Rayleigh Mixture Distribution

Abstract

This paper presents Rayleigh mixtures of distributions in which the weight functions are assumed to be chi-square, 𝑡 and 𝐹 sampling distributions. The exact probability density functions of the mixture of two correlated Rayleigh random variables have been derived. Different moments, characteristic functions, shape characteristics, and the estimates of the parameters of the proposed mixture distributions using method of moments have also been provided.

1. Introduction

In statistics, a mixture distribution is expressed as a convex combination of other probability distributions. It can be used to model a statistical population with subpopulations, where components of mixture probability densities are the densities of the subpopulations, and the weights are the proportion of each subpopulation in the overall population. Mixture distribution may suitably be used for certain data set where different subsets of the whole data set possess different properties that can best be modeled separately. They can be more mathematically manageable, because the individual mixture components are dealt with more nicely than the overall mixture density. The families of mixture distributions have a wider range of applications in different fields such as fisheries, agriculture, botany, economics, medicine, genetics, psychology, paleontoogy, electrophoresis, finance, communication theory, sedimentology/geology, and zoology.

Pearson [1] is considered as the torch bearer in the field of mixtures distributions. He studied the estimation of the parameters of the mixture of two normal distributions. After a long period of time, some basic properties of mixture distributions were studied by Robins (1948). Some of other researchers [2–5] have studied in greater detail the finite mixture of distributions. Roy et al. [6–12] defined and studied poisson, binomial, negative binomial, gamma, chi-square and Erlang mixtures of some standard distributions. In the light of the above-mentioned distributions, here we have studied Rayleigh mixtures of distributions in which the weight functions are assumed to be chi-square, 𝑡- and 𝐹-distribution, and the moments, characteristic function, and shape characteristics of these mixtures distributions have also been studied.

2. Preliminaries

Suppose the random variable 𝑋 has a probability density function (pdf) 𝑓(𝑥∣𝜃) and if the parameter space Θ is a discrete random variable containing parameter values 𝜃₁,𝜃₂,…,𝜃_𝑘 such that the distribution of Θ is 𝑃(Θ=𝜃_𝑖)=𝑝_𝑖, then the unconditional distribution of 𝑋 is𝑚(𝑥)=^𝑘_𝑖=1𝑝_𝑖𝑓𝑥∣𝜃_𝑖. (2.1)

This is called a mixture of the distributions 𝑓(𝑥∣𝜃_𝑖) with weight 𝑝_𝑖,𝑖=1,2,…,𝑘. The above definition may be extended to the case for large 𝑘.

It can be generalized to the case when the parameter space Θ is absolutely continuous random variable having pdf  𝜏(𝜃). We will have, then, a continuous mixture of densities 𝑓(𝑥∣𝜃) with weight function 𝜏(𝜃). In this case, the unconditional distribution of 𝑋 is 𝑚(𝑥)=_Θ𝑓(𝑥∣𝜃)𝜏(𝜃)𝑑𝜃. (2.2)

3. Main Results

In this paper we first define the general form of Rayleigh mixture distribution. Then we furnished the Rayleigh mixture of some well-known sampling distributions such as chi-square, 𝑡- and 𝐹-distributions. The exact distribution of the mixture of two correlated Rayleigh distributions has been studied.

The main results of this study have been presented in the form of some definitions and theorems.

Definition 1. A random variable 𝑋 is said to have Rayleigh mixture distribution if its probability density function is defined by 𝑓𝑥;𝜎²,𝑛=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝜏(𝑥,𝑟;𝑛)𝑑𝑟, (3.1) where 𝜏(𝑥,𝑟;𝜏,𝑛) is a probability density function or any sampling distribution such as chi-square, 𝑡- and 𝐹-distribution.
The name Rayleigh mixture distributions is given due to the fact that the derived distribution (3.1) is the weighted sum of 𝜏(𝑥,𝑟;𝜏,𝑛) with weight factor equal to the probabilities of Rayleigh distribution.

3.1. Formulation of Rayleigh Mixture Distribution

The Rayleigh mixtures of distributions in which the weight functions are assumed to be chi-square, 𝑡- and 𝐹-distribution. In a statistical theory, we will use chi-square distribution as a weight function if sampling statistic follows chi-square distribution. For example, sampling variance is followed by chi-square distribution and we can use chi-square distribution as a weight function. Similarly, we will use 𝑡-distribution and 𝐹-distribution if and only if sampling statistic follows 𝑡-distribution and 𝐹-distribution, respectively. For example, if population variance is unknown and sample size is very small, then the sampling mean follows 𝑡-distribution and the ratio of sampling variances follows 𝐹-distribution. Now we define Rayleigh mixtures of distributions for different weight functions as follows.

