## Generalized Rayleigh Distribution Revisited

### by:Mohammad El-Moniem Soleha(1), Iman A. Sewilam(2)
.

**Abstract:**
Some generalizations and basic properties of the Rayleigh density function are reviewed. An entropy-like trnasformation g (x) is proposed, using the distribution function F(x) and the Reliability function R(x) of a continuous Rayleigh random variable (RRV) X, modeling the time-to-failure of a given component. The derivative of g` (x) = u (x; ?), is found to be a “density” function, which represnets a peculiar form of “two” of the reviwed generalizations. Some characteristics of the “derived density” function are provided, like the reliability and the hazard (failure) rate functions. In this context, the behavior of the failure rate function is presented, and the mean time-to-failure (MTF) u(x; ?) function is evaluated. The parameter of the derived density is estimated using two different methods of estimation, namely, the method of moments and the maximum likelihood method. The resulting estimates and are not coinciding. Considering the mean square error as a criterion for comparing and , the superiority of over - as it is expected – is confirmed by a Monte-Carlo simulation-based numerical example. In this context, the desired effect of increasing the sample size on the value of MSE for the two estimators is graphically demonstrated. **Key Words: **
Generalized Gomma distribution, Generalized Weibull distributin, Rayleigh random variable (RRV), reliability function, hazard function, Moment and
Maximum likelihood estimators, the mean square error (MSE)

**Authors:**

Mohammad El-Moniem Soleha, drsoleha@Yahoo.com

Iman A. Sewilam,

**Editor:**
Weiming Ke, Weiming.ke@sdstate-edu

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