ON THE RATIO OF PEARSON TYPE VII AND BESSEL RANDOM VARIABLES

For given random variables X and Y , the distribution of the ratio X/Y is of interest in biological and physical sciences, econometrics, and ranking and selection. Examples include Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, and inventory ratios in economics. Another important example is the stress-strength model in the context of reliability. It describes the life of a component which has a random strength Y and is subjected to random stress X . The component fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily whenever Y > X . Thus, Pr(X < Y) is a measure of a component reliability. It has many applications especially in engineering concepts such as structures, deterioration of rocket motors, static fatigue of ceramic components, fatigue failure of aircraft structures and aging of concrete pressure vessels. The distribution of X/Y has been studied by several authors especially when X and Y are independent random variables and come from the same family. For instance, see Marsaglia [14] and Korhonen and Narula [9] for normal family, Press [18] for Student’s t family, Basu and Lochner [1] for Weibull family, Shcolnick [23] for stable family, Hawkins and Han [5] for non-central chi-squared family, Provost [19] for gamma family, and Pham-Gia [17] for beta family. However, there is relatively little work of this kind when X and Y belong to different families. In the applications mentioned above, it is indeed quite possible that X and Y could arise from different but similar distributions. Pearson type VII distributions (which contain Student’s t distributions as particular cases) are becoming of increasing importance in classical as well as in Bayesian statistical


Introduction
For given random variables X and Y , the distribution of the ratio X/Y is of interest in biological and physical sciences, econometrics, and ranking and selection.Examples include Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, and inventory ratios in economics.Another important example is the stress-strength model in the context of reliability.It describes the life of a component which has a random strength Y and is subjected to random stress X.The component fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily whenever Y > X.Thus, Pr(X < Y) is a measure of a component reliability.It has many applications especially in engineering concepts such as structures, deterioration of rocket motors, static fatigue of ceramic components, fatigue failure of aircraft structures and aging of concrete pressure vessels.
The distribution of X/Y has been studied by several authors especially when X and Y are independent random variables and come from the same family.For instance, see Marsaglia [14] and Korhonen and Narula [9] for normal family, Press [18] for Student's t family, Basu and Lochner [1] for Weibull family, Shcolnick [23] for stable family, Hawkins and Han [5] for non-central chi-squared family, Provost [19] for gamma family, and Pham-Gia [17] for beta family.However, there is relatively little work of this kind when X and Y belong to different families.In the applications mentioned above, it is indeed quite possible that X and Y could arise from different but similar distributions.
Pearson type VII distributions (which contain Student's t distributions as particular cases) are becoming of increasing importance in classical as well as in Bayesian statistical modeling.These distributions have been perhaps unjustly overshadowed-for at least seventy years-by the normal distribution.Pearson type VII distributions are of central importance in statistical inference.Their applications are a very promising path to take.Classical analysis is soundly bend on the normal distribution while Pearson type VII distributions (in particular, Student's t distributions) offer a more viable alternative with respect to real-world data particularly because its tails are more realistic.We have seen unexpected applications in novel areas such as cluster analysis, discriminant analysis, multiple regression, robust projection indices and missing data imputation.
Pearson type VII distributions (in particular, Student's t distributions) for the past fifty years have also played a crucial role in Bayesian analysis.They serve as the most popular prior distribution (because elicitation of prior information in various physical, engineering and financial phenomena is closely associated with Student's t distributions) and generate meaningful posterior distributions.For further discussion of applications, the reader is referred to Kotz and Nadarajah [11].
On the other hand, Bessel function distributions (which contain logistic distributions as particular cases) have found applications in a variety of areas that range from image and speech recognition and ocean engineering to finance.They are rapidly becoming distributions of first choice whenever "something" with heavier than Gaussian tails is observed in the data.For further discussion of applications, the reader is referred to Kotz et al. [10].
There has been considerable work on the relationship between the Student's t and logistic distributions (which are particular cases of Pearson type VII and Bessel function distributions, resp.).The similarities in the shapes of the logistic and the normal distributions have been noted by several authors.An excellent summary of these properties is found in Johnson and Kotz [7].However, Mudholkar and George [15] showed that the Student's t distribution function with 9 degrees of freedom, when standardized to have variance one provides a better fit of a standardized logistic distribution than the standard normal.George and Ojo [3] and George et al. [2] extended this result by showing that where T is a Student's t random variable with ν degrees of freedom and a=π (ν−2)/(3ν).This approximation (1.1) was obtained by equating the cumulant of the Student's t distribution with that of the logistic distribution.The approximation was found to be accurate to two decimal places for middle values of t and to three decimal places at the tails.A better approximation between the Student's t and logistic distributions has been recently proposed by Li and De Moor [13].
The above discussion naturally raises the important question: what is the exact distribution of the ratio of Student's t and logistic random variables?This question does not appear to have been addressed in the literature.In this paper, we discuss the more general problem: the exact distribution of the ratio |X/Y | when X and Y are independent random S. Nadarajah and S. Kotz 193 variables having the Pearson type VII and Bessel function distributions with the pdfs where is the modified Bessel function of the second kind.We also provide an application section and compute "correction factors" for the approximation provided by (1.1).The Pearson type VII distribution is related to the Student's t distribution as follows: then U is a Student's t random variable with degrees of freedom a.Note that the pdf of a Student's t random variable with degrees of freedom ν is given by for −∞ < x < ∞.Nadarajah and Kotz [16] have shown that the cdf corresponding to (1.6) can be expressed as The representations in (1.5) and (1.7) will be crucial for the calculations of this note.The calculations involve the Euler psi function defined by the Struve function defined by the Bessel function of the first kind defined by the hypergeometric functions defined by and, the Lommel functions defined by where (e) k = e(e + 1)•••(e + k − 1) denotes the ascending factorial.We also need the following important lemma. (1.13) Further properties of the above special functions can be found in Prudnikov et al. [20,21] and Gradshteyn and Ryzhik [4].

