1-1)!(m2-1)!(m1+j-1)!(m2-j-1)!(t1t2)j,t1<t2.
Similarly, we can show that for the case t1>t2, the UMVUE of P(X<Y) is
(47)P~=1-∑i=0m1-1(-1)i(m1-1)!(m2-1)!(m1-i-1)!(m2+i-1)!(t2t1)it1>t2.


## 5. Exact Confidence Interval forpth Quantile

Theorem 2.
Under progressive Type II censored data an exact(1-α)100% confidence interval for pth quantile, 0<p<1 is given by,
(48)[1clog{2m[ecζ~p-1]χ2m,1-α/22+1},1clog{2m[ecζ~p-1]χ2m,α/22+1}].Proof.
Using (2) and (35), we have
(49)ecξ~p=1+(tm)clog[1(1-p)].
Hence, t=m[ecξ~p-1]/clog[1/(1-p)]. Now, t has gamma distribution with parameters θ and m.
Using the relationclog[1/(1-p)]=θ[ecξp-1], we make the transformation Q=2Tθ where
(50)Q=2m[ecξ~p-1][ecξp-1].
Now, Q has χ2 distribution with 2m degrees of freedom. Thus,
(51)P{χ2m,α/22<Q<χ2m,1-α/22}=1-α.
Using (50) and rearranging (51), we get (48).


## 6. Illustrative Example

In this section, we illustrate the use of the estimation methods given in this paper.The following are the numbers of tumor-free days of 30 rats fed with unsaturated diet, see King et al. [28]. The data are
(52)112,68,84,109,153,143,60,70,98,164,63,63,77,91,91,66,70,77,63,66,66,94,101,105,108,112,115,126,161,178.
These data are presented by Lee [29] and studied by Chen [14] and Wu et al. [20]. Chen [14] and Wu et al. [20] assumed a Gompertz distribution for tumor-free times. Wu et al. [20] obtained the MLE of c as c^=0.0505 for progressive Type II censored data. We generate a progressive Type II censored data with binomial removals from these data, assuming c=0.0505. The progressive censored sample size is m=18. The dropout numbers have been generated using MYSTAT software as follows: r1 from B(12,0.1) and ri/r1,r2,…,ri-1 have B(12-∑j=1i-1rj,0.1) distribution for i=2,3,…17 and set
(53)r18={12-∑j=117rjif12-∑j=117rj>00otherwise.Using the results given in Section3, the UMVU estimates of different parametric functions of θ based on data given in Table 1 are given in Table 2.Table 1
Progressive Type II censored sample with binomial removals generated from the tumor-free time data.
i
x
i
r
i
1
60
1
2
63
1
3
66
0
4
66
0
5
66
0
6
68
1
7
70
1
8
77
1
9
91
1
10
94
0
11
98
0
12
101
0
13
105
0
14
108
1
15
112
0
16
112
1
17
126
0
18
143
4Table 2
The UMVU estimates of different parametric functions ofθ based on data given in Table 1.
Serial Number
Parametric function
UMVU estimate
1
exp
[
-
θ
]
H
1,18
=
0.999914
2
Variance ofH1,18
V
~
ar
[
H
1,18
]
=
4.38788
E
-
10
3
exp
[
θ
]
H
-
1,18
=
1.000086378
4
Variance ofH-1,18
V
~
ar
[
H
-
1,18
]
=
4.38932
E
-
10
5
1
/
θ
G
1,18
=
10934.27603
6
Variance ofG1,18
V
~
ar
[
G
1,18
]
=
6292546.967
7
θ
G
-
1,18
=
8.63747
E
-
5
8
Variance ofG-1,18
V
~
ar
[
G
-
1,18
]
=
4.38858
E
-
10
9
Hazard function atx=95.
h
~
(
x
)
=
0.010469216
10
Density atx=95
ϕ
95,18
=
0.008617136
11
Variance ofϕ95,18
V
~
ar
[
ϕ
95,18
]
=
2.80714
E
-
6
12
Reliability atx=95
R
~
(
x
)
=
0.81314
13
Variance ofR~(x) at x=95
V
~
ar
[
R
~
(
x
)
]
=
0.001683
14
c.d.f atx=95
F
~
(
x
)
=
0.186862An exact 95% confidence interval for the third quartile, that is,p=0.75, is (123.35, 141.87).



---

*Source: 237940-2013-02-25.xml*