3.1.1. Rayleigh Mixtures of Chi-Square Distribution

Definition 2 .1. A random variable 𝜒² is said to have a Rayleigh mixture of chi-square distribution with parameter 𝜎² with degrees of freedom 𝑛 if its probability density function is defined by 𝑓𝜒²;𝜎²,𝑛=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑒^−𝜒2/2𝜒^{2(𝑛/2)+𝑟−1}2^{(𝑛/2)+𝑟})𝑛/2+𝑟𝑑𝑟;0<𝜒²<∞, (3.2) where the weight function 𝜏(𝑥,𝑟;𝜏,𝑛) in (3.1) is the chi-square sampling distribution. Here the notation “)” in (3.2) is a gamma function such that )𝑎=(𝑎−1)!=(𝑎−1)(𝑎−2)⋯3⋅2⋅1.

3.1.2. Rayleigh Mixtures of 𝑡-Distribution

Definition 3. A random variable 𝑡 is defined to have a Rayleigh mixture of 𝑡-distributions with parameter 𝜎² and degrees of freedom 𝑛 if its probability density function is defined as 𝑓𝑡;𝜎²,𝑛=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑡^2𝑟𝑛^(1/2)+𝑟𝐵(1/2+𝑟,𝑛/2)1+𝑡²/𝑛^{(𝑛+1)/2+𝑟}𝑑𝑟,−∞<𝑡<∞, (3.3) where the weight function 𝜏(𝑥,𝑟;𝜏,𝑛) in (3.1) is the student 𝑡-distribution.

3.1.3. Rayleigh Mixtures of 𝐹-Distribution

Definition 4. A random variable 𝐹 is defined to have a Rayleigh mixture of 𝐹-distributions with parameter 𝜎² and degrees of freedom 𝑛₁ and 𝑛₂, if its probability density function is defined as 𝑓𝐹;𝜎²,𝑛₁,𝑛₂=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²(𝑛₁/𝑛₂)^{(𝑛1/2)+𝑟}𝐹^{𝑛1/2+𝑟−1}𝐵𝑛₁/2+𝑟,𝑛₂/21+(𝑛₁/𝑛₂)𝐹^{(𝑛1+𝑛2)/2+𝑟}𝑑𝑟,0<𝐹<∞, (3.4) where the weight function 𝜏(𝑥,𝑟;𝜏,𝑛) in (3.1) is the 𝐹-distribution.

3.1.4. Mixture of Two Correlated Rayleigh Distributions

Let 𝑋 and 𝑌 be two independent Rayleigh variables with probability density function (pdf). The joint distribution of 𝑋 and 𝑌 with correlation coefficient 𝜌(−1≤𝜌≤1) can be constructed by the following formula:𝑓𝑥,𝑦=𝑓(𝑥)𝑔𝑦1+𝜌(1−2𝐹(𝑥))×1−2𝐺𝑦 (3.5) which was developed by Farlie-Gumbl-Morgenstern (1979).

Using the formula, the mixture of two correlated distributions is as follows:𝑓𝑥,𝑦;𝜎₁,𝜎₁;𝜌=𝑥𝑦𝜎₁𝜎₂²×𝑒^{−(1/2)((𝑥2/𝜎21)+(𝑦2/𝜎22))}+𝜌𝑒^{−(1/2)((𝑥2/𝜎21)+(𝑦2/𝜎22))}−2𝜌𝑒^{−(1/2)((2𝑥2/𝜎21)+(𝑦2/𝜎22))}−2𝜌𝑒^{−(1/2)((𝑥2/𝜎21)+(2𝑦2/𝜎22))}+4𝜌𝑒^{−((𝑥2/𝜎21)+(𝑦2/𝜎22))}, (3.6)

where 𝑥>0, 𝑦>0; 𝜎₁,𝜎₂>0 and −1≤𝜌≤1.

3.2. Derivation of Characteristics of Rayleigh Mixture Distribution

Moments and different characteristics of the Rayleigh mixture of distributions are presented by the following theorems.

Theorem 3.1. If 𝜒² follows a Rayleigh mixture of chi-square distribution with parameter 𝜎² with degrees of freedom 𝑛, then the 𝑠th raw moment of this mixture distribution about origin is given by 𝜇^′_𝑠=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²2^𝑠)𝑛/2+𝑟+𝑠)𝑛/2+𝑟𝑑𝑟. (3.7) Hence, the mean and the variance of this mixture distribution are as follows: Mean=𝑛+𝜎√√2𝜋,Variance=2𝑛+2𝜎2𝜋+2𝜎²(4−𝜋). (3.8)

