On the Distribution of the Product of Independent Beta Random Variables — Applications

Abstract

A first approach, based on recently obtained asymptotic expansions of ratios of gamma functions, enables the obtention of the distribution of the product of independent and identically distributed random variables in a much manageable form. However, for the general case, this approach leads to a form which although being very manageable and in line with some previous results, suffers from serious problems of precision and convergence, which have been completely overlooked by other authors and which in most cases prevent its practical use. Nevertheless, it is based on these first results that the authors, using the concept of near-exact distribution, are able to obtain highly manageable but extremely accurate approximations for all cases of the distribution of the product of independent Beta random variables. These near-exact approximations, given their high manageability, accuracy, and proximity to the exact distribution, may in practice be used instead of the exact distribution.

Appendix A: Simple Proofs for Expressions (11) and (45)

The asymptotic expansions in (39) and (45) were first proved by Burić and Elezović (2011). We propose here simpler and straightforward demonstrations.

Barnes (1899) established an asymptotic expansion for the logarithm of the gamma function in the form

$$\displaystyle \begin{aligned} \begin{array}{rc} &\log \,\varGamma (x + h) \approx \log \,\sqrt {2\pi } + \left( {x + h - \frac{1}{2}} \right)\log \,x - x - \sum\limits_{r = 1}^\infty {( - 1)^r \frac{{B_{r + 1} (h)}}{{r(r + 1)x^r }}} ,\\ \end{array}\end{aligned} $$

$$\displaystyle \begin{aligned} \qquad \qquad \left( {x \to \infty } \right) {} \end{aligned} $$

(37)

for any $x,h\in \mathbb {C}$ and where B _r(⋅) is the Bernoulli polynomial of degree r.

Assuming $z,\alpha ,\beta \in \mathbb {C}$ and $m\in \mathbb {R}$, the application of (37) leads to

$$\displaystyle \begin{aligned} \log \left[ {\frac{{\varGamma (z + \alpha )}}{{\varGamma (z + \beta )}}} \right]^m \approx m\,(\alpha - \beta )\log \,z + \sum\limits_{r = 1}^\infty {\delta _{r,m} (\alpha ,\beta )} \;z^{ - r} ,\quad \quad \left( {z \to \infty } \right) \end{aligned}$$

where

$$\displaystyle \begin{aligned} \delta _{r,m} (\alpha ,\beta ) = ( - 1)^r m\,\frac{{B_{r + 1} (\beta ) - B_{r + 1} (\alpha )}}{{r(r + 1)}}\,. {} \end{aligned} $$

(38)

Therefore, we may write

$$\displaystyle \begin{aligned} \left[ {\frac{{\varGamma (z + \alpha )}}{{\varGamma (z + \beta )}}} \right]^m \approx z^{m\;(\alpha - \beta )}\, e^{\sum_{r = 1}^\infty {\delta_{r,m}(\alpha,\beta)\,z^{ - r} } } ,\quad \quad \left( {z \to \infty } \right)\,, \end{aligned}$$

from which, expanding the exponential function according to expressions (2.7) and (2.8) in Moschopoulos (1985) we obtain the asymptotic expansion

$$\displaystyle \begin{aligned} \left[ {\frac{{\varGamma (z + \alpha )}}{{\varGamma (z + \beta )}}} \right]^m \approx \sum_{k = 0}^\infty {\nu _{k,m} (\alpha ,\beta )\,z^{m\;(\alpha - \beta ) - k} } ,\quad \quad \left( {z \to \infty } \right)\,, {} \end{aligned} $$

(39)

for the power of a ratio of two gamma functions, with ν _k,m(α, β) given by (12).

However, it is indeed possible to improve the series expansion in (39), achieving a faster convergence series after a convenient parameter manipulation, in case we are willing to use a power basis which is also function of α and β.

