Estimation under a Finite Mixture of Exponentiated Exponential Components Model and Balanced Square Error Loss

By exponentiating each of the components of a finite mixture of two exponential components model by a positive parameter, several shapes of hazard rate functions are obtained. Maximum likelihood and Bayes methods, based on square error loss function and objective prior, are used to obtain estimators based on balanced square error loss function for the parameters, survival and hazard rate functions of a mixture of two exponentiated exponential components model. Approximate interval estimators of the parameters of the model are obtained.


Introduction
The study of homogeneous populations was the main concern of statisticians along history.However, Newcomb [1] and Pearson [2] were two pioneers who approached heterogeneous populations with 'finite mixture distributions '.
With the advent of computing facilities, the study of heterogeneous populations, which is the case with many real World populations (see Titterington et al. [3]), attracted the interest of several researchers during the last sixty years.Monographs and books by Everitt and Hand [4], Titterington et al. [3], McLachlan and Basford [5], Lindsay [6] and McLachlan and Peel [7], collected and organized the research done in this period, analyzed data and gave examples of possible practical applications in different areas.Reliability and hazard based on finite mixture models were surveyed by AL-Hussaini and Sultan [8].
In this paper, concentration will be on the study of a finite mixture of two exponentiated exponential components.Due to the exponentiation of each component by a positive parameter, the model is so flexible that it shows different shapes of hazard rate function.
Maximum likelihood estimates (MLEs) and Bayes estimates (BEs), using square error loss (SEL) function are obtained and used in finding the estimates of the parameters, survival function (SF) and hazard rate function (HRF) using the balanced square error loss (BSEL) function.
Approximate interval estimators of the parameters are obtained by first finding the approximate Fisher information matrix.
The cumulative distribution function (CDF), denoted by   H x  , of a finite mixture of k components, de- noted by   , 1, , where, for    0 p  , , , , , , where is a parameter space.
The case  , in (1), is of practical importance and so,we shall restrict our study to this case.In such case, the population consists of two sub-populations, mixed with proportions p and .We shall write the CDF of a mixture of two components as  [9] obtained the information matrix for a mixture of two Inverse Gaussian components.AL-Hussaini and Abd-El-Hakim ( [10][11][12]) studied the failure rate of a finite mixture of two components, one of which is Inverse Gaussian and the other is Weibull.They estimated the parameters of such model and studied the efficiency of schemes of sampling.AL-Hussaini [13] predicted future observables from a mixture of two exponential components.AL-Hussaini [14] obtained Bayesian predictive density function when the population density is a mixture of general components.Other references on finite mixtures may be found in AL-Hussaini and Sultan [8].
In this paper, the components are assumed to be exponentiated exponential whose CDFs are of the form where So that the corresponding probability density function (PDF) components take, for , the form The PDF, SF and HRF, of the mixture (2), denoted by where In ( 5)-( 8), shall be given by ( 3) and ( 4), respectively, so that , , , .

Hazard Rate Function of the Mixture
It is well-known that the exponential distribution has a constant HRF on the positive half of the real line.A finite mixture of two exponential components has a decreasing hazard rate function (DHRF) on the positive half of the real line.See, for example, AL-Hussaini and Sultan [8].If each of the exponential components is exponentiated by a positive parameter, more flexible model is obtained in that several shapes of the HRF of the mix-ture are obtained.Figure 1 shows six different shapes of CDFs and their corresponding HRFs of the vector of parameters.Examples of such shapes are given as follows:  

Point Estimation Using Balanced Square Error Loss Function
It is well-known that the Bayes estimator of a function where the integrals are taken over the m-dimensional space and   The SEL function has probably been the most popular loss function used in literature.The symmetric nature of SEL function gives equal weight to over-and under-estimation of the parameters under consideration.However, in life-testing, over-estimation may be more serious than under-estimation or vice-versa.Consequently, research has been directed towards asymmetric loss functions.Varian [15] suggested the use of linear exponential (LINEX) loss functions.Thompson and Basu [16] suggested the use of quadratic exponential (QUADREX) loss function.Ahmadi et al. [17] suggested the use of the so called balanced loss function (BLF), which was originated by Zellner [18], to be of the form
The BLF (10) specializes to various choices of loss functions such as the absolute error loss, entropy, LINEX and generalizes SEL functions.
is substituted in (10), we obtain the balanced square error loss (BSEL) function, given by The estimator BSEL u   u of a function  , using BSEL may be given by   where ML u   is the MLE of u and its Bayes ˆSEL u p = 0.9,  1 = 0.5,  2 =1.5,  1 = 2,  2 = 3 estimator using SEL function.The estimator of a function, using BSEL is actually a mixture of the MLE of the function and the BE, using SEL.Other estimators, such as the least squares estimator may replace the MLE.Also, a LINEX or QUADREX loss function could be used . Having obtained the MLE and BE based on SEL, the estimates based on BSEL function are given, from (11), by Suppose that r units have failed during the interval   0, r x : 1 units from the first sub-population and 2 units from the second such that 1 2 and nr units , which cannot be identified as to sub-population, are still functioning.Let, for and where  is the vector of parameters involved and , , , , , , given by ( 4), in ( 13), we obtain

Maximum Likelihood Estimation
The log-LF is given by , ln , ln ln ln ln ln ln e 1 ln 1 e ln .
ˆML , of the five parameters are obtained by solving the following system of likelihood equations The invariance property of MLEs enables us to obtain the MLEs by replacing the parameters by their MLEs in and 2) It can be numerically shown that the vector of parameters , satisfying the likelihood equations actually maximizes the LF (14).This is done by applying Theorem (7-9) on p. 152 of Apostol [20].
3) The parameters of the components are assumed to be distinct, so that the mixture is identifiable.For the concept of identifiability of finite mixtures and examples, see Everitt and Hand [4], AL-Hussainiand Ahmad [21] and Ahmad and AL-Hussaini [22].