Proof. We know the 𝑠th raw moment defined by 𝜇^′_𝑠=𝐸𝜒^2𝑠=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑒^−𝜒2/2𝜒^{2𝑛/2+𝑟+𝑠−1}2^𝑛/2+𝑟)𝑛/2+𝑟𝑑𝑟𝑑𝜒²=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²2^𝑠)𝑛/2+𝑟+𝑠)𝑛/2+𝑟𝑑𝑟. (3.9) If we put 𝑠=1 in (3.9), we get 𝜇^′₁=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²2)𝑛/2+𝑟+1√)𝑛/2+𝑟𝑑𝑟=𝑛+𝜎2𝜋. (3.10) If we put 𝑠=2 in (3.9), we have 𝜇^′₂=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²2²)𝑛/2+𝑟+2)𝑛/2+𝑟𝑑𝑟=4^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²(𝑛/2+𝑟+1)(𝑛/2+𝑟)𝑑𝑟=𝑛²+4𝜎(𝑛+1)√2)32+2𝑛+8𝜎²Onsimplification=𝑛²+𝜎(𝑛+1)2√2𝜋+2𝑛+8𝜎². (3.11) Hence, the variance is defined by 𝜇₂=𝜇^′₂−𝜇^′₁²=2𝑛+2𝜎√2𝜋+2𝜎²(4−𝜋). (3.12) This completes the proof.

Theorem 3.2. If 𝜒² follows a Rayleigh mixture of chi-square distributions with parameter 𝜎² and degrees of freedom 𝑛, then its characteristic function is given by Φ_𝜒2(𝑡)=(1−2𝑖𝑡)^−𝑛/2∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²(1−2𝑖𝑡)^−𝑟𝑑𝑟. (3.13)

Proof. The characteristic function is defined as Φ_𝜒2(𝑡)=𝐸𝑒^{𝑖𝑡𝜒2}=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑒^{−(𝜒2/2)(1−2𝑖𝑡)}𝜒^{2𝑛/2+𝑟−1}2^𝑛/2+𝑟)𝑛/2+𝑟𝑑𝑟𝑑𝜒²=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²1(1−2𝑖𝑡)^𝑛/2+𝑟=()𝑑𝑟1−2𝑖𝑡^−𝑛/2∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²(1−2𝑖𝑡)^−𝑟𝑑𝑟 (3.14) and hence proved

Theorem 3.3. If 𝑡 follows a Rayleigh mixture of 𝑡-distributions with parameter 𝜎² and degrees of freedom 𝑛, then the 𝑠th raw moment about origin is 𝜇^′_2𝑠+1=𝜇_2𝑠+1𝜇=0,_′2𝑠=𝜇_2𝑠=𝑛^𝑠∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑟+𝑠+1/2)𝑛/2−𝑠)𝑟+1/2)𝑛/2𝑑𝑟. (3.15) And hence 𝑛Mean=0,Variance=()√𝑛−21+𝜎2𝜋for𝑛>2. (3.16) Therefore, Skewness:𝛽₁=0,Kurtosis:𝛽₂=(𝑛−2)2𝜎²+2𝜎√2𝜋+3(𝑛−4)1+𝜎√2𝜋²,𝑛>4. (3.17)

