1. Introduction

The probability Rayleigh distribution naturally arises in cases when the wind speed data is analysis into two-orthogonal dimensional vector components, where the magnitude of components is independent and normally distributed with equal variances. Also this distribution arises in the case of random complex numbers whose real and imaginary components are as normal.

The two parameters parameters, Burr Type , which introduced by[7], is called Generalized Rayleigh distribution. In this paper, we introduce a new two-parameter Generalized Rayleigh in presence of one outlier generated from another distribution, after definition of proposed distribution, its Moment-Generating function is derived, to help us in finding the first and second moments for this distribution, these moments are used to obtain the mixture estimator of parameters, as well as the Maximum Likelihood estimators. The paper is organized as follows: In section (2) we present the and its properties, and the provides its Then section (3), discuss finding moment generting function and the methods of estimating parameters which are Maximum Likelihood and Least Squares and Mixture of estimators are derived. These estimators are compared using

(MSE), through simulation programs, prepared for this purpose.

In this paper, we introduce a new family of continues distribution, called a new – two parameter generalized Rayleigh GR in presence of outlier generated distribution from another distribution i.e the distribution depend on mixing the distribution of random variables, distributed as Rayleigh with scale parameter , and shape parameter while the random variable which represents the outlier is one random variable that is uniformly distributed So the aim of research is how to find the marginal of this type of distribution, in precence of outlier, and also how to derive its cumulative distribution function, and it's moment generating function to help in obtaining the moments after all required derivation three methods are introduced include moment estimators and least squares estimators, and maximum likelihood estimators.

2. Definition of Distribution

Let the random variables are such that of them are distributed as Rayleigh with scale parameter , and shape parameter (), i.e.

(1)

With cumulative is:

(2)

The random variables have the and one random variable is distributed with

(3)

(4)

And cumulative distribution function is:

(5)

3. Moment Generating Function

The moments of may be determined direct or using Moment generating function technique. Now we derive for the distribution defined in (1):

(6)

(7)

after simplification of (7), it can be written as:

Therefore equation (7) can be written as (8):

(8)

According to Beta formula:

(9)

Therefore :

(10)

(11)

Differentiating and evaluating at we get and as:

(12)

Taking the second derivative , we have as:

Then:

Then:

(13)

Where

Therefore:

where A, B are functions of , the Moment estimator's of are obtained as follows:

(14)

(15)

To estimate we can solve (15) by Newton Raphson method. Hence the solution of equation (15) is:

(16)

Where:

To estimate we can solve equation (3) by Newton Raphson method.

Since:

Where:

If known then:

(17)

This value can be considered as initial value for solving equation (16) by Newton Raphson method.

Also to proceed in finding the moment estimator for the parameter , the following equation is applied as follows:

(18)

Also the estimate can be obtained by solving equation (18) by Newton Raphson method:

where;

4. Leat Squares Estimators

In this section we provide the regression based method estimators of unknown parameters, which was originally suggested by[16] to estimate the parameters of Beta distribution.

Suppose denotes the ordered sample, where:

The estimators for the unknown parameters of the proposed of Generalized Rayleigh with two parameters in case of outlier (i.e equation 3), and its (equation 5), can be obtained by minimizing the quantity T, defined in equation (19):

(19)

with respect to and :

(20)

Also we find :

(21)

Where is estimated from equation (20) which is non linear equation can be solved using Newton Raphson method, or we can use Moment estimator of (equation 17) to obtain least square estimator of parameter (), and then obtain the mixed estimator.

5. Maximum Likelihood Estimators

Let the random variables are such that of them are distributed with and one variable is distributed with , so the joint distribution of are:

(21)

(22)

(23)

Where

(24)

(25)

Non linear equation (25) can be solved by using Newton Raphson method:

to obtai for.

Summary

1. The of in presence of outlier is defined in equation (1).

2. The is obtained in closed from equation (5).

3. The Moment Generating function is derived and written in a closed from equation (10).

4. First and second Moments of this distribution are obtained equation (12) and equation (13).

5. Moment's estimator's can be obtained from equation (14) and (15) using Newton Raphson formula (16) and (18).

6. Since the was found in closed from equation (5), it can be used to find the LS estimator, through minimizing equation (19), the estimators of LS are obtained from equations (20, 21).

7. The Likelihood function is defined in equations (21), (22) and (23), and then the Maximum Likelihood estimator's are explained in equations (24) and (25).

8. Any estimators of Moments, LS can be used in MLE to obtain the Mixture estimators.

6. Conclusions

This paper offers a new family of distributions, the two - parameter Rayleigh distribution in the presence of one outlier, which is important for analysis lifetime data. The distribution has two parameters (scale parameter , and shape parameter ), and consist of mixing the distribution of random variables with the distribution of random variable (), i.e (the distribution of is Rayleigh distribution with () and uniformly with , and the replicate each experiment (R=1000), and to use mean square error (MSE) or integrated mean square error (IMSE) for comparison.

We have studied various method for estimating the parameters, (Least Squares, Moments, Mixture), and derived the Moment Generating function, which is used to obtain the first and second Moment of this , and then used as a possible alternative method for estimating Parameters.

This work will be done in another suggestion in future, to apply this method on another distribution.