This paper presents Rayleigh mixtures of distributions in which the weight functions are assumed to be chi-square,

In statistics, a mixture distribution is expressed as a convex combination of other probability distributions. It can be used to model a statistical population with subpopulations, where components of mixture probability densities are the densities of the subpopulations, and the weights are the proportion of each subpopulation in the overall population. Mixture distribution may suitably be used for certain data set where different subsets of the whole data set possess different properties that can best be modeled separately. They can be more mathematically manageable, because the individual mixture components are dealt with more nicely than the overall mixture density. The families of mixture distributions have a wider range of applications in different fields such as fisheries, agriculture, botany, economics, medicine, genetics, psychology, paleontoogy, electrophoresis, finance, communication theory, sedimentology/geology, and zoology.

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Suppose the random variable

This is called a mixture of the distributions

It can be generalized to the case when the parameter space

In this paper we first define the general form of Rayleigh mixture distribution. Then we furnished the Rayleigh mixture of some well-known sampling distributions such as chi-square,

The main results of this study have been presented in the form of some definitions and theorems.

A random variable

The name Rayleigh mixture distributions is given due to the fact that the derived distribution (

The Rayleigh mixtures of distributions in which the weight functions are assumed to be chi-square,

A random variable

A random variable

A random variable

Let

Using the formula, the mixture of two correlated distributions is as follows:

where

Moments and different characteristics of the Rayleigh mixture of distributions are presented by the following theorems.

If

We know the

If

The characteristic function is defined as

If

The

Now, the

Hence,

If

The

If

From here we may get the mean and variance of this distribution.

The characteristic function of the random variable

The different moments of the random variable which is the resultant of the product of two correlated Rayleigh random variables are obtained by following theorem.

For

We know that

Special findings of the above theorem if

If

Under the transformation

Combining all of the obtained results for the integrals in (

For nonnegative integer

According to definition of expectation we can obtain the following result for the

The moment generating function of

The moment generating function of

We know the well-known method of the maximum likelihood estimation is very complicated for the parameter estimation of mixture distribution and method of moment is very suitable in these cases. Hence, we used method of moments (MoMs) estimation of technique for estimation of the parameter of the Rayleigh mixture distribution.

Let

The first sample raw moment is

And as we already got

Hence

Therefore,

Let

The second sample raw moment is obtained as

We have already found

Hence, by the method of moments, we get

Therefore,

Let

The first sample raw moment is

And we already get

Hence, from the method of moments estimator is

Therefore,

In this paper, we have presented the Rayleigh mixtures of distributions in which the weight functions are assumed to be chi-square,

The authors would like to thank the editor and referee for their useful comments and suggestions which considerably improved the quality of the paper.