Proof. The (2𝑠+1)th raw moment (odd order moments) about origin is given by 𝜇^′_2𝑠+1=𝐸𝑡^2𝑠+1=^∞_−∞^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑡^{2𝑟+2𝑠+1}𝑛^1/2+𝑟𝐵(1/2+𝑟,𝑛/2)1+𝑡²/𝑛^{(𝑛+1)/2+𝑟}=𝑑𝑟𝑑𝑡_∞0𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑛^1/2+𝑟𝐵(1/2+𝑟,𝑛/2)^∞_−∞𝑡^{2𝑟+2𝑠+1}1+𝑡²/𝑛^{(𝑛+1)/2+𝑟}𝑑𝑡𝑑𝑟=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑛^1/2+𝑟𝐵(1/2+𝑟,𝑛/2)^∞_−∞𝜓(𝑡)(𝑡)=𝑡𝑑𝑡𝑑𝑟=0Since,𝜓^{2𝑟+2𝑠+1}1+𝑡²/𝑛^{(𝑛+1)/2+𝑟}isanoddfunctionof𝑡. (3.18) If 𝑠=0, then 𝜇^′₁=Mean=0. If 𝑠=1 then 𝜇^′₃=𝜇₃=0. So, 𝜇^′_2𝑠+1=𝜇_2𝑠+1=0.
Now, the (2𝑠)th raw moment about origin is given by 𝜇_2𝑠=𝜇^′_2𝑠=𝐸𝑡^2𝑠=^∞_−∞^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑡^2𝑟+2𝑠𝑛^1/2+𝑟𝐵(1/2+𝑟,𝑛/2)1+𝑡²/𝑛^{(𝑛+1)/2+𝑟}=𝑑𝑟𝑑𝑡_∞0𝑟𝑒^{−𝑟2/2𝜎2}𝑛^{𝑠+𝑟+1/2}𝜎²𝑛^1/2+𝑟𝐵(1/2+𝑟,𝑛/2)^∞₀𝑡²/𝑛^{𝑠+𝑟+1/2−1}1+𝑡²/𝑛^{(𝑛+1)/2+𝑟}𝑑𝑡²𝑛𝑑𝑟=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝑛^{𝑠+𝑟+1/2}𝜎²𝑛^1/2+𝑟𝐵(1/2+𝑟,𝑛/2)^∞₀𝑢^{𝑠+𝑟+1/2−1}(1+𝑢)^{(𝑛+1)/2+𝑟}𝑡𝑑𝑢𝑑𝑟;Putting𝑢=²𝑛=𝑛^𝑠∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑟+𝑠+1/2)𝑛/2−𝑠)𝑟+1/2)𝑛/2𝑑𝑟. (3.19) If 𝑠=1 then, 𝜇^′₂=𝜇₂=𝑛^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑟+3/2)𝑛/2−1)𝑟+1/2=𝑛)𝑛/2𝑑𝑟+𝑛−22𝑛√𝑛−22𝜎)32=𝑛;𝑛>2√𝑛−21+𝜎2𝜋;∵)32=√𝜋2. (3.20) If 𝑠=2, then,𝜇^′₄=𝜇₄=𝑛^2∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑟+5/2)𝑛/2−2)𝑟+(1/2)=𝑛)𝑛/2𝑑𝑟₂(𝑛−2)(𝑛−4)^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑟²+4𝑟+3𝑑𝑟Since,)𝑟+5/2)𝑛/2−2)𝑟+1/2=())𝑛/2𝑟+3/2)(𝑟+1/2()=𝑟𝑛/2−1)(𝑛/2−2²+4𝑟+3().𝑛−2)(𝑛−4 (3.21)
Hence, 𝜇^′₄=𝜇₄=𝑛²(𝑛−2)(𝑛−4)2𝜎²)2+4𝜎√2)32=𝑛+3₂(𝑛−2)(𝑛−4)2𝜎²+2𝜎√2𝜋+3,𝑛>4. (3.22) To find the Skewness and Kurtosis of this mixture distribution Skewness:𝛽₁=𝜇²₃𝜇³₂=0,Kurtosis:𝛽₂=𝜇₄𝜇²₂=(𝑛−2)2𝜎²+2𝜎√2𝜋+3(𝑛−4)1+𝜎√2𝜋². (3.23) This completes the proof.

Theorem 3.4. If 𝐹 follows a Rayleigh mixture of 𝐹-distributions having parameter 𝜎² with degrees of freedom 𝑛₁ and 𝑛₂, respectively, then the 𝑠th raw moment about origin is given by 𝜇^′_𝑠=𝑛₂𝑛₁^𝑠∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑛₁/2+𝑟+𝑠)𝑛₂/2−𝑠)𝑛₁/2+𝑟)𝑛₂/2𝑑𝑟. (3.24) Hence, the mean and variance of this distribution are 𝑛Mean=²𝑛₁𝑛₂𝑛−2¹+𝜎√,𝑛2𝜋Variance=²𝑛₁²𝑛²₁+2𝑛₁+2𝑛₁+1𝜎√2𝜋+8𝜎²𝑛₂−2𝑛₂−𝑛−4¹+𝜎√2𝜋²𝑛₂−2², (3.25) respectively.

Proof. The 𝑠th raw moment about origin is given by 𝜇^′_𝑠=𝐸𝐹^𝑠=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑛₁/𝑛₂^𝑛1/2+𝑟𝐹^{𝑛1/2+𝑟+𝑠−1}𝐵𝑛₁/2+𝑟,𝑛₂/21+𝑛₁/𝑛₂𝐹^{(𝑛1+𝑛2)/2+𝑟}=𝑑𝑟𝑑𝐹_∞0𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑛₁/𝑛₂^𝑛1/2+𝑟𝐵𝑛₁/2+𝑟,𝑛₂/2^∞₀𝐹^{𝑛1/2+𝑟+𝑠−1}1+(𝑛₁/𝑛₂)𝐹^{(𝑛1+𝑛2)/2+𝑟}=𝑛𝑑𝐹𝑑𝑟₂𝑛₁^𝑠∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑛₁/2+𝑟+𝑠)𝑛₂/2−𝑠)𝑛₁/2+𝑟)𝑛₂/2𝑑𝑟. (3.26) If we put 𝑠=1, we get Mean=𝜇^′₁=𝑛₂𝑛₁^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑛₁/2+𝑟+1)𝑛₂/2−1)𝑛₁/2+𝑟)𝑛₂=𝑛/2𝑑𝑟₂𝑛₁𝑛₂𝑛−2¹+𝜎√2𝜋. (3.27) Putting 𝑠=2, we get 𝜇^′₂=𝑛₂𝑛₁^2∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑛₁/2+𝑟+2)𝑛₂/2−2)𝑛₁/2+𝑟)𝑛₂=𝑛/2𝑑𝑟₂𝑛₁²1𝑛₂−2𝑛₂𝑛−4²¹+2𝑛₁+2𝑛₁+1𝜎√2𝜋+8𝜎². (3.28) Then the variance, 𝜇₂=𝜇^′₂−𝜇^′₁²=𝑛₂𝑛₁²𝑛²₁+2𝑛₁+2𝑛₁+1𝜎√2𝜋+8𝜎²𝑛₂−2𝑛₂−𝑛−4¹+𝜎√2𝜋²𝑛₂−2², (3.29) hence proved.

