Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application

Hayrinisa Demirci Biçer

doi:10.1515/math-2019-0057

Hayrinisa Demirci Biçer

Open Access

https://doi.org/10.1515/math-2019-0057

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Abstract

Abstract In the present paper, we introduce a new form of generalized Rayleigh distribution called the Alpha Power generalized Rayleigh (APGR) distribution by following the idea of extension of the distribution families with the Alpha Power transformation. The introduced distribution has the more general form than both the Rayleigh and generalized Rayleigh distributions and provides a better fit than the Rayleigh and generalized Rayleigh distributions for more various forms of the data sets. In the paper, we also obtain explicit forms of some important statistical characteristics of the APGR distribution such as hazard function, survival function, mode, moments, characteristic function, Shannon and Rényi entropies, stress-strength probability, Lorenz and Bonferroni curves and order statistics. The statistical inference problem for the APGR distribution is investigated by using the maximum likelihood and least-square methods. The estimation performances of the obtained estimators are compared based on the bias and mean square error criteria by a conducted Monte-Carlo simulation on small, moderate and large sample sizes. Finally, a real data analysis is given to show how the proposed model works in practice.

Highlights

The famous distribution families have been successfully used in modeling real-world data sets, until recently
In the present paper, we introduce a new form of generalized Rayleigh distribution called the Alpha Power generalized Rayleigh (APGR) distribution by following the idea of extension of the distribution families with the Alpha Power transformation
The behavior of the pdf of APGR distribution is displayed in Figure 1 for di erent values of the model parameters

Summary

Introduction

The famous distribution families have been successfully used in modeling real-world data sets, until recently. In the aim of this context, in the study, a new three-parameter family of Rayleigh distribution which is named alpha power generalized Rayleigh distribution (APGR) is derived using the alpha power transform (APT) method recently introduced by Mahdavi and Kundu [5]. A random variable X is said to have a APGR distribution with parameters α, β and λ, if it has the following pdf and cdf ln α α−. We present an analysis on a real-life data set called the coal mining disaster data set to illustrate the modeling behavior of the APGR distribution in comparison with Rayleigh and generalized Rayleigh distributions. We apply the Kolmogorov-Smirnov (KS) test statistic to check whether this data set follows the APGR and most popular lifetime distributions such as Rayleigh, generalized Rayleigh, exponential, Gamma, Weibull, Log-Normal.

Conclusion

B Appendix