1. Introduction [3][6][7]

A mixture distribution is a compounding of statistical distributions, which arise when sampling frominhomogeneous populations (or mixed populations) with a different probability density function in each component.For example the distribution of times to failure in a mixture of good and defective items, and the distribution of some diagnostic measure in a mixed population of patients, some of whom have a given disease and some of them do not have.

The application of finite mixture models arise in economics, medicine, psychology, agriculture, life testing and reliability. In many applications, the available data can be considered as data coming from a mixture population of two or more distributions.This idea enables us to mix statistical distribution to get a new distribution carrying the properties of its components. The distribution constructed here is denoted by (), mixture of two parameters Pareto and two parameters Weibull.

2. Distribution

Here the distribution is a mixture of some distributions with Weibull distribution, like exponential Weibull, Pareto Weibull, ErlangWeibull, Gamma Weibull. Now we explain distribution which is a mixture of Pareto distribution with Weibull distribution and we shall denoted it by

The statistical properties of () are explained, then we want to estimate its parameters;

a. The probability density function is;

(1)

() parameters of vector of parameters.

b. The cumulative distribution function is;

(2)

Also this function can be drawn for various values of parameters.The reliability function of () distribution denoted by;

(3)

c. The hazard function of () is;

(4)

d. The moment about origin of () distribution which can be used in estimation method are defined as;

(5)

Where;

existing only for

The mean and variance of () distribution can be obtained from (5), where;

(6)

While the variance of () is;

(7)

To estimate parameters by moments we must find, and another four moments.

(8)

When is integer for example then solving these equations yield moment estimator.

(9)

(10)

(11)

(12)

(13)

Here we assumed if we assume the parameters reduced to since so we construct three equations for moment methods;

Assume

(14)

(15)

(16)

From equation (14), find

If , substitute in (15);

By using fixed point method where get;

(17)

Using simple program, the moment estimator of is obtained, therefore;

(18)

From we can estimate by moment estimator from solving;

And from equation (8) since;

Use to obtain

3. Maximum Likelihood Method[5]

To apply the ML for the mixture distribution from two subpopulation (), where () is Pareto population and is Weibull population and we assume is mixing proportion parameters, where . Let is the time failure for random sample taken from mixed distribution if we can determine the units of each sub population then choosing the random variable which is the time through it the failure time of units can be determined before such that;

Therefore the sample units are of type (time censored sampling Type 1) and the conditional function for time failure from for units that failed before time is;

The probability that unit from failed before unit from failed before also, and is remained until time , is;

(9)

Then the likelihood function for sample is;

The parameters here are

Therefore;

4. Simulation Procedure

We have;

To find the estimators of (MOM & MLE), we perform simulation experiments using Monte Carlo assuming that;

Chosen values of parameters are;

Also chosen values for predetermined censoring time (T).

Also the total number of time intervals are;

The results for estimators of parameters are explained in the following tables;

Table (1). Values of (MLE) estimatorsfor parameters of       distribution for model I, (L=1000)

Table (2). Values of (MOM) estimatorsfor parameters of       distribution for model I, (L=1000)

Table (3). Values of MSE for MLE estimators of       distribution for model I, (L=1000)

Table (4). Values of MSE for MOMestimators of       distribution for model I, (L=1000)

Table (5). Values of reliability function estimator by MLE , MOM & MSE       for model I

Table (6). Values of reliability function estimator by MLE, MOM & MSE       for model I

Table (7). Values of reliability function estimator by MLE , MOM & MSE       for model I

5. Conclusions

From simulation experiments, the results indicate that:

1-The maximum likelihood estimator for reliability function is better than the moment estimator of reliability function.

2- When the values of estimator by MLE and MOM, and also of MSE , are convergent inspite of increasing sample size.

3- The estimator of by MLE, and MOM approximated from real values of R by increasing the size of sample from .

4- The values of real function (R) are increasing with (T).

5- The values of mean square errors for estimated parameters are decreased when (T) increasing, this is due to decreasing to failure for units drawn from each sub – population.