Theorem 3.5. If 𝐹 follows a Rayleigh mixture of 𝐹-distributions having parameter 𝜎² with degrees of freedom 𝑛₁ and 𝑛₂, respectively, then its characteristic function is given by Φ_𝐹(𝑡)=^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎^2∞_𝑥=01⋅𝑛𝑥!²𝑖𝑡𝑛₁^𝑥)𝑛₁/2+𝑟+𝑥)𝑛₂/2−𝑥)𝑛₁/2+𝑟)𝑛₂/2𝑑𝑟. (3.30)

From here we may get the mean and variance of this distribution.

Proof. The characteristic function of the random variable 𝐹 is given by Φ_𝐹(𝑡)=𝐸𝑒^𝑖𝑡𝐹=^∞₀𝑒^{𝑖𝑡𝐹∞}₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²𝑛₁/𝑛₂^𝑛1/2+𝑟𝐹^{𝑛1/2+𝑟−1}𝐵𝑛₁/2+𝑟,𝑛₂/21+(𝑛₁/𝑛₂)𝐹^{(𝑛1+𝑛2)/2+𝑟}=𝑑𝑟𝑑𝐹_∞0𝑟𝑒^{−𝑟2/2𝜎2}𝜎^2∞_𝑥=01⋅𝑛𝑥!²𝑖𝑡𝑛₁^𝑥)𝑛₁/2+𝑟+𝑥)𝑛₂/2−𝑥)𝑛₁/2+𝑟)𝑛₂/2𝑑𝑟. (3.31) Hence the 𝑠th raw moment about origin is 𝜇^′_𝑠()=coefficientof𝑖𝑡^𝑠𝑠!inΦ_𝐹(𝑡)=𝑛₂𝑛₁^𝑠∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑛₁/2+𝑟+𝑠)𝑛₂/2−𝑠)𝑛₁/2+𝑟)𝑛₂/2𝑑𝑟. (3.32) If 𝑠=1, then 𝜇^′₁=𝑛₂𝑛₁^∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑛₁/2+𝑟+1)𝑛₂/2−1)𝑛₁/2+𝑟)𝑛₂=𝑛/2𝑑𝑟₂𝑛₁𝑛₂𝑛−2¹+𝜎√2𝜋. (3.33) If we put 𝑠=2, we get 𝜇^′₂=𝑛₂𝑛₁^2∞₀𝑟𝑒^{−𝑟2/2𝜎2}𝜎²)𝑛₁/2+𝑟+2)𝑛₂/2−2)𝑛₁/2+𝑟)𝑛₂=𝑛/2𝑑𝑟₂𝑛₁²1𝑛₂−2𝑛₂𝑛−4²¹+2𝑛₁+2𝑛₁+1𝜎√2𝜋+8𝜎². (3.34) Therefore, Variance:𝜇₂=𝜇^′₂−𝜇^′₁²=𝑛₂𝑛₁²𝑛²₁+2𝑛₁+2𝑛₁+1𝜎√2𝜋+8𝜎²𝑛₂−2𝑛₂−𝑛−4¹+𝜎√2𝜋²𝑛₂−2². (3.35) Driving coefficient of Skewness = 𝛽₁ and coefficient of Kurtosis =𝛽₂ is a tedious job; we have avoided the task here.

The different moments of the random variable which is the resultant of the product of two correlated Rayleigh random variables are obtained by following theorem.

Theorem 3.6. For  −1≤𝜌≤1, the (𝑎,𝑏)th product moment of the mixture of two correlated Rayleigh random variables is denoted by 𝜇′(𝑎,𝑏;𝜌) and given by 𝜇^′𝑎,𝑏;𝜌=𝜎^𝑎₁𝜎^𝑏₂𝑎×Γ2Γ𝑏+12+1×2^𝑎+𝑏/2+𝜌2^𝑎/2−12^𝑏/2−1. (3.36)

