Abstract
Specifying a proper statistical model to represent asymmetric lifetime data with high kurtosis is an open problem. In this paper, the three-parameter, modified, slashed, generalized Rayleigh family of distributions is proposed. Its structural properties are studied: stochastic representation, probability density function, hazard rate function, moments and estimation of parameters via maximum likelihood methods. As merits of our proposal, we highlight as particular cases a plethora of lifetime models, such as Rayleigh, Maxwell, half-normal and chi-square, among others, which are able to accommodate heavy tails. A simulation study and applications to real data sets are included to illustrate the use of our results.
Highlights
Academic Editor: Jinyu LiReceived: 3 June 2021Accepted: 4 July 2021Published: 8 July 2021Publisher’s Note: MDPI stays neutralVodua [1] proposed a two-parameter distribution for the analysis of positive data, the well-known generalized Rayleigh distribution
The proposed distribution can be written as the ratio of two independent random variables
We derived some special cases that can be seen as twoparameter extensions of the Rayleigh, half normal, Maxwell and chi square distributions
Summary
Vodua [1] proposed a two-parameter distribution for the analysis of positive data, the well-known generalized Rayleigh distribution. We focus on the approach proposed in [8], where an extension of the generalized Rayleigh distribution called the slashed generalized Rayleigh distribution was introduced. This model has a kurtosis coefficient, which exhibits a wider range of values than the kurtosis coefficient in the GR distribution; it is appropriate to fit the data sets with outliers. Recall that a rv T follows a slashed generalized Rayleigh distribution, denoted as T ∼ SGR(θ, α, q), if its stochastic representation is given by the following: T=. We follow the idea proposed by [10], who introduced an extension of the normal distribution called modified slash (MS) distribution.
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