Abstract
We propose an alternative approach to the classical “nonparametric” test problems, such as the goodness of fit test and the two-sample “nonparametric” test. In this approach, those problems are reviewed from the viewpoint of the estimation of the underlying population distributions and are formulated as the problem of model selection between Bayesian models which were recently proposed by the present authors. The model selection can be easily realized by choosing a model with the smallest ABIC, Akaike Bayesian information criterion. The approach provides the estimates of the density of the underlying population distribution(s) of any shape as well as the evaluation of the goodness of fit or the check of homogeneity of distributions. The practical utility of the present procedure is demonstrated by numerical examples. The difference in behavior between the present procedure and a density estimator GALTHY proposed by Akaike and Arahata is also briefly discussed.
Similar content being viewed by others
References
Akaike H. (1973). Information theory and an extension of the maximum likelihood principle, 2nd Internat. Symp. Inform. Theory, (eds. B. N. Petrov and F. Csaki), 267–806, Akademiai Kiado, Budapest.
Akaike H. (1977). On entropy maximization principle, Applications of Statistics, (ed. P. R. Krishnaiah), 27–41, North-Holland, Amsterdam.
Akaike H. (1980). Likelihood and Bayes procedure, Bayesian Statistics, (eds. J. M. Bernardo, M. H. DeGroot, D. U. Lindley and A. F. M. Smith), University Press, Valencia, Spain.
Akaike H. and Arahata E. (1978). GALTHY, a probability density estimation, Comput. Sci. Monographs, No. 9, The Institute of Statistical Mathematics, Tokyo.
Fisher R. A. (1936). The use of multiple measurements in taxonomic problems, Ann. Eugenics, 7, 179–188.
Ishiguro M. and Sakamoto Y. (1984). A Bayesian approach to the probability density estimation, Ann. Inst. Statist. Math., 36, 523–538.
Leurgans S. (1980). Evaluating laboratory measurement techniques, Biostatistics Casebook, (eds. R. G. MillerJr., B. Efron, B. W. M. BrownJr. and L. E. Moses), 190–219, Wiley, New York.
Neyman J. (1937). ‘Smooth’ test for goodness of fit, Skandinavisk Aktuarietidskrift, 20, 149–199.
Sakamoto Y., Ishiguro M. and Kitagawa G. (1986). Akaike Information Criterion Statistics, Reidel, Dordrecht, Holland.
Author information
Authors and Affiliations
Additional information
This paper was originally read at the Conference on “Graphical Models to Analyze Structures” (Organizer: N. Wermuth, Johannes Gutenberg University), June 30-July 2, 1986, Wiesbaden, West Germany.
About this article
Cite this article
Sakamoto, Y., Ishiguro, M. A bayesian approach to nonparametric test problems. Ann Inst Stat Math 40, 587–602 (1988). https://doi.org/10.1007/BF00053067
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00053067