Approximations for the Likelihood Ratio Statistic for Hypothesis Testing Between Two Beta Distributions
Tóm tắt
In this paper, the likelihood ratio to test between two Beta distributions is addressed. The exact distribution of the likelihood ratio statistic, for simple hypotheses, is obtained in terms of Gamma or Generalized Integer Gamma distributions, when the first or the second of the two parameters of the Beta distributions are equal and integers. In the remaining cases addressed, near-exact or asymptotic approximations are developed for the likelihood ratio statistic.
Both the exact, asymptotic or near-exact representations are obtained using a logarithm transformation of the likelihood ratio statistic and by working with the corresponding characteristic function. The numerical studies illustrate the precision of the approximations developed. Simulations are developed to analyse the power and the reproducibility probability of the tests.
Tài liệu tham khảo
Arts GRJ, Coolen FPA, van der Laan P (2004) Nonparametric predictive inference in statistical process control. Qual Technol Quant Manag 1:201–216
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 distribution of statistics which are the product of an odd number of independent beta random variables. J Multivar Anal 89:191–218
Coelho CA, Alberto RP (2012) On the Distribution of the Product of Independent Beta Random Variables Applications. Technical Report, CMA, p 12
Coelho CA, Marques FJ (2012) Near-exact distributions for the likelihood ratio test statistic to test equality of several variance-covariance matrices in elliptically contoured distributions. Comput Stat 27:627–659
Coelho CA, Arnold BC, Marques FJ (2010) Near-exact distributions for certain likelihood ratio test statistics. J Stat Theory Pract 4:711–725
Coolen FPA, Alqifari HN (2018) Nonparametric predictive inference for repro- ducibility of two basic tests based on order statistics. REVSTAT Stat J 16:167–185
Coolen FPA, Bin Himd S (2014) Nonparametric predictive inference for reproducibility of basic nonparametric tests. J Stat Theory Pract 8:591–618
Gil-Pelaez J (1951) Note on the inversion theorem. Biometrika 38:481–482
Goodman SN (1992) A comment on replication, p-values and evidence. Stat Med 11:875–879
Griffiths D (1973) Maximum likelihood estimation for the beta-binomial distribution and an application to the household distribution of the total number of cases of a disease. Biometrics 29:637–648
Gupta AK, Nadarajah S (2004) Handbook of beta distribution and its applications. Statistics: a series of textbooks and monographs. Taylor & Francis
Hill BM (1968) Posterior distribution of percentiles: Bayes’ theorem for sampling from a population. J Am Stat Assoc 63:677–691
Luke YL (1969) The special functions and their approximations. Academic Press Inc, London
Marques FJ, Coelho CA, de Carvalho M (2015) On the distribution of linear combinations of independent Gumbel random variables. Stat Comput 25:683–701
Marques FJ, Coelho CA, Rodrigues PC (2017) Testing the equality of several linear regression models. Comput Stat 32(4):1453–1480
Marques FJ, Coolen FPA, Coolen-Maturi T (2018) Introducing nonparametric predictive inference methods for reproducibility of likelihood ratio tests. J Stat Theory Pract, submitted for publication
Pham TV, Piersma SR, Warmoes M, Jimenez CR (2010) On the beta-binomial model for analysis of spectral count data in label-free tandem mass spectrometry-based proteomics. Bioinformatics 26:363–369
Rachev ST, Hsu JSJ, Bagasheva BS, Fabozzi FJ (2008) Bayesian methods in Finance. Fabozzi series. Wiley, Frank J, Hoboken
Tricomi FG, Erdélyi A (1951) The asymptotic expansion of a ratio of Gamma functions. Pac J Math 1:133–142
Wiley JA, Martin JL, Herschkorn SJ, Bond J (2015) A new extension of the binomial error model for responses to items of varying difficulty in educational testing and attitude surveys. PLoS ONE 10:e0141981
Wilks SS (1983) The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann Math Stat 9:60–62