Confluent hypergeometric slashed-Rayleigh distribution: Properties, estimation and applications

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Abstract

This article proposes a new distribution, the Confluent hypergeometric slashed-Rayleigh distribution. The new distribution can be seen as an alternative to the slashed-Rayleigh distribution. It arises as quotient of two independent random variables, one being a Rayleigh distribution in the numerator the other a square root of the beta distribution in the denominator. Several structural properties (such as the density function, hazard rate function and moments) are derived. Parameters estimation is performed based on the moment and maximum likelihood methods. Finally, two applications are presented in which the utility of the new model in the analysis of real data is illustrated.

Introduction

The Rayleigh distribution plays an important role in the modeling of random phenomena in various areas, for example: in communications theory to model multiple paths of dense scattered signals reaching a receiver; in the physical sciences to model wind speed, wave heights and sound/light radiation; in engineering to measure the lifetime of an object where the lifetime depends on the objects age, e.g. resistors, transformers, and capacitors in aircraft radar sets; in medical imaging science to model noise variance in magnetic resonance imaging.

This model was originally derived by Lord Rayleigh (John William Strutt, 1880–1919) in connection with a problem in the field of acoustics. It is said that a random variable X follows a Rayleigh (R) distribution, denoted as XR(σ), if its probability density function (pdf) is given by fX(x;σ)=xσex22σ,x>0,where σ>0 is a scale parameter.

The rth moment for a random variable XR(σ) is given by E(Xr)=(2σ)r2Γr+22,where Γ(α)=0uα1eudu is the gamma function. For more details of the Rayleigh distribution see Johnson et al. [1].

A different distribution related with the normal distribution is the slash distribution. It is represented as the quotient between two independent random variables, one normal and the other a power of uniform distribution (U(0,1)). Thus we say that S has a slash distribution if: S=ZU1q,where ZN(0,1), UU(0,1), Z is independent of U and q>0. When q=1 we obtain the canonic slash distribution, and when q the standard normal distribution. The density of the canonic slash is: p(x)=ϕ(0)ϕ(x)x2x012ϕ(0)x=0where ϕ represents the expression of the standard normal distribution, (see Johnson et al. [1]). This distribution has heavier tails than normal distribution, i.e. it has greater kurtosis. Properties of this family are discussed by Rogers and Tukey [2] and Mostelles and Tukey [3]. The maximum likelihood (ML) estimates of location and scale are discussed in Kadafar [4]. Wang and Genton [5] offered a multivariate version of the slash distribution and a multivariate skew version. Gómez et al. [6] and Gómez and Venegas [7] extended slash distribution using the family of univariate and multivariate elliptical distributions. Gómez et al. [8] used this family to extend the Birnbaum–Saunders distribution. Recently Gómez et al. [9] used the slash method to extend the Gumbel distribution.

Iriarte et al. [10] introduced an extension of the R distribution namely the slashed-Rayleigh (SR) distribution. This extension presents heavier tails than the R distribution, i.e. greater kurtosis. Consequently it is appropriate for use in the analysis of data with atypical observations, in contexts in which the R model is used. Gómez-Déniz et al. [11] used the SR model for modeling fading–shadowing wireless channels. It is said that a random variable T follows a SR distribution, denoted as TSR(σ,q), if it can be represented as T=XU1q,q>0,where XR(σ) and UU(0,1) are independent. Notice that if q, then TX.

The pdf associated with Eq. (5) is given by fT(t;σ,q)=q(2σ)q2tq+1Γq+22Ft22σ;q+22,1,t>0,where σ>0 is a scale parameter, q>0 is a kurtosis parameter and F(x;α,β) is the cumulative distribution function (cdf) of the gamma distribution.

A distribution that plays an important role in this paper is the confluent hypergeometric (CH) distribution considered in Gordy [12] with density function given by CH(x;p,q,s)=xp1(1x)q1esxB(p,q)1F1(p,p+q,s),0<x<1,p,q,s>0where B(,) is the beta function and 1F1 is the confluent hypergeometric function (see Abramowitz and Stegun [13]).

In this article, we introduce the confluent hypergeometric slashed-Rayleigh (CHSR) distribution, which is based on the confluent hypergeometric function. The CHSR distribution is an alternative to the SR distribution used to model data sets with atypical observations. We define a random variable that follows a CHSR distribution as a quotient of independent random variables, one being a R distribution in the numerator and the other the square root of a Beta(q,q) distribution in the denominator. The decision to consider a Beta distribution with only one parameter was taken to preserve the parsimony of the model and allow it to compete with the SR model or other two-parameter models supported on the positive real numbers. The result of this construction is a two-parameter model that is more flexible than the R distribution in terms of the kurtosis coefficient. Thus the proposed distribution can serve as an alternative when analyzing data sets that present atypical observations, in contexts where the SR distribution is used.

The article is organized as follows. In Section 2 we present the stochastic representation, the density function and some mathematical properties of the new model. In Section 3, we discuss the estimation of model parameters. Section 4 presents two simulation studies to illustrate the behavior of ML estimates and the CHSR model. In Section 5, two applications to real data sets are presented. Finally, our conclusions are reported.

Section snippets

CHSR distribution

In this section we introduce the stochastic representation, density function and some properties of the CHSR distribution.

Inference

In this section, the moment and ML estimators for the CHSR distribution are discussed.

Simulation studies

In this section we will analyze two simulation studies, the first study is to analyze the behavior of parameters in different sample sizes, and the second study to assess the performance R, SR and CHSR models among the Akaike information criterion (AIC) introduced by Akaike [15] and Bayesian information criterion (BIC) proposed by Schwarz [16].

Applications

In this section, we analyze two real data sets. The first is a set of Waiting times (in minutes) and the second a set of Failure times of components. In the first application we compare the CHSR model with the SR and R models, and in the second application we compare it with others two-parameters models like the Rayleigh–Lindley (RL) (see Gómez et al. [17]), Weibull (W) and gamma (G) distributions.

Conclusions

This paper presents a new continuous distribution with two parameters, called the CHSR model. The CHSR model is constructed using the slash-type methodology. The new distribution has two parameters and is an alternative for modeling data sets with atypical observations. We study the properties of the CHSR distribution including moments, asymmetry and kurtosis coefficients and stochastic representation. The parameters are estimated by the moments and ML methods. Results of a small scale

Acknowledgments

The authors thank the two anonymous referees and the associate editor for their thorough suggestions and comments that significantly improved the presentation of the paper.

The research of N.M. Olmos was supported by MINEDUC-UA project, Chile, Code ANT 1756 and SEMILLERO, Chile UA-2020. The research of Yolanda M. Gómez was support by proyecto DIUDA programa de inserción N 22367 of the Universidad de Atacama, Chile. The research of Osvaldo Venegas was support by Vicerrectoría de Investigación y

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