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Zbl 1110.62020
Grilo, Lu{\'\i}s M.; Coelho, Carlos A.
Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent beta random variables. (English)
[J] J. Stat. Plann. Inference 137, No. 5, 1560-1575 (2007). ISSN 0378-3758

Summary: Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.

MSC 2000:: *62E17 Approximations to statistical distributions (nonasymptotic)
62H10 Multivariate distributions of statistics

Keywords: near-exact approximations; generalized near-integer gamma distribution; independent random variables; mixtures; multivariate analysis; tables

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