Proof. We know that 𝜇′𝑎,𝑏;𝜌=𝐸𝑋^𝑎𝑌^𝑏=^∞₀𝑥^𝑎𝑦^𝑏×𝑥𝑦𝜎₁𝜎₂²×𝑒^{−(1/2)(𝑥2/𝜎21+𝑦2/𝜎22)}+𝜌𝑒^{−(1/2)(𝑥2/𝜎21+𝑦2/𝜎22)}−2𝜌𝑒^{−(1/2)(2𝑥2/𝜎21+𝑦2/𝜎22)}−2𝜌𝑒^{−(1/2)(𝑥2/𝜎21+2𝑦2/𝜎22)}+4𝜌𝑒^{−(𝑥2/𝜎21+𝑦2/𝜎22)}=𝑑𝑥𝑑𝑦_∞0𝑥^𝑎+1𝑦^𝑏+1𝜎₁𝜎₂²×𝑒^{−(1/2)(𝑥2/𝜎21+𝑦2/𝜎22)}𝑑𝑥𝑑𝑦+𝜌^∞₀𝑥^𝑎+1𝑦^𝑏+1𝜎₁𝜎₂²×𝑒^{−(1/2)(𝑥2/𝜎21+𝑦2/𝜎22)}𝑑𝑥𝑑𝑦−2𝜌^∞₀𝑥^𝑎+1𝑦^𝑏+1𝜎₁𝜎₂²×𝑒^{−(1/2)(2𝑥2/𝜎21+𝑦2/𝜎22)}𝑑𝑥𝑑𝑦−2𝜌^∞₀𝑥^𝑎+1𝑦^𝑏+1𝜎₁𝜎₂²×𝑒^{−(1/2)(𝑥2/𝜎21+2𝑦2/𝜎22)}𝑑𝑥𝑑𝑦+4𝜌^∞₀𝑥^𝑎+1𝑦^𝑏+1𝜎₁𝜎₂²×𝑒^{−(𝑥2/𝜎21+𝑦2/𝜎22)}𝑑𝑥𝑑𝑦. (3.37) Now taking the first integral from (3.37)^∞₀𝑥^𝑎+1𝑦^𝑏+1𝜎₁𝜎₂²×𝑒^{−(1/2)(𝑥2/𝜎21+𝑦2/𝜎22)}𝑑𝑥𝑑𝑦 (3.38) and making a transformation 𝑝=(1/2)(𝑥²/𝜎²₁) and 𝑞=(1/2)(𝑦²/𝜎²₂) we obtain the following result: 𝜎₁√2^𝑎∞₀𝑝^𝑎/2𝑒^−𝑝𝑑𝑝×𝜎₂√2^𝑏∞₀𝑞^𝑏/2𝑒^−𝑞𝑑𝑞=2^{(𝑎+𝑏)/2}𝜎^𝑎₁𝜎^𝑏₂𝑎×Γ2Γ𝑏+12+1. (3.39) Using the similar mathematical simplification we get the following results for the 2nd integral 2^{(𝑎+𝑏)/2}𝜌𝜎^𝑎₁𝜎^𝑏₂𝑎×Γ2Γ𝑏+12+1, (3.40) for the 3rd integral 2^𝑏/2𝜌𝜎^𝑎₁𝜎^𝑏₂𝑎×Γ2Γ𝑏+12+1, (3.41) for the 4th integral 2^𝑎/2𝜌𝜎^𝑎₁𝜎^𝑏₂𝑎×Γ2Γ𝑏+12+1, (3.42) for the 5th integral 𝜌𝜎^𝑎₁𝜎^𝑏₂𝑎×Γ2Γ𝑏+12+1. (3.43) Now putting all of these values in (3.37) and simplifying the result we get our desired result stated in the theorem.
Special findings of the above theorem if 𝜌=0 then the product moment of the two correlated Rayleigh variables is nothing but the product of 𝑎th and 𝑏th moments of two independent Rayleigh variables. In such case the product moment is as follows: 𝐸𝑋^𝑎𝑌^𝑏=2^{(𝑎+𝑏)/2}×𝜎^𝑎₁𝜎^𝑏₂𝑎×Γ2Γ𝑏+12+1=2^𝑎/2𝜎^𝑎₁Γ𝑎2+1×2^𝑏/2𝜎^𝑏₂Γ𝑏2𝑋+1=𝐸^𝑎⋅𝐸𝑌^𝑏. (3.44)

Theorem 3.7. If 𝑋 and 𝑌 are two correlated Rayleigh variates having joint density given in (3.6), then probability density function of 𝑊=𝑋/𝑌 is given by 𝑓𝑤;𝜎₁,𝜎₂;𝜌=𝜋2×𝑤²𝜎₁𝜎₂×1+𝜌1+√2𝑤²𝜎²₂+𝜎²₁^−(3/2)−2𝜌2𝑤²𝜎²₂+𝜎²₁^−(3/2)+𝑤²𝜎²₂+2𝜎²₁^−(3/2), (3.45) where, 𝑤>0;  𝜎₁>0,  𝜎₂>0 and −1≤𝜌<1.

