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) Zbl 1110.62020
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:
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 62H10 | Multivariate distribution of statistics |
Keywords:
near-exact approximations; generalized near-integer gamma distribution; independent random variables; mixtures; multivariate analysis; tablesReferences:
| [1] | Abramowitz, M., Stegun, I.A., 1974. Handbook of Mathematical Functions, 9th print. Dover, New York.; Abramowitz, M., Stegun, I.A., 1974. Handbook of Mathematical Functions, 9th print. Dover, New York. |
| [2] | Anderson, T. W., An Introduction to Multivariate Statistical Analysis (1984), Wiley: Wiley New York · Zbl 0651.62041 |
| [3] | Berry, A., The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc., 49, 122-136 (1941) · JFM 67.0461.01 |
| [4] | Coelho, C. A., The generalized integer Gamma distribution—a basis for distributions in multivariate statistics, J. Multivariate Anal., 64, 86-102 (1998) · Zbl 0893.62046 |
| [5] | Coelho, C. A., Addendum to the paper the generalized integer Gamma distribution—a basis for distributions in multivariate statistics, J. Multivariate Anal., 69, 281-285 (1999) · Zbl 0937.62054 |
| [6] | Coelho, C. A., The generalized near-integer Gamma distribution. A basis for ‘near-exact’ approximations to the distributions of statistics which are the product of an odd number of independent Beta random variables, J. Multivariate Anal., 89, 191-218 (2004) · Zbl 1047.62014 |
| [7] | Esseen, C.-G., Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian Law, Acta Mathematica, 77, 1-125 (1945) · Zbl 0060.28705 |
| [8] | Gupta, R. D.; Richards, D., Exact distributions of Wilks’ \( \Lambda\) under null and non-null (linear) hypotheses, Statistica, 39, 333-342 (1979) · Zbl 0407.62046 |
| [9] | Gupta, R.D., Richards, D., 1983. Application of results of Kotz, Johnson and Boyd to the null distribution of Wilks’ criterion. In: Sen, P.K. (Ed.), Contributions to Statistics: Essays in Honour of Norman L. Johnson, pp. 205-210.; Gupta, R.D., Richards, D., 1983. Application of results of Kotz, Johnson and Boyd to the null distribution of Wilks’ criterion. In: Sen, P.K. (Ed.), Contributions to Statistics: Essays in Honour of Norman L. Johnson, pp. 205-210. · Zbl 0528.62044 |
| [10] | Hwang, H.-K., On convergence rates in the central limit theorems for combinatorial structures, European J. Combin., 19, 329-343 (1998) · Zbl 0906.60024 |
| [11] | Johnson, N.L., Kotz, S., Balakrishnan, N., 1995. Continuous Univariate Distributions, vol. 2, second ed. Wiley, New York.; Johnson, N.L., Kotz, S., Balakrishnan, N., 1995. Continuous Univariate Distributions, vol. 2, second ed. Wiley, New York. · Zbl 0821.62001 |
| [12] | Kabe, D. G., On the exact distribution of a class of multivariate test criteria, Ann. Math. Statist., 33, 1197-1200 (1962) · Zbl 0121.36803 |
| [13] | Loève, M., 1977. Probability Theory, vol. I, fourth ed. Springer, New York.; Loève, M., 1977. Probability Theory, vol. I, fourth ed. Springer, New York. · Zbl 0359.60001 |
| [14] | Nandi, S. B., The exact null distribution of Wilks’ criterion, Sankhyã, 39, 307-315 (1977) · Zbl 0408.62047 |
| [15] | Tretter, M. J.; Walster, G. W., Central and noncentral distributions of Wilks’ statistic in Manova as mixtures of incomplete Beta functions, Ann. Statist., 3, 467-472 (1975) · Zbl 0305.62033 |
| [16] | Wilks, S. S., Certain generalizations in the analysis of variance, Biometrika, 24, 471-494 (1932) · Zbl 0006.02301 |
| [17] | Wilks, S. S., On the independence of \(k\) sets of normally distributed statistical variables, Econometrica, 3, 309-326 (1935) · Zbl 0012.02903 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
