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Fixed-α and fixed-βefficiencies
Published online by Cambridge University Press: 08 February 2013
Abstract
Consider testingH0 : F ∈ ω0against H1 : F ∈ ω1for a random sampleX1, ..., Xnfrom F, where ω0 andω1 are two disjoint sets of cdfs onℝ = (−∞, ∞). Two non-local types of efficiencies, referred to as thefixed-α and fixed-β efficiencies, are introduced forthis two-hypothesis testing situation. Theoretical tools are developed to evaluate theseefficiencies for some of the most usual goodness of fit tests (including theKolmogorov–Smirnov tests). Numerical comparisons are provided using several examples.
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- © EDP Sciences, SMAI, 2013
References
Abrahamson, I.G., Exact Bahadur efficiences
for they Kolmogorov–Smirnov and Kiefer one- and two-sample statistics.
Ann. Math. Stat.
38 (1967)
1475–1490. Google Scholar
Anderson, T.W. and
Darling, D.A., Asymptotic theory of certain
‘goodness of fit’ criteria based on stochastic processes.
Ann. Math. Stat.
23 (1952)
193–212. Google Scholar
Bahadur, R.R., An optimal property of the
likelihood ratio statistic, Proc. of the 5th Berkeley
Symposium
1 (1966) 13–26.
Google Scholar
Bahadur, R.R., Rates of convergence of
estimates and test statistics. Ann. Math.
Stat.
38 (1967)
303–324. Google Scholar
Brown, L.D., Non-local asymptotic optimality
of appropriate likelihood ratio tests. Ann. Math.
Stat.
42 (1971)
1206–1240. Google Scholar
Chernoff, H., A measure of asymptotic
efficiency for tests of a hypothesis based on the sum of observations.
Ann. Math. Stat.
23 (1952)
493–507. Google Scholar
R. Courant and D. Hilbert, Methods of
Mathematical Physics I. Wiley, New York (1989).
A.B. Hoadley, The theory of large deviations with
statistical applictions. University of Califonia, Berkeley, Unpublished dissertation
(1965).
Hoadley, A.B., On the probability of large
deviations of functions of several empirical cumulative distribution
functions. Ann. Math. Stat.
38 (1967)
360–382. Google Scholar
Hodges, J.L. and Lehmann, E.L., The efficiency of some
nonparametric competitors of the t-test.
Ann. Math. Stat.
27 (1956)
324–335. Google Scholar
Kac, M., Kiefer, J. and Wolfowitz, J., On tests of normality and
other tests of goodness of fit based on distance methods.
Ann. Math. Stat.
26 (1955) 189–11.
Google Scholar
Kallenberg, W.C.M. and
Koning, A.J., On Wieand’s
theorem. Stat. Probab. Lett.
25 (1995)
121–132. Google Scholar
Kallenberg, W.C.M. and
Ledwina, T., On local and nonlocal measures
of efficiency. Ann. Stat.
15 (1987)
1401–1420. Google Scholar
Kolmogorov, A.N., Confidence limits for an
unknown distribution function. Ann. Math.
Stat.
12 (1941)
461–463. Google Scholar
Litvinova, V.V. and
Nