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Register a new userA Bayesian Nonparametric Test for Assessing Multivariate Normality
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Added by
Luai Al-Labadi Dr.
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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In this paper, a novel Bayesian nonparametric test for assessing multivariate normal models is presented. While there are extensive frequentist and graphical methods for testing multivariate normality, it is challenging to find Bayesian counterparts. The proposed approach is based on the use of the Dirichlet process and Mahalanobis distance. More precisely, the Mahalanobis distance is employed as a good technique to transform the $m$-variate problem into a univariate problem. Then the Dirichlet process is used as a prior on the distribution of the Mahalanobis distance. The concentration of the distribution of the distance between the posterior process and the chi-square distribution with $m$ degrees of freedom is compared to the concentration of the distribution of the distance between the prior process and the chi-square distribution with $m$ degrees of freedom via a relative belief ratio. The distance between the Dirichlet process and the chi-square distribution is established based on the Anderson-Darling distance. Key theoretical results of the approach are derived. The procedure is illustrated through several examples, in which the proposed approach shows excellent performance.
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Bayesian nonparametric statistics is an area of considerable research interest. While recently there has been an extensive concentration in developing Bayesian nonparametric procedures for model checking, the use of the Dirichlet process, in its simplest form, along with the Kullback-Leibler divergence is still an open problem. This is mainly attributed to the discreteness property of the Dirichlet process and that the Kullback-Leibler divergence between any discrete distribution and any continuous distribution is infinity. The approach proposed in this paper, which is based on incorporating the Dirichlet process, the Kullback-Leibler divergence and the relative belief ratio, is considered the first concrete solution to this issue. Applying the approach is simple and does not require obtaining a closed form of the relative belief ratio. A Monte Carlo study and real data examples show that the developed approach exhibits excellent performance.
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