Bayes Estimation Using SEL Function
Suppose that an objective (non-informative) prior is used, in which p,  , 1  , 2  , 2  are independent and that p-Uniform on , so that the prior PDF is given by The following theorem gives expressions for the Bayes estimators using the SEL function.
Theorem The Bayes estimators of the parameters, SF and HRF, assuming that the prior belief of the expeimenter has PDF (16), are given by , 1 , j j n r j j j j j j j j j j j j j j j j j j j (20 , , j j j j j j j j j r j r j j r j j r j j j j ln 1 e ln 1 e ln 1 e ln r r l n1 e .

Approximate Confidence Intervals
Let   , , The proof of the theorem is given . The F (see Nelson [23]), etric ma is observed Fisher information matrix for the MLEs of the parameters is the 5 × 5 symm trix of the negative second partial derivatives of log-LF (15) with respect to the parameters.That , evaluated at the vector of MLEs  .
The second partial derivatives are given in Appendix 2. The inverse of F is the local estimate V of the asymptotic variance-covariance matrix of V    , the ith diagonal element of the matrix V, given by ( 34).
An approximate two-sided

, SF and HRF
A sample is generated from the mixture We generate 100 samm a finite mixture of two 1) te 1 u and 2 u from Uniform (0,l) distribu-For given values of p,  1   ples of size n = 50 each, fro exponentiated exponential components, whose PDF is given by ( 5) and ( 4), as follows: Genera tion. 2) ,  2 ,  1 ,  2 , generate x according to the expression: , where the sample is generated from the mixture in such a way that 0 , 1,2, 1, , x is chosen to be equal to 1.

The estimates of
and absolute biases are computed by using the ML and Bayes methods.The Bayes estimates are obtained under SEL function.An stimator of a function, using BSEL, is actua e lly a mixture ( 2, 0.4, 0.6, 0.8) of e h u in MATL function knowns, by using some iteration scheme.Neverthless, the MLE of th function and the BE, using SEL.The MLEs are computed using t e b ilt-AB ® "ga" to find the maximum of the log-LF (15) using the genetic algorithm.This is better than solving the system of five likelihood equations in the five un- mators of the parameters are given by: It then follows that the ate 95% nce intervals of the parameters p, approxim confide 1 , 2 , 1 and 2     , ven by ( given, r ly, by: ( p < 0.561), (0.5287 <  < 4.7599), (1.3832 <  2 < 5.3226), 1.8039 <  2 < 4.56).

Concluding Remarks
In this article, we have considered point and interval estim xp nt co nts f-fit.This asymmetric loss function may be considered as a comnon-Bayesian estimates.gi 35), are espective 0.271 < 1 (0.9294 <  1 < 4.18) and ( ation.Point estimation, of the parameters of a finite mixture of two exponentiated e one ial mpone , SF and HRF is based on BLEF which is a weighted average of two losses: one of which reflects precision of estimation and the other reflects goodness-o timates under SEL(B-SEL) function.

Interval Estimation of the Param
The asymptotic variance-covariance matrix (34), based on the generated data, is found to be promise between Bayesian and 0.00546 0.0106 0.00941 0.01345 0.01   1.1651 0.3266 0.66019 0.23199 1.0099 We have also estimated the parameters of the mixture by obtaining the asymptotic variance-covariance matrix and hence the approximate confidence intervals.x x H r j n r j j j x j j j j R x p q C p q j j j where .
So that LF ( 14) can be written in the form are given by (31).Suppose that the prior PDF is as in (16).It then follows that the posterior PDF is given by are given by (32).The normalizing constant A is given by where is given by (17), in which I is given by (22).
, is given in (18), in which I is given by (23).
given in (18), in which I is given by (24).
is given in (19), in which I is given by (25).
, is given in (19), in which I is given by (26).
I is given by ( 27) and is given in (20), in which 6 I by (28).
, and using  π x given by (34), it can be shown that where  are given by (33).By integrating both sides of (35) with respect to the five parameters, we where S 7 is given by ( 21), in which I 7 is given by ( 29) and I 8 by (30).

Appendix 2
Second Partial Derivatives of the Log-likelihood Function

.
the failure time of the ith unit belonging to the subThe likelihood function (LF) is given by Mendenhall and Hader[19], as


is the percentile of the standard normal distribution with right-tale probab5.Numeri al Exampleility of c.

c 5 . 1 .
Point Estimation of the Parameters in such a way that , 1,2, 1, , you get a sample of size n.The observations are ordered and only the first r = n) out of the = 50 observations are assumed to wn.Now we have 1 r observations from the first component of the mixture and 2 r observations from the r  ).  The value of 0