Type-1 Beta Distribution and its Connections to Likelihood Ratio Test

Authors

  • Nicy Sebastian St Thomas College Thrissur, Calicut University, Kerala, India
  • T. Princy Cochin University of Science and Technology, Kerala, India

DOI:

https://doi.org/10.6092/issn.1973-2201/16837

Keywords:

Likelihood ratio criteria, Type-1 beta matrix, General structures, Real and complex cases, Asymptotic chi-square, Asymptotic normal, Exact distribution

Abstract

In many cases involving hypothesis testing for parameters in multivariate Gaussian populations and certain other populations, likelihood ratio criteria, or their one-to-one functions, can be expressed in terms of the determinant of a real type-1 beta matrix. In geometrical probability problems, when the random points are type-1 beta distributed, the volume content of the parallellotope generated by these points is also associated with the determinant of a real type-1 beta matrix. These problems in the corresponding complex domain do not seem to have been discussed in the literature. It is well-known that the determinant of a real type-1 beta matrix can be written as a product of statistically independently distributed real scalar type-1 beta random variables. This paper addresses the general h-th moments of a scalar random variable, in either the real or complex domain, for any arbitrary h. The structure of these moments is quite general, and the paper provides exact distribution results, asymptotic gamma function results, and asymptotic normal results for both the real and complex domains.

References

T. W. ANDERSON (1958). An Introduction to Multivariate Statistical Analysis. John Wiley & Sons, New York.

Z. BAI, D. JIANG, J.-F. YAO, S. ZHENG (2009). Corrections to LRT on large-dimensional covariance matrix by RMT. Annals of Statistics, 37, no. 6B, p. 3822–3840.

A. BENAVOLI, A. FACCHINI, M. ZAFFALON (2016). Quantum mechanics: the Bayesian theory generalized to the space of Hermitian matrices. Physical Review A, 94, no. 4, p. 042106.

X. DENG (2016). Texture Analysis and Physical Interpretation of Polarimetric SAR Data. Universitat Politècnica de Catalunya.

H. DETTE, N. DÖRNEMANN (2020). Likelihood ratio tests for many groups in high dimensions. Journal of Multivariate Analysis, 178, p. 104605.

N. DÖRNEMANN (2023). Likelihood ratio tests under model misspecification in high dimensions. Journal of Multivariate Analysis, 193, p. 105122.

H. JIANG, S. WANG (2017). Moderate deviation principles for classical likelihood ratio tests of high-dimensional normal distributions. Journal of Multivariate Analysis, 156, pp. 57–69.

T. JIANG, Y. QI (2015). Likelihood ratio tests for high-dimensional normal distributions. Scandinavian Journal of Statistics, 42, no. 4, pp. 988–1009.

T. JIANG, F. YANG (2013). Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. The Annals of Statistics, 41, no. 4, pp. 2029–2074.

E. L. LEHMANN (2012). On likelihood ratio tests. Selected works of EL Lehmann, pp. 209–216.

A. LEMONTE (2016). The Gradient Test: Another Likelihood-Based Test. Academic Press.

A. J. LEMONTE (2013). On the gradient statistic under model misspecification. Statistics & Probability Letters, 83, no. 1, pp. 390–398.

J. LIM, E. LI, S.-J. LEE (2010). Likelihood ratio tests of correlated multivariate samples. Journal of Multivariate Analysis, 101, no. 3, pp. 541–554.

X. LUO,W. Y. TSAI (2012). A proportional likelihood ratio model. Biometrika, 99, no. 1, pp. 211–222.

B. F. MANLY, J. RAYNER (1987). The comparison of sample covariance matrices using likelihood ratio tests. Biometrika, 74, no. 4, pp. 841–847.

A. M. MATHAI (1993). A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Clarendon Press, Oxford, UK.

A. M. MATHAI (1999). An Introduction to Geometrical Probability: Distributional Aspects with Applications, vol. 1. CRC Press, Boca Raton, FL, USA.

A. M.MATHAI, S. B. PROVOST (2022). On the singular gamma, wishart, and beta matrixvariate density functions. Canadian Journal of Statistics, 50, no. 4, pp. 1143–1165.

A. M. MATHAI, S. B. PROVOST, H. J. HAUBOLD (2022). Multivariate Statistical Analysis in the Real and Complex Domains. Springer Nature, Cham, Switzerland.

J. NEYMAN, E. PEARSON (1933). On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London A, 231, pp. 289–337.

J. NEYMAN, E. S. PEARSON (1928). On the use and interpretation of certain test criteria for purposes of statistical inference: Part I. Biometrika, pp. 175–240.

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Published

2024-12-10

How to Cite

Sebastian, N., & Princy, T. (2023). Type-1 Beta Distribution and its Connections to Likelihood Ratio Test. Statistica, 83(2), 153–164. https://doi.org/10.6092/issn.1973-2201/16837

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