Proof. Under the transformation 𝑥=𝑧, 𝑦=𝑧/𝑤 in (3.6) with the Jacobean 𝐽⟶()=𝑤𝑥,𝑦𝑤,𝑧²𝑧, (3.46) the pdf of 𝑤 and 𝑧 is given by 𝑓()=𝑧𝑤,𝑧²𝑤𝜎₁𝜎₂²1+𝜌𝑒^{−𝑧2/2(1/𝜎21+1/𝑤2𝜎22)}−2𝜌𝑒^{−𝑧2/2(2/𝜎21+1/𝑤2𝜎22)}−2𝜌𝑒^{−𝑧2/2(1/𝜎21+2/𝑤2𝜎22)}+4𝜌𝑒^{−𝑧2(1/𝜎21+1/𝑤2𝜎22)}. (3.47) Now integrating over 𝑧 we get the marginal distribution of 𝑤 as 𝑓(𝑤)=1+𝜌𝑤𝜎₁𝜎₂^2∞0𝑧²𝑒^{−𝑧2/2(1/𝜎21+1/𝑤2𝜎22)}−𝑑𝑧2𝜌𝑤𝜎₁𝜎₂^2∞0𝑧²𝑒^{−𝑧2/2(2/𝜎21+1/𝑤2𝜎22)}−𝑑𝑧2𝜌𝑤𝜎₁𝜎₂^2∞0𝑧²𝑒^{−𝑧2/2(1/𝜎21+2/𝑤2𝜎22)}+𝑑𝑧4𝜌𝑤𝜎₁𝜎₂^2∞0𝑧²𝑒^{−𝑧2(1/𝜎21+1/𝑤2𝜎22)}𝑑𝑧. (3.48) Taking each of the integral separately and making transformation 𝑧=12𝑝𝜎²₁+1𝑤²𝜎²₂^−1/2forthefirstintegral,𝑧=22𝑝𝜎²₁+1𝑤²𝜎²₂^−1/2forthesecondintegral,𝑧=12𝑝𝜎²₁+2𝑤²𝜎²₂^−1/2√forthethirdintegral,𝑧=𝑝1𝜎²₁+1𝑤²𝜎²₂^−1/2forthefourthintegral. (3.49) We have got the following results: 𝜋21𝜎²₁+1𝑤²𝜎²₂^−3/2,𝜋22𝜎²₁+1𝑤²𝜎²₂^−3/2,𝜋21𝜎²₁+2𝑤²𝜎²₂^−3/2,√𝜋41𝜎²₁+1𝑤²𝜎²₂^−3/2 (3.50) for the first, second, third, and fourth integration, respectively.

Combining all of the obtained results for the integrals in (3.48) we achieve the result stated as in the theorem.

Theorem 3.8. For nonnegative integer 𝑎 and −1<𝜌<1 the 𝑎th moment for 𝑊=𝑋/𝑌 is 𝐸𝑊^𝑎=1𝜎₂√2𝜎₁𝜎₂^𝑎+1×Γ𝑎+32Γ−𝑎2×1+𝜌1−2√21+2^𝑎+2. (3.51)

Proof. According to definition of expectation we can obtain the following result for the 𝑎th moment: 𝐸𝑊^𝑎=^∞₀𝑤^𝑎+2×𝜋21+𝜌1+√2𝑤²𝜎²₂+𝜎²₁^−3/2−2𝜌2𝑤²𝜎²₂+𝜎²₁^−3/2+𝑤²𝜎²₂+2𝜎²₁^−3/2𝑑𝑤. (3.52) By making transformation 𝑤=(𝜎₁/𝜎₂)√𝑚, 𝑤=(𝜎₁/𝜎₂)𝑚/2 and 𝑤=(𝜎₁/𝜎₂)√2𝑚 for the first, second, and third parts of the component containing the integral we have 𝐸𝑊^𝑎=√𝜋2^3/2×𝜎₁^𝑎+1𝜎₂^𝑎+21+𝜌1+√2×𝐵𝑎+32𝑎,−2−𝜌√𝜋2^(𝑎+4)/2×𝜎₁^𝑎+1𝜎₂^𝑎+21+𝜌1+√2×𝐵𝑎+32𝑎,−2−𝜌√𝜋2^{−(𝑎+2)/2}×𝜎₁^𝑎+1𝜎₂^𝑎+21+𝜌1+√2×𝐵𝑎+32𝑎,−2. (3.53) Simplifying this we have the stated result of the theorem.

