Abstract
In this paper we first show how the exact distributions of the most common likelihood ratio test (l.r.t.) statistics, that is, the ones used to test the independence of several sets of variables, the equality of several variance-covariance matrices, sphericity, and the equality of several mean vectors, may be expressed as the distribution of the product of independent Beta random variables or the product of a given number of independent random variables whose logarithm has a Gamma distribution times a given number of independent Beta random variables. Then, we show how near-exact distributions for the logarithms of these statistics may be expressed as Generalized Near-Integer Gamma distributions or mixtures of these distributions, whose rate parameters associated with the integer shape parameters, for samples of size n, all have the form (n−j)/n for j=2,…,p, where for three of the statistics, p is the number of variables involved, while for the other one, it is the sum of the number of variables involved and the number of mean vectors being tested. What is interesting is that the similarities exhibited by these statistics are even more striking in terms of near-exact distributions than in terms of exact distributions. Moreover all the l.r.t. statistics that may be built as products of these basic statistics also inherit a similar structure for their near-exact distributions. To illustrate this fact, an application is made to the l.r.t. statistic to test the equality of several multivariate Normal distributions.
Similar content being viewed by others
References
Anderson TW (2003) An introduction to multivariate statistical analysis, 3rd edn. Wiley, New York
Anderson TW, Fang K-T (1990) Inference in multivariate elliptically contoured distributions based on maximum likelihood. In: Fang K-T, Anderson TW (eds) Statistical inference in elliptically contoured and related distributions, pp 201–216
Anderson TW, Fang K-T, Hsu H (1986) Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions. Can J Stat 14:55–59
Aslam S, Rocke DM (2005) A robust testing procedure for the equality of covariance matrices. Comput Stat Data Anal 49:863–874
Berry A (1941) The accuracy of the Gaussian approximation to the sum of independent variates. Trans Am Math Soc 49:122–136
Coelho CA (1992). Generalized canonical analysis. PhD thesis, The University of Michigan, Ann Arbor, MI
Coelho CA (1998) The Generalized Integer Gamma distribution—a basis for distributions in Multivariate Statistics. J Multivar Anal 64:86–102
Coelho CA (2004) 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 Multivar Anal 89:191–218
Coelho CA (2007) The wrapped Gamma distribution and wrapped sums and linear combinations of independent Gamma and Laplace distributions. J Stat Theory Practice 1:1–29
Coelho CA, Marques FJ (2009a) The advantage of decomposing elaborate hypotheses on covariance matrices into conditionally independent hypotheses in building near-exact distributions for the test statistics. Linear Algebra Appl 430:2592–2606
Coelho CA, Marques FJ (2009b). Near-exact distributions for the likelihood ratio test statistic for testing equality of several variance–covariance matrices (revisited). The New University of Lisbon, Mathematics Department, technical report #9/2009, submitted for publication. http://www.dmat.fct.unl.pt/fct/servlet/DownloadFile?file=1162&sub=investiga&modo=prepub
Coelho CA, Marques FJ (2010) Near-exact distributions for the independence and sphericity likelihood ratio test statistics. J Multivar Anal 101:583–593
Coelho CA, Alberto RP, Grilo LM (2006) A mixture of Generalized Integer Gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables: applications. J Interdiscip Math 9:229–248
Esseen C-G (1945) Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law. Acta Math 77:1–125
Hsieh HK (1979) On asymptotic optimality of likelihood ratio tests for multivariate normal distributions. Ann Stat 7:592–598
Hwang H-K (1998) On convergence rates in the central limit theorems for combinatorial structures. Eur J Combin 19:329–343
Jensen DR, Good IJ (1981) Invariant distributions associated with matrix laws under structural symmetry. J R Stat Soc Ser B 43:327–332
Kariya T (1981) Robustness of multivariate tests. Ann Stat 9:1267–1275
Kshirsagar AM (1972) Multivariate analysis. Dekker, New York
Loève M (1977) Probability theory, 4th edn, vol I. Springer, New York
Marques FJ, Coelho CA (2008) Near-exact distributions for the sphericity likelihood ratio test statistic. J Stat Plan Inference 138:726–741
Mauchly JW (1940) Significance test for sphericity of a normal n-variate distribution. Ann Math Stat 11:204–209
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Domingo Morales.
Rights and permissions
About this article
Cite this article
Marques, F.J., Coelho, C.A. & Arnold, B.C. A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis. TEST 20, 180–203 (2011). https://doi.org/10.1007/s11749-010-0193-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-010-0193-3
Keywords
- Wilks lambda statistic
- Independence test
- Sphericity test
- Generalized Integer Gamma distribution
- Generalized Near-Integer Gamma distribution
- Mixtures