The main purpose of this paper is to obtain estimates of parameters, reliability and hazard rate functions of a heterogeneous population represented by finite mixture of two general components. The doubly Type II censoring of generalized order statistics scheme is used. Maximum likelihood and Bayes methods of estimation are used for this purpose. The two methods of estimation are compared via a Monte Carlo Simulation study.
Generalized Order Statistics; Bayes Estimation; Heterogeneous Population; Monte Carlo Integration; Monte Carlo Simulation1. Introduction
Let the random variable (rv) T follow a class including some known lifetime models, its cumulative distribution function (CDF) is given by
and its probability density function (PDF) is given by
where is the derivative of with respect to t and is a nonnegative continuous function of t and may be a vector of parameters, such that as and as
The reliability function (RF) and hazard rate function (HRF) are given, respectively, by
where
Bayesian inferences based on finite mixture distribution have been discussed by several authors. Bayesian estimation of the mixing parameter, mean and reliability function of a mixture of two exponential lifetime distributions based on right censored samples considered by [1,2] estimated the survival and hazard functions of a finite mixture of two Gompertz components by using type I and type II censored samples, using the maximum likelihood (ML) and Bayes methods. Based on type I censored samples from a finite mixture of two truncated type I generalized logistic components, [3] computed the Bayes estimates of parameters, reliability and hazard rate functions. [4] considered estimation for the mixed exponential distribution based on record statistics. [5] considered Bayes inference under a finite mixture of two compound Gompertz components model. [6] studied some properties of the mixture of two inverse Weibull distributions and obtained the estimates of the unknown parameters via the EM Algorithm.
[7] introduced the generalized order statistics (gos’s). Ordinary order statistics, ordinary record values and sequential order statistics are, among others, special cases of gos’s. The gos’s have been considered extensively by many authors, among others, they are [8-20].
Mixtures of distributions arise frequently in life testing, reliability, biological and physical sciences. Some of the most important references that discussed different types of mixtures of distributions are a monograph by [21-23].
The PDF, CDF, RF and HRF of a finite mixture of two components of the class under study are given, respectively,
where, for, the mixing proportions are such that and are given from (1), (2), (3) after using and instead of and.
The property of identifiability is an important consideration on estimating the parameters in a mixture of distributions. Also, testing hypothesis, classification of random variables, can be meaning fully discussed only if the class of all finite mixtures is identifiable. Idenifiability of mixtures has been discussed by several authors, including [24-26].
Our aim of this paper is the estimation of the parameters and functions of these parameters of a class of finite mixture distributions based on doubly Type II censoring gos’s using ML and Bayes methods. Illustrative example of Gompertz distribution is given and compared with the results obtained by previous researchers.
2. Maximum Likelihood Estimation
Let
be the (r - s) gos’s drawn from a mixture of two components of the class (2). Based on this doubly censored sample, the likelihood function can be written [27] as
(9)
where is the parameter space, and
For definition and various distributional properties of gos’s, see [7, 28].
The likelihood function (9) and maximum likelihood estimates (MLE’s) can be obtained by using (1) and (5) in two cases, regarding to m value, as follows.
2.1 MLE’s When <img src="3-1240108\3f0b0295-dfbe-4a4f-8454-38f301df6bfc.jpg" />
In this case, substituting (1), (5) in (9), the likelihood function takes the form
Take the logarithm of (10), we have
where,
Differentiating (11) with respect to the parameters and (involved in) and equating to zero gives the following likelihood equations
where, for j = 1,2
The solution of the five nonlinear likelihood Equations (12) using numerical method, yields the MLE’s and.
2.2. MLE’s When <img src="3-1240108\b2c82fac-8e60-413d-b631-dec8e3b60b09.jpg" />
The likelihood function takes the form
So, from (14)
Differentiating (15) with respect to the parameters and and equating to zero gives the following likelihood equations
where
The solution of the five nonlinear likelihood Equations (16) using numerical method, yields the MLE’s and.
3. Bayes Estimation
In this section, Bayesian estimation for the parameters of a class of finite mixture distributions is considered under squared error and Linex (Linear-Exponential) loss functions.
We shall use the conjugate prior density, that was suggested by [29], in the following form
where is the hyperparameter space.
3.1. Bayes Estimates When <img src="3-1240108\3fd3c75c-3db2-44d3-838f-818811e28a87.jpg" />
It follows, from (10) and (18), that the posterior density function is given by
where
The Bayes estimator of a function, say, under squared error and Linex loss functions is given, respectively, by
where the integral is taken over the five dimensional space and.
To compute the integral, we can use the Monte Carlo Integration (MCI) method in the form
where is generated from the PDF, for more details see [30].
Under squared error and Linex loss functions, we can obtain the Bayes estimator of the parameter p, by generating
from the prior (18) and setting in (23) and (24). The Bayes estimates of and can be similarly computed.
