Uniformly Powerful Goodness of Fit Tests

The simple hypothesis is tested that the distribution of independent random variables X1, X2,..., Xn is a given probability measure PO. Let 7r,, be any sequence of partitions. The alternative hypothesis is the set of probability measures P with EA, IP(A) Po(A)I ? 8, where 8 > 0. Note the dependence of this set of alternatives on the sample size. It is shown that if the effective cardinality of the partitions is of the same order as the sample size, then sequences of tests exist with uniformly exponentially small probabilities of error. Conversely, if the effective cardinality is of larger order than the sample size, then no such sequence of tests exists. The effective cardinality is the number of sets in the partition which exhaust all but a negligible portion of the probability under the null hypothesis.

[1]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[2]  H. Chernoff.  LARGE-SAMPLE THEORY: PARAMETRIC CASE' , 1956 .

[3]  W. Hoeffding,et al.  Distinguishability of Sets of Distributions , 1958 .

[4]  L. Schwartz,et al.  A Necessary and Sufficient Condition for the Existence of Consistent Estimates , 1960 .

[5]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[6]  L. Schwartz.  On Bayes procedures , 1965 .

[7]  S. Kullback,et al.  A lower bound for discrimination information in terms of variation (Corresp.) , 1967, IEEE Trans. Inf. Theory.

[8]  R. R. Bahadur.  An optimal property of the likelihood ratio statistic , 1967 .

[9]  L. Brown.  Non-Local Asymptotic Optimality of Appropriate Likelihood Ratio Tests , 1971 .

[10]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[11]  Lars Holst,et al.  Asymptotic normality and efficiency for certain goodness-of-fit tests , 1972 .

[12]  L. Lecam.  Convergence of Estimates Under Dimensionality Restrictions , 1973 .

[13]  P. J. Huber,et al.  Minimax Tests and the Neyman-Pearson Lemma for Capacities , 1973 .

[14]  G. Tusnády.  On Asymptotically Optimal Tests , 1977 .

[15]  I. Csiszár.  Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem , 1984 .

[16]  Wilbert C.M. Kallenberg,et al.  On Moderate and Large Deviations in Multinomial Distributions , 1985 .

[17]  M. Quine,et al.  EFFICIENCIES OF CHI-SQUARE AND LIKELIHOOD RATIO GOODNESS-OF-FIT TESTS , 1985 .

[18]  D. Freedman,et al.  On the consistency of Bayes estimates , 1986 .