Goodness of fit tests for nonincreasing densities on real positive data with nonparametric Bayesian methods

Document Type : Original Scientific Paper

Author

Department of Statistics‎, ‎Razi University‎, ‎Kermanshah‎, ‎Iran

Abstract

In this paper‎, ‎we study a nonparametric Bayesian inference on the family of nonincreasing density functions on real positive data‎. ‎One interesting problem is the goodness of fit test in such a context‎. ‎In other words‎, ‎we consider nonparametric Bayesian testing on the family of nonincreasing density in this domain‎. ‎So‎, ‎we define nonparametric hypothesis testing and compare two different testing approaches‎. ‎The first approach is given based on the Bayes factor‎. ‎This approach is the well-known Bayesian approach for testing‎, ‎although its computation is complicated‎. ‎Decision-theoretic considerations with the loss function drive the second approach for a given distance‎. ‎This second approach has the advantage of considering the distance to the null hypothesis but needs the definition of a threshold‎. ‎When no threshold is known as a priori‎, ‎a possibility exists to calculate a p-value‎, ‎and the method becomes more complicated to compute‎. ‎We propose a hybrid algorithm to accelerate the computation of the p-value‎. ‎The comparison of both approaches is performed based on a simulation study.

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References


Al Labadi, L. and Berry, S. (2022). Bayesian estimation of extropy and goodness of fit tests. Journal of Applied Statistics, 49(2):357–370.
Al Labadi, L. and Zarepour, M. (2013). A Bayesian nonparametric goodness of fit test for right censored data based on approximate samples from the beta-Stacy process. Canadian Journal of Statistics, 41(3):466–487.
Basu, S. and Chib, S. (2003). Marginal likelihood and Bayes factors for Dirichlet process mixture models. Journal of the American Statistical Association, 98(461):224–235.
Balabdaoui, F. and Wellner, J.A. (2007). Estimation of a k-monotone density: limit distribution theory and the spline connection. The Annals of Statistics, 35(6):2536–2564.
Berger, J.O. and Delampady, M. (1987). Testing precise hypotheses. Statistical Science, 2(3):317–352.
Blackwell, D. and MacQueen, J.B. (1973). Ferguson distributions via Polya urn schemes. The Annals of Statistics, 1(2):353–355.
Brunner, L.J. and Lo, A.Y. (1989). Bayes methods for a symmetric unimodal density and its mode. The Annals of Statistics, 17(4):1550–1566.
Calderhead, B. and Girolami, M. (2009). Estimating Bayes factors via thermodynamic integration and population MCMC. Computational Statistics and Data Analysis, 48(12):4028–4045.
Damien, P. and Walker, S.G. (2001). Sampling truncated normal, beta, and gamma densities. Journal of Computational and Graphical Statistics, 10(2):206–215.
Dass, S.C. and Lee, J. (2006). A note on the consistency of Bayes factors for testing point null versus non-parametric alternatives. Journal of Statistical Planning and Inference, 119(1):143–152.
Ferguson, T. (1973). Bayesian analysis of some nonparametric problems. The Annals of Statistics, 1(2):209–230.
Guglielmi, A. and Tweedie, R.L. (2001). Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process. Bernoulli, 7(4):573–592.
Hart, J.D. and Choi, T. (2017). Nonparametric goodness of fit via cross-validation Bayes factors. Bayesian Analysis, 12(3):653–677.
Kalli, M., Griffin, J.E. and Walker, S.G. (2011). Slice sampling mixture models. Statistics and Computing, 21(1):93–105.
Kong, A., Liu, J.S. and Wong, W.H. (1994). Sequential imputations and Bayesian missing data problems. Journal of the American Statistical Association, 89(425):278–288.
Lévy, P. (1962). Extensions d'un théorème de D. Dugué et M. Girault. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 1:159–173.
Lo, A.Y., Brunner, L.J. and Chan, A.T. (1996). Weighted Chinese restaurant processes and Bayesian mixture models. Research Report, Hong Kong: The University of Science and Technology, ISMT Department.
MacEachern, S.N., Clyde, M. and Liu, J.S. (1999). Sequential importance sampling for non-parametric Bayes models: the next generation. Canadian Journal of Statistics, 27(2):251–267.
McVinish, R., Rousseau, J. and Mengersen, K. (2009). Bayesian goodness-of-fit testing with mixtures of triangular distributions. Scandinavian Journal of Statistics, 36(2):337–354.
Papaspiliopoulos, O. and Roberts, G.O. (2008). Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models. Biometrika, 95(1):169–186.
Quintana, F.A. and Newton, M.A. (1998). Assessing the order of dependence for partially exchangeable binary data. Journal of the American Statistical Association, 93(441):194–202.
Robert, C.P. and Rousseau, J. (2003). A mixture approach to Bayesian goodness of fit. Les cahiers du CEREMADE (2005-31). http://www.ceremade.dauphine.fr/preprints/CMD/2002-9.ps.gz
Rousseau, J. (2007). Approximating interval hypotheses: P-values and Bayes factors. Bayesian Statistics, 8:417–452.
Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica, 4:639–650.
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). The Annals of Statistics, 22(4):1701–1786.
Verdinelli, I. and Wasserman, L. (1998). Bayesian goodness-of-fit testing using infinite-dimensional exponential families. The Annals of Statistics, 26(4):1215–1241.
Williamson, R.E. (1956). Multiply monotone functions and their Laplace transforms. Duke Mathematical Journal, 23(2):189–207.
Woodroofe, M. and Sun, J. (1999). Testing uniformity versus a monotone density. The Annals of Statistics, 27(1):338–360.
Journal of Statistical Modelling: Theory and Applications
Volume 5, Issue 1
January 2024
Pages 15-32

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History

  • Receive Date: 09 April 2024
  • Revise Date: 11 July 2024
  • Accept Date: 23 July 2024

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