Abstract
This paper establishes essentially complete class theorems and gives conditions for admissibility of tests for parametric families of distributions having some kinds of regular variation properties. A complete treatment of topologically contiguous one-dimensional bounded and unbounded hypotheses is given. Examples of applications to well known families of distributions are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ash, R. B. (1972).Real Analysis and Probability, Academic Press, New York.
Bauer, H. (1981).Probability Theory and Elements of Measure Theory, Academic Press, London.
Billingsley, P. (1968).Convergence of Probability Measures, Wiley, New York.
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987).Regular Variation, Cambridge University Press, Cambridge.
Birnbaum, A. (1955). Characterizations of complete classes of tests of some multiparametric hypotheses with applications to likelihood ratio tests,Ann. Math. Statist.,26, 21–36.
Blyth, C. R. (1951). On minimax statistical decision problems and their admissibility,Ann. Math. Statist.,22, 22–42.
Brown, L. D. and Marden, J. I. (1989). Complete class results for hypothesis testing problems with simple null hypotheses,Ann. Statist.,17, 209–235.
Eaton, M. L. (1970). A complete class theorem for multidimensional one-sided alternatives,Ann. Math. Statist.,41, 1884–1888.
Farrell, R. H. (1968). Towards a theory of generalized Bayes tests,Ann. Math. Statist.,39, 1–22.
Feller, W. (1966).An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York.
Ferguson, T. S. (1967).Mathematical Statistics, Academic Press, New York.
Ghia, G. D. (1976). Truncated generalized Bayes tests, Ph.D. Dissertation, Yale University.
Gumbel, E. J. (1944). Ranges and midranges,Ann. Math. Statist.,15, 414–422.
Johnson, N. L. and Kotz, S. (1970).Continuous Univariate Distributions-2, Wiley, New York.
Karamata, J. (1930). Sur un mode de croissance regulière,Mathematica (Cluj),4, 38–53.
Kudô, H. (1961). Locally complete classes of tests,Bulletin of the International Statistical Institute,38, 173–180.
Lehmann, E. L. (1947). On families of admissible tests,Ann. Math. Statist.,18, 97–104.
Marden, J. I. (1982). Minimal complete classes of test of hypotheses with multivariate one-sided alternatives,Ann. Math. Statist.,10, 962–970.
Matthes, T. K. and Truax, D. R. (1967). Tests of composite hypotheses for multivariate exponential family,Ann. Math. Statist.,38, 681–697.
Neyman, J. and Pearson, E. S. (1936). Contributions to the theory of testing statistical hypotheses. I. Unbiased critical regions of type A and type A1, II. Certain theorems on unbiased critical regions of type A,Statist. Res. Mem.,1, 1–37.
Neyman, J. and Pearson, E. S. (1938). Contributions to the theory of testing statistical hypotheses. III. Unbiased tests of simple statistical hypotheses specifying the values of more than unknown parameter,Statist. Res. Mem.,2, 25–27.
Parthasarathy, K. R. (1978).Introduction to Probability and Measure, The Macmillan Press, London.
Perks, W. F. (1932). On some experiments in the graduation of mortality statistics,J. Inst. Actuar.,58, 12–57.
Rider, P. R. (1957). Generalized Cauchy distributions,Ann. Inst. Statist. Math.,9, 215–223.
Talacko, J. (1956). Perks' distributions and their role in the Theory of Wiener's stochastic variables,Trabajos de Estadistica,7, 159–174.
Wald, A. (1950).Statistical Decision Functions, Wiley, New York.
Author information
Authors and Affiliations
About this article
Cite this article
Kowalski, J.P. Complete classes of tests for regularly varying distributions. Ann Inst Stat Math 47, 321–350 (1995). https://doi.org/10.1007/BF00773466
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00773466