Cdf
where I(•) denotes the integral where F(•) inside the integrals denotes the cdf of a Student's t random variable with degrees of freedom a. Substituting the form for F given by (1.7) for odd degrees of freedom, (2.3) can be reduced to where J(k) denotes the integral By direct application of Lemma 1.1, one can calculate (2.5) as The result of the theorem follows by substituting (2.6) into (2.4).
Theorem 2.2 is the analogue of Theorem 2.1 for the case when the degrees of freedom 2(M − 1) is an even integer.Theorem 2.2.Suppose X and Y are distributed according to (1.2) and (1.3), respectively.If a = 2(M − 1) is an even integer then the cdf of Z = |X/Y | can be expressed as where r = √ a/Nz.
Proof.Substituting the form for F given by (1.7) for odd degrees of freedom, (2.3) can be reduced to where J(k) denotes the integral By direct application of Lemma 1.1, one can calculate (2.9) as (2.10) The result of the theorem follows by substituting (2.10) into (2.8).

Theorem 2 .
1 derives an explicit expression for the cdf of |X/Y | in terms of the Lommel function.S. Nadarajah and S. Kotz 195 Theorem 2.1.Suppose X and Y are distributed according to (1.2) and (1.3), respectively.If a = 2(M − 1) is an odd integer then the cdf of Z = |X/Y | can be expressed as

Figure 2 .
Figure 2.1 illustrates possible shapes of the pdf of |X/Y | for M = 6, N = 1 and a range of values of m.Note that the shapes are unimodal and that the value of m largely dictates the behavior of the pdf near z = 0.Table2.1 provides percentage points of the random variable Z = |X/Y |, where X is Student's t random variable and Y is the logistic random variable (see the next section).

Table 2 .
1 provides percentage points of the random variable Z = |X/Y |, where X is Student's t random variable and Y is the logistic random variable (see the next section).

Table 2 .
1. Percentage points of Z = |X/Y |, where X is Student's t and Y is logistic.