On the Product and Ratio of Pearson Type VII and Laplace Random Variables

The distributions of the product |XY | and the ratio|X/Y | are derived whenX andY are Pearson type VII and Laplace random variables distributed independently of each other. Extensive tabulations of the associated percentage points are also given. Zusammenfassung:Die Verteilungen des Produkts |XY | und des Quotienten |X/Y | sind hergeleitet, fallsX undY unabḧangig verteilte Pearson type VII und Laplace Zufallsvariablen sind. Umfangreiche Tabellen der dazugeḧorenden Perzentile werden auch angegeben.


Introduction
For given random variables X and Y , the distributions of the product |XY | and the ratio |X/Y | are of interest in many areas of the sciences, engineering and medicine.Examples of |XY | include traditional portfolio selection models (Grubel, 1968), relationship between attitudes and behavior (Rokeach and Kliejunas, 1972), number of cancer cells in tumor biology (Ladekarl et al., 1997) and stream flow in hydrology (Cigizoglu and Bayazit, 2000).Examples of |X/Y | include Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, inventory ratios in economics and safety factor in engineering systems (see, for example, Kotz et al., 2003).
The distributions of |XY | and |X/Y | have been studied by several authors especially when X and Y are independent random variables and come from the same family.With respect to |XY |, see Sakamoto (1943) for the uniform family, Harter (1951) and Wallgren (1980) for the Student's t family, Springer and Thompson (1970) for the normal family, Stuart (1962) and Podolski (1972) for the gamma family, Steece (1976), Bhargava and Khatri (1981) and Tang and Gupta (1984) for the beta family, Abu-Salih (1983) for the power function family, and Malik and Trudel (1986) for the exponential family (see also Rathie and Rohrer, 1987, for a comprehensive review of known results).With respect to |X/Y |, see Marsaglia (1965) and Korhonen and Narula (1989) for the normal family, Press (1969) for the Student's t family, Basu and Lochner (1971) for the Weibull family, Shcolnick (1985) for the stable family, Hawkins and Han (1986) for the non-central chisquared family, Provost (1989) for the gamma family, and Pham-Gia (2000) for the 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 quite possible that X and Y could Austrian Journal of Statistics, Vol. 34 (2005), No. 1, 11-23 arise from different but similar distributions.In this note, we study the distributions of |XY | and |X/Y | when X and Y are independent random variables having the Pearson type VII and Laplace distributions with pdfs and Extensive tabulations of the associated percentage points are also provided.Since λ is just a scale parameter we shall assume without loss generality that λ = 1.
The results of this note use the following relationship between the Pearson type VII distribution and the well-known Student's t distribution: if M = 1 + a/2 and then U is a Student's t random variable with a degrees of freedom.Note that the pdf of a Student's t random variable with degrees of freedom ν is given by Nadarajah and Kotz (2003) have shown that the cdf corresponding to (4) can be expressed as (5) This result will be crucial for the calculations of this note.The calculations involve the Meijer G-function defined by where (e) k = e(e + 1) • • • (e + k − 1) denotes the ascending factorial and L denotes an integration path defined as one of: • the path L running from −∞ to +∞ in such a way that the poles of the functions Γ(1 − a k + t) lie to the left, and the poles of the functions Γ(b j − t) lie to the right of L for j = 1, . . ., m and k = 1, . . ., n.
• a loop, beginning and ending at +∞, that encircles the poles of the functions Γ(b j − t) for j = 1, . . ., m once in the negative direction.All the poles of the functions Γ(1 − a k + t) must remain outside this loop.
• a loop L, beginning and ending at −∞, that encircles the poles of the functions Γ(1 − a k + t) for k = 1, . . ., n once in the positive direction.All the poles of the functions Γ(b j − t) for j = 1, . . ., m must remain outside this loop.
In all three cases, the poles of Γ(b j − t) must not coincide with the poles of Γ(1 − a k + t) for any j = 1, . . ., m and k = 1, . . ., n (see Section 9.3 in Gradshteyn and Ryzhik, 2000, for further details).

Product
The cdf of the Student's t distribution takes different forms depending on whether its degrees of freedom parameter is an odd integer or even integer (see Nadarajah and Kotz, 2003, and references therein).Note that in (3) the degrees of freedom parameter a = 2(M − 1).Thus, one would expect the cdf of |XY | to be different depending on whether a is an odd integer or not.Theorems 1 and 2 derive explicit expressions for the cdf of |XY | for these two cases.Theorem 1 Suppose X and Y are independent random variables distributed according to ( 1) and ( 2), respectively.If a = 2(M − 1) is an odd integer then the cdf of Z = |XY | can be expressed as where r = a/mz and I(•) denotes the integral Proof: Using the relationship (3), one can write the cdf as , which can be expressed as 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 ( 5) for odd degrees of freedom, (8) can be reduced to where J(k) denotes the integral By direct application of Lemma 1, one can calculate (10) as The result of the theorem follows by substituting ( 11) into (9).
Theorem 2 Suppose X and Y are independent random variables distributed according to ( 1) and ( 2), respectively.If a = 2(M − 1) is an even integer then the cdf of Z = |XY | can be expressed as where r = a/mz.Proof: Substituting the form for F given by ( 5) for even degrees of freedom, (8) can be reduced to where J(k) denotes the integral By direct application of Lemma 1, one can calculate (14) as The result of the theorem follows by substituting (15) into (13).

Ratio
For the reasons mentioned in Section 2, the cdf of |X/Y | will be different depending on whether a is an odd integer or not.Theorems 3 and 4 derive explicit expressions for the cdf of |X/Y | for these two cases.Theorem 3 Suppose X and Y are independent random variables distributed according to (1) and ( 2), respectively.If a = 2(M − 1) is an odd integer then the cdf of Z = |X/Y | can be expressed as where r = a/mz and 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 ( 5) for odd degrees of freedom, (18) can be reduced to where J(k) denotes the integral By direct application of Lemma 1, one can calculate (20) as The result of the theorem follows by substituting ( 21) into (19).
Theorem 4 Suppose X and Y are independent random variables distributed according to ( 1) and ( 2), respectively.If a = 2(M − 1) is an even integer then the cdf of Z = |X/Y | can be expressed as where r = a/mz.Proof: Substituting the form for F given by (5) for even degrees of freedom, (18) can be reduced to where J(k) denotes the integral By direct application of Lemma 1, one can calculate (24) as The result of the theorem follows by substituting ( 25) into (23).

Percentiles
Almost every textbook in statistics has tables of percentage points of the Student's t distribution for integer values of its degrees of freedom parameter.In this section, we provide tabulations of percentage points z p of |XY | and |X/Y | for integer values of the degrees of freedom parameter a = 2(M − 1); see equation (3).We feel tables (and not pictures) are the right way to present percentage points.Of course, it is of no use to have pictures of percentile points if one needs to use them.
We have used the results in Theorems 1 to 4 to compute the percentile points.The percentage points z p of |XY | are obtained by numerically solving the equations where r = a/mz p , a = 2(M − 1) and I(a) is given by the integral in (7).The percentage points z p of |X/Y | are obtained by numerically solving the equations Austrian Journal of Statistics, Vol. 34 (2005), No. 1, 11-23 and √ a where a = 2(M − 1), r = a/mz p , and I(a) is given by the integral in (17).Evidently, this involves computation of the Meijer G function and routines for this are widely available.We used the function MeijerG (•) in the algebraic manipulation package MAPLE.

)Proof:
Using the relationship (3), one can write the cdf as Pr(|X/Y | ≤ z) = Pr(|U/Y | ≤ r), which can be expressed as