Goodness-of-fit test statistics that dominate the Kolmogorov statistics
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Two statistics are proposed for the simple goodness-of-fit problem. These are derived from a general principle for combining dependent test statistics that has been discussed elsewhere by the authors. It is shown that these statistics are relatively optimal in the sense of Bahadur efficiency and consequently, are more efficient than any weighted Kolmogorov statistic at every alternative. A curious pathology occurs: Under certain alternatives, the sequence of statistics has a Bahadur efficacy or exact slope only in the weak sense of convergence in law.
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- Abrahamson, I.: Exact Bahadur efficiences for the Kolmogorov-Smirnov and Kuiper one- and two-sample statistics. Ann. Math. Statist. 38, 1475–1490 (1967)Google Scholar
- Bahadur, R.R.: Some Limit Theorems in Statistics. SIAM; Philadelphia.Google Scholar
- Berk, R.H., Jones, D.H.: Relatively optimal combinations of test statistics. Scand. J. Statist. 5, 158–162 (1978)Google Scholar
- Chernoff, H.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, 493–507 (1952)Google Scholar
- Hoeffding, W.: Asymptotically optimal tests for multinomial distributions. Ann. Math. Statist. 36, 369–408 (1965)Google Scholar
- Jaeschke, D.: The limit distribution for the maximum of the standardized sample distribution function. [Unpublished 1977]Google Scholar
- Kiefer, J.: Skorohod embedding of multivariate RV's and the sample DF. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 24, 1–35 (1972)Google Scholar
- Kiefer, J.: Iterated logarithm analogues for sample quantiles when p n↓0. Proc. 6th Berkeley Sympos. Math. Statist. Probab. Univ. Calif. 1, 227–244 (1973)Google Scholar
- Lai, T.L.: On Chernoff-Savage statistics and sequential rank tests. Ann. Statist. 3, 825–845 (1975)Google Scholar
- Robbins, H.: A one-sided confidence interval for an unknown distribution function (Abstract). Ann. Math. Statist. 25, 409 (1954)Google Scholar
- Tusnády, G.: On asymptotically optimal tests. Ann. Statist. 5, 385–393 (1977)Google Scholar