Proposed Methods for Estimating Parameters of the Generalized Raylieh Distribution in the Presence of One Outlier

In this paper, the probability d istribution (Generalized Ray leigh) with two parameters (θ, σ2 ), in case of outlier, is developed, where the probability density function (pp. dd. ff) is defined, and its moment generating function is derived, to help us in finding the moments, also its cumulative distribution function is found to be used, in obtaining the least squares estimator of the parameters σ2 and θ. The parameters are estimated also by method of moments and method of least square, and also mixture of the estimators are derived, and explained, the estimators of maximum likelihood for θ, σ2 are also obtained.


Introduction
The probability Ray leigh distribution naturally arises in cases when the wind speed data is analysis into two-orthogonal dimensional vector components, where the magnitude of co mponents is independent and normally distributed with equal variances. Also this distribution arises in the case of random comp lex numbers whose real and imaginary co mponents are as normal. The two parameters parameters, Burr Type X , 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 fro m another distribution, after definition of proposed distribution, its Moment-Generat ing function is derived, to help us in finding the first and second mo ments for this distribution, these mo ments are used to obtain the mixture estimator of parameters, as well as the Maximu m Likelihood estimators. The paper is organized as follows: In section (2) we present the ( , 2 ) and its properties, and the provides its . . . Then section (3), discuss finding mo ment generting function and the methods of estimating parameters which are Maximu m Likelihood and Least Squares and Mixture of estimators are derived. These estimators are compared using (MSE), through simulat ion programs, prepared for this purpose.
In this paper, we introduce a new family of continues distribution, called a new -two parameter generalized Rayleigh GR( , 2 ) in presence of outlier generated distribution fro m another distribution i.e the distribution depend on mixing the distribution of ( 1 , 2 , … , −1 ) random variables, d istributed as Rayleigh with scale parameter ( 2 = 1), and shape parameter while the ( ) random variab le wh ich represents the outlier is one random variable that is uniformly distributed ~ (0, 2 ). So the aim o f research is how to find the marginal . . of this type of distribution, in precence of outlier, and also how to derive its cumu lative distribution function, and it's mo ment generating function to help in obtaining the mo ments after all required derivation three methods are introduced include mo ment estimators and least squares estimators, and maximu m likelihood estimators.

Moment Generating Function
The ℎ mo ments of may be determined direct or using Moment generating function technique. Now we derive ( ) for the distribution defined in (1): The second term integral can be evaluated as follows: after simplificat ion of (7), it can be written as: Therefore equation (7) can be written as (8): According to Beta formula: Therefore ( ) : Differentiating ( ) and evaluating at = 0 we get ( ) and ( 2 ) as: � in the Presence of One Outlier Taking the second derivative ′′ ( ) , we have ( 2 ) as: Then: Then: To estimate we can solve (15) by Newton Raphson method. Hence the solution of equation (15) is: This value can be considered as init ial value for solving equation (16) by Newton Raphson method.
Also to proceed in finding the mo ment estimator for the parameter 2 , the following equation is applied as follows: Also the estimate � 2 can be obtained by solving equation (18) by Newton Raphson method:

Leat Squares Estimators
In this section we provide the regression based method estimators of unknown parameters, which was orig inally suggested by [16] to estimate the parameters of Beta with respect to and 2 : 2 )] Also we find 2 : Where � is estimated fro m equation (20) wh ich is non linear equation can be solved using Newton Raphson method, or we can use Mo ment estimator of (equation 17) to obtain least square estimator o f parameter ( 2 ), and then obtain the mixed estimator.

Maximum Likelihood Estimators
Let the random variab les Non linear equation (25) can be solved by using Newton Raphson method: for ( � ).

Conclusions
This paper offers a new family of distributions, the twoparameter Rayleigh distribution in the presence of one outlier, wh ich is important for analysis lifet ime data. The distribution has two parameters (scale parameter σ 2 , and shape parameter θ), and consist of mixing the distribution of ( 1 , 2 , … , −1 ) random variables with the distribution of random variab le ( ), i.e (the distribution of 1 , 2 , … , −1 ) is Rayleigh distribution with ( σ 2 = 1) and ~ uniformly with (0, θσ 2 ), and the replicate each experiment (R=1000), and to use mean square error (MSE) or integrated mean square error (IM SE) for co mparison.
We have studied various method for estimating the parameters, (Least Squares, Moments, Mixture), and derived the Moment Generat ing function, which is used to obtain the first and second Moment of this GR(θ, σ 2 ), 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.