From the property of the Bernoulli polynomials stated in expression 23.1.8 in Abramowitz and Stegun (1972), we note that, when n is even,

$$\displaystyle \begin{aligned} B_n (1 - x) = B_n (x)\,. {} \end{aligned} $$

(40)

Consider now ${c\in \mathbb {C}}$ and let z ^∗ = z + c, α ^∗ = α − c, and β ^∗ = β − c. Then, from (39) we may write

$$\displaystyle \begin{aligned} \left[ {\frac{{\varGamma (z^* + \alpha ^* )}}{{\varGamma (z^* + \beta ^* )}}} \right]^m \approx \sum_{k = 0}^\infty {\nu _{k,m} (\alpha ^* ,\beta ^* )\,(z^* )^{m\;(\alpha - \beta ) - k} } ,\quad \quad \left( {z \to \infty } \right)\,. {} \end{aligned} $$

(41)

Now, the proper choice of c will allow us to reduce the number of terms in the above series. For this purpose we will force ν _1,m(α ^∗, β ^∗) = 0. Since ν _1,m(α ^∗, β ^∗) = δ _1,m(α ^∗, β ^∗), it is enough to determine c such that

$$\displaystyle \begin{aligned} B_2 (\beta - c) = B_2 (\alpha - c)\,. {} \end{aligned} $$

(42)

But then, according to (40), we have

$$\displaystyle \begin{aligned} B_2 (\beta - c) = B_2 (1 - (\beta - c))\,, \end{aligned}$$

which, together with (42), entails

$$\displaystyle \begin{aligned} c = (\alpha + \beta - 1)/2\,. {} \end{aligned} $$

(43)

Moreover, this choice of c implies that, for every odd j

$$\displaystyle \begin{aligned} \delta _{j,m} (\alpha - k,\beta - k) = 0\,. {} \end{aligned} $$

(44)

Thus, from (43) and (44), we may prove that, for every odd k

$$\displaystyle \begin{aligned} \nu _{k,m} (\alpha - c,\beta - c) = 0\,, \end{aligned}$$

and that therefore all the odd terms of the series represented in (41) vanish.

The proof is done by induction. As induction basis, the statement ν _1,m(α − c, β − c) = 0 holds, according to the choice of c in (43). Assuming as induction hypothesis that, for any given odd k, the statement

$$\displaystyle \begin{aligned} \begin{array}{l} \nu _{1,m} (\alpha - c,\beta - c) = \nu _{3,m} (\alpha - c,\beta - c) = \ldots = \nu _{k - 2,m} (\alpha - c,\beta - c)\\ {} \hskip 7.75cm = \nu _{k,m} (\alpha - c,\beta - c) = 0 \end{array} \end{aligned}$$

is true, we have, by (38),

$$\displaystyle \begin{aligned} \nu _{k + 2,m} (\alpha - c,\beta - c) = \frac{1}{{k + 2}}\;\sum\limits_{j = 1}^{k + 2} {j\,\delta _{j,m} (\alpha - c,\beta - c)\,} \nu _{k + 2 - j,m} (\alpha - c,\beta - c)\,. \end{aligned}$$

Now all the terms in the summation are zero since, when j is odd the factor δ _j,m(α − c, β − c) is zero, according to (44). Otherwise, when j is even, the index k + 2 − j is odd and, by the induction hypothesis, the correspondent coefficient ν _k+2−j,m(α − c, β − c) is also zero.

Finally, expression (41) may be written as

$$\displaystyle \begin{aligned} \left[ {\frac{{\varGamma (z + \alpha )}}{{\varGamma (z + \beta )}}} \right]^m \approx \sum_{k = 0}^\infty {\nu _{2k,m} (\alpha - c,\beta - c)\,(z + {\textstyle{{\alpha + \beta - 1} \over 2}})^{m(\alpha - \beta ) - 2k} } ,\quad \left( {z \to \infty } \right)\,, {}\end{aligned} $$

(45)

where each coefficient ν _2k,m(α − c, β − c) is given by (12), with c = (α + β − 1)∕2.

Setting m = 1 in (39) we obtain a power expansion which is equivalent to the asymptotic series expansion for the ratio of two gamma functions proposed by Tricomi and Erdélyi (1951), here in the form given by Fields (1966), while by setting m = 1 in (45) we obtain the equivalent to the asymptotic series expansion for the ratio of two gamma functions proposed by Fields (1966).