The quantile-based generalised Rayleigh distribution

Abstract

The Rayleigh distribution is a special case of different families of distributions, one of these being the Weibull distribution. This mini-dissertation considers the development of a new distributional family, the quantile-based generalized Rayleigh distribution, henceforth denoted GRD. The Rayleigh distribution acts as the parent distribution that will be used in the construction of the GRD. That is, the quantile function of the GRD is obtained by taking the weighted sum of the quantile function of the Rayleigh distribution and the quantile function of the reflected Rayleigh distribution. Compared to the Rayleigh distribution, the GRD is more flexible in terms of distributional shape in that, depending on the value of its shape parameter, it can be negatively skewed, symmetric or positively skewed. The GRD furthermore possesses the advantageous property of skewness-invariant measures of kurtosis. The GRD is characterized through its L-moments. These measures are used to describe the location, spread and shape of this distribution. Quantile-based measures of location, spread and

shape are also considered in this mini-dissertation. Using method of L-moments estimation, closed- form expressions for the estimators of the unknown parameters of the GRD are derived and the GRD

is then fitted to two observed data sets.