3.2. Bayes Estimates When <img src="3-1240108\a001d06d-60e3-4ca9-b910-8eb86bff3dfb.jpg" />
The posterior density function can be obtained from (14) and (18), as
where
Under squared error and Linex loss functions, we can obtain the Bayes estimator of the parameter p, by generating
from the prior (18) and setting in (23) and (24). The Bayes estimates of and can be similarly computed.
4. Example4.1. Gompertz Components4.1.1. Maximum Likelihood Estimation
Suppose that, for and
so
.
In this case, the subpopulation is Gompertz distribution with parameter
For by substituting and in (12), we have the following nonlinear equations
where, for
and are the solution of the above nonlinear equations.
Also, for substituting and in (13), (16) and (17), we have the following nonlinear equations:
and are the solution of the above nonlinear equations.
wang#title3_4:spSpecial caseswang#title3_4:sp1) Upper order statistics
If we put and in (10),
the likelihood function takes the form
Substituting and in (27), we have the following nonlinear equations
where.
The solution of the nonlinear likelihood equations (31) gives the MLE’s and.
wang#title3_4:sp2) Upper record values
If we put in (14), the likelihood function takes the form
Substituting in (29), we have the following nonlinear equations
The solution of the nonlinear likelihood Equations (33) gives the MLE’s and.
4.1.2. Bayes Estimation
Let and are independent random variables such that and for, to follow a left truncated exponential density with parameter , as used by [2]. A joint prior density function is then given by
where
and
For the posterior density function then takes the form
(35)
For m = −1 the posterior density function then takes the form
(36)
Under squared error and Linex loss functions, we can obtain the Bayes estimator of the parameter by generating from the prior (34) and setting in (23) and (24). The Bayes estimates of and can be similarly computed.
wang#title3_4:spSpecial caseswang#title3_4:sp1) Upper order statistics
If we put and in (35), the posterior density function takes the form
(37)
wang#title3_4:sp2) Upper record values
If we put in (36), the posterior density function takes the form
(38)
Under squared error and Linex loss functions, we can obtain the Bayes estimator of the parameter by generating from the prior (34) and setting in (23) and (24). The Bayes estimates of and can be similarly computed.
5. Simulation Study
A comparison between ML and Bayes estimators, under either a squared error or a Linex loss functions, is made using a Monte Carlo simulation study in the two cases upper order statistics and upper record values according to the following steps:
1) For a given values of the prior parameters generate a random value from the distribution.
2) For a given values of the prior parameters for generate a random value from the distribution.
3) Using the generated values of and we generate a random sample of size from a mixture of two components, as follows:
• generate two observations from
• if then
otherwise
•
• repeat above steps times to get a sample of size.
4) The sample obtained in Step 3 is ordered.
5) The MLE’s of the parameters and are obtained by solving the nonlinear Equations (31), by using Mathematica 6.
6) Using the generated values of and we generate upper record values of size from a mixture of two components.
7) The MLE’s of the parameters and are obtained by solving the nonlinear Equations (33), by using Mathematica 6.
8) The Bayes estimates under squared error and Linex loss functions (BES, BEL), of and are computed, by using MCI forms (23) and (24), respectively.
9) The squared deviations are computed for different samples and censoring sizes, where stands for the parameter and its estimate (ML or Bayes).
10) The above Steps (3)-(9) are repeated 1000 times. The averages and the estimated risks (ER) are computed over the 1000 repetitions by averaging the estimates and the squared deviations, respectively.
The computational (our) results were computed by using Mathematica 6.0. In all above cases the prior parameters chosen as, which yield the generated values of and (as the true values). The true values of and when, are computed to be and The value of the shape parameter of the Linex loss function is. The averages and the estimated risks (ER) are displayed in Tables 1-4. Figures 1 and 2 represent the estimated risks of the estimates in the case of upper order statistics. Figures 3 and 4 represent the estimated risks of the estimates
(Upper order statistics) Averages and Estimated Risks (ER) of the estimates of <img src="3-1240108\b219ef49-4504-4d63-9e98-7fd311b8d68a.jpg" /> for different samples and censoring sizes
(Upper order statistics) averages and estimated risks (ER) of the estimates of <img src="3-1240108\01e67ccf-2957-4261-be47-0e7ce277252c.jpg" /> and <img src="3-1240108\cb3033a8-5473-44ad-a1cb-30eecf043023.jpg" /> for different sample and censoring sizes
(Upper record values) Averages and Estimated Risks (ER) of the estimates of <img src="3-1240108\64a7f07e-128d-47da-ada8-71a5d4dddb93.jpg" /> for different sampl
(Upper record values) Averages and Estimated Risks (ER) of the estimates of <img src="3-1240108\5e486f25-36f5-4d3a-9640-056fb0273682.jpg" /> and <img src="3-1240108\2d762263-0735-4092-ad5c-f5b9c51ee9a5.jpg" /> for different sample and censoring sizes
in the case of upper record values.
6 Concluding Remarks
1) Estimation of the parameters of the finite mixture model of two Gompertz distributions are considered from a Bayesian approach based on gos’s. A compareson between ML and Bayes estimators, under either a squared error loss or a Linex loss, is made by using a Monte Carlo simulation study in both two cases considering order statistics and upper record values cases.