Theorem 3.9. The moment generating function of 𝑊 is 𝑀_𝑊(𝑡)=^∞_𝑎=𝑜𝑡^𝑎×1𝑎!𝜎₂√2𝜎₁𝜎₂^𝑎+1×Γ𝑎+32Γ−𝑎2×1+𝜌1−2√21+2^𝑎+2. (3.54)

Proof. The moment generating function of 𝑉 at 𝑡 is given by 𝑀_𝑊(𝑡)=𝐸𝑒^𝑡𝑤=^∞_𝑎=0𝑡^𝑎𝐸𝑊𝑎!^𝑎=^∞_𝑎=𝑜𝑡^𝑎×1𝑎!𝜎₂√2𝜎₁𝜎₂^𝑎+1×Γ𝑎+32Γ−𝑎2×1+𝜌1−2√21+2^𝑎+2. (3.55)

3.3. Parameter Estimation of Rayleigh Mixture Distribution

We know the well-known method of the maximum likelihood estimation is very complicated for the parameter estimation of mixture distribution and method of moment is very suitable in these cases. Hence, we used method of moments (MoMs) estimation of technique for estimation of the parameter of the Rayleigh mixture distribution.

3.3.1. Parameter Estimation of Rayleigh Mixture of Chi-Square Distribution

Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be a random sample from the distribution defined in (3.2) where the parameter 𝜎² is unknown.

The first sample raw moment is𝑚^′₁=1𝑛^𝑛_𝑖=1𝑋_𝑖=𝑋say. (3.56)

And as we already got𝜇^′₁=𝑛+𝜎√2𝜋. (3.57)

Hence𝑛+𝜎√2𝜋=𝑋. (3.58)

Therefore,𝜎²=𝑋−𝑛²2𝜋. (3.59)

3.3.2. Parameter Estimation of Rayleigh Mixture of 𝑡-Distribution

Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be a random sample from the distribution defined in (3.3) where the parameter 𝜎² is unknown. We want to estimate this parameter by method of moment.

The second sample raw moment is obtained as 𝑚^′₂=1𝑛^𝑛_𝑖=1𝑋²_𝑖=𝑆²say. (3.60)

We have already found 𝜇^′₂=𝜇₂=𝑛√𝑛−21+𝜎2𝜋,𝑛>2. (3.61)

Hence, by the method of moments, we get𝑛√𝑛−21+𝜎2𝜋=𝑆²,𝑛>2. (3.62)

Therefore,𝜎²=1𝑆2𝜋²(𝑛−2)𝑛−1². (3.63)

3.3.3. Parameter Estimation of Mixture of 𝐹-Distribution

Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be a random sample from the distribution as specified in (3.4) where the parameter 𝜎² is unknown. We want to estimate this parameter by method of moments.

The first sample raw moment is𝑚^′₁=1𝑛^𝑛_𝑖=1𝑋_𝑖=𝑋say. (3.64)

And we already get𝜇^′₁=𝑛₂𝑛₁𝑛₂𝑛−2¹+𝜎√2𝜋. (3.65)

Hence, from the method of moments estimator is𝑛₂𝑛₁𝑛₂𝑛−2¹+𝜎√2𝜋=𝑋. (3.66)

Therefore,𝜎²=1𝑛2𝜋𝑛¹₂²𝑋𝑛₂−2−𝑛₁². (3.67)

4. Concluding Remarks

In this paper, we have presented the Rayleigh mixtures of distributions in which the weight functions are assumed to be chi-square, 𝑡- and 𝐹-distributions, and the mixture of two correlated Rayleigh distributions has been presented. The moments, characteristic function and shape characteristics of these mixtures distributions have also been studied. The Rayleigh distribution is frequently used to model wave heights in oceanography and in communication theory to describe hourly median and instantaneous peak power of received radio signals. It could also be used to model the frequency of different wind speeds over a year at wind turbine sites. The Rayleigh mixture of sampling distribution may be used in the similar nature but with some additional informative environment. Suppose we want to know the distribution of the average fish caught by fisherman in the Bay of Bengal of a particular day. Fishing depends on height of the wave and wind speed in that zone. As we know the average amount fish catch by the fisherman depends on the weather of the Sea. If the wave heights are very high the fishermen are prohibited to go to the sea for fishing if it is not so much dangerous but still the sea is unstable they are asked to be very careful during fishing. This means that average amount of fishing and standard deviation of the amount fish catch by the fishermen varies based on heights of the wave. The distribution of wave heights follows Rayleigh distribution and distribution of average catch fish at a normal situation follows 𝑡-distribution but at the Bay of Bangle it is seriously affected by height of wave; hence the average number of fish catch at the Bay of Bangle will follow Rayleigh mixture of 𝑡-distribution. Similarly, the distribution of the variability of the number of fishes catch by the fishermen at the Bay of Bangle follows Rayleigh mixture of chi-square distribution. We hope the findings of the paper will be useful for the practitioners that have been mentioned above.

Acknowledgment

The authors would like to thank the editor and referee for their useful comments and suggestions which considerably improved the quality of the paper.