Use of Beta-Polynomial Approximations for Variance Homogeneity Test and a Mixture of Beta Variates

  • Published : 2009.03.30


Approximations for the null distribution of a test statistic arising in multivariate analysis to test homogeneity of variances and a mixture of two beta distributions by making use of a product of beta baseline density function and a polynomial adjustment, so called beta-polynomial density approximant, are discussed. Explicit representations of density and distribution approximants of interest in each case can easily be obtained. Beta-polynomial density approximants produce good approximation over the entire range of the test statistic and also accommodate even the bimodal distribution using an artificial example of a mixture of two beta distributions.


  1. Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions, John Wiley & Sons, London
  2. Bhattacharya, C. G. (1967). A simple method of resolution of a distribution into Gaussian compo-nents, Biometrics, 23, 115-135
  3. Butler, R. W. and Wood, A. T. A. (2002). Laplace approximations for hypergeometric functions with matrix argument, The Annals of Statistics, 30, 1155-1177
  4. Butler, R. W. and Wood, A. T. A. (2004). Mixture representations of noncentral distributions in multivariate analysis with application to a dimensional CLT, Scandinavian Journal of Statistics, 31, 631-650
  5. Gupta, A. K. and Rathie, A. K. (1982). Distribution of the likelihood ratio criterion for the problem of K samples, Metron, 40, 147-156
  6. Ha, H. T. and Provost, S. B. (2007). A viable alternative to resorting to statistical tables, Communica-tion in Statistics: Simulation and Computation, 36, 1135-1151
  7. Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, John Wiley & Sons, New York
  8. Kolassa, J. E. (2003). Multivariate saddlepoint tail probability approximations, The Annals of Statistics, 31, 274-286
  9. Neyman, J. and Pearson, E. S. (1931). On the problem of k-samples, Bulletin of Academic Polonaise Science Letters, 3, 460-481
  10. Provost, S. B. and Rudiuk, E. M. (1995). Moments and densities of test statistics for covariance structures, International Journal of Mathematical and Statistical Sciences, 4, 85-104
  11. Reid, N. (1988). Saddlepoint methods and statistical inference, Statistical Science, 3, 213-227
  12. Tretter, M. J. and Walster, G. W. (1975). Central and noncentral distributions of Wilks’ statistic in Manova as mixtures of incomplete Beta functions, The Annals of Statistics, 3, 467-472