2) From Tables 1 and 2, we see that in most of the considered cases, the ER’s of the estimates decrease as n increases. In complete sample case, the Bayes estimates of p, and HRF under Linex loss function have the smallest ER’s as compared with their corresponding estimates under squared error loss function or MLE’, while the ER’s of the Bayes estimates of and RF under squared error loss functions are the smallest estimated risks. For censored samples, the Bayes estimates of p under Linex loss function have the smallest ER’s as compared with their corresponding estimates under squared error loss function or MLE’s. While, the Bayes estimates (against the proposed prior) of and HRF under squared error loss function have the smallest ER’s as compared with their corresponding estimates. It is observed that MLE’s for HRF perform best when sample size n is increased. Also, we note that the MLE’s of and RF have the smallest ER’s as compared with Bayes estimates.
3) From Tables 3 and 4, we see that the Bayes estimates (against the proposed prior) of the parameters and HRF under Linex loss function have the smallest ER's as compared with their corresponding estimates under squared error loss function or MLE’s. While, the Bayes estimates of (for complete sample) and RF under squared error loss function have the smallest ER’s as compared with both Bayes estimates under Linex loss function or the MLE’s. Also, it is observed that MLE's for RF perform best when sample size n is increased.
4) If the mixing proportion p is known, [2] estimated the parameters reliability and hazard rate functions based on Types I and II censored samples.
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is the derivative of 
with respect to t and 
is a nonnegative continuous function of t and 
may be a vector of parameters, such that
as 
and 
as 
, the mixing proportions 
are such that 
and
are given from (1), (2), (3) after using 
and 
instead of 
and
.
(9)
is the parameter space, and
, 
and 
(involved in
) and equating to zero gives the following likelihood equations
and
.
and 
and equating to zero gives the following likelihood equations
and
.
is the hyperparameter space.
, under squared error and Linex loss functions is given, respectively, by
.
is generated from the PDF
, for more details see [
in (23) and (24). The Bayes estimates of 
and 
can be similarly computed.
in (23) and (24). The Bayes estimates of 
and 
can be similarly computed.
and 
.
subpopulation is Gompertz distribution with parameter 
by substituting 
and 
in (12), we have the following nonlinear equations
and 
are the solution of the above nonlinear equations.
substituting 
and 
in (13), (16) and (17), we have the following nonlinear equations:
and 
are the solution of the above nonlinear equations.
and 
in (10),
and 
in (27), we have the following nonlinear equations
.
and
.
in (14), 
the likelihood function takes the form
in (29), we have the following nonlinear equations
and
.
and 
are independent random variables such that 
and for
, 
to follow a left truncated exponential density with parameter 
, as used by [
the posterior density function 
then takes the form
(35)
then takes the form
(36)
by generating 
from the prior (34) and setting 
in (23) and (24). The Bayes estimates of 
and 
can be similarly computed.
and 
in (35), 
the posterior density function takes the form
(37)
in (36), 
the posterior density function takes the form
(38)
by generating 
from the prior (34) and setting 
in (23) and (24). The Bayes estimates of 
and 
can be similarly computed.
generate a random value 
from the 
distribution.
for 
generate a random value 
from the 
distribution.
and 
we generate a random sample of size 
from a mixture of two 
components, 
as follows:
from 
then
times to get a sample of size
.
and 
are obtained by solving the nonlinear Equations (31), by using Mathematica 6.
and 
we generate upper record values of size 
from a mixture of two 
components.
and 
are obtained by solving the nonlinear Equations (33), by using Mathematica 6.
and 
are computed, by using MCI forms (23) and (24), respectively.
are computed for different samples and censoring sizes, where 
stands for the parameter and 
its estimate (ML or Bayes).
, which yield the generated values of 
and 
(as the true values). The true values of 
and 
when
, are computed to be 
and 
The value of the shape parameter 
of the Linex loss function is
. The averages and the estimated risks (ER) are displayed in Tables 1-4. Figures 1 and 2 represent the estimated risks of the estimates in the case of upper order statistics. Figures 3 and 4 represent the estimated risks of the estimates
and HRF under Linex loss function have the smallest ER’s as compared with their corresponding estimates under squared error loss function or MLE’, while the ER’s of the Bayes estimates of 
and RF under squared error loss functions are the smallest estimated risks. For censored samples, the Bayes estimates of p under Linex loss function have the smallest ER’s as compared with their corresponding estimates under squared error loss function or MLE’s. While, the Bayes estimates (against the proposed prior) of 
and HRF under squared error loss function have the smallest ER’s as compared with their corresponding estimates. It is observed that MLE’s for HRF perform best when sample size n is increased. Also, we note that the MLE’s of 
and RF have the smallest ER’s as compared with Bayes estimates.
(for complete sample) and RF under squared error loss function have the smallest ER’s as compared with both Bayes estimates under Linex loss function or the MLE’s. Also, it is observed that MLE's for RF perform best when sample size n is increased.
reliability and hazard rate functions based on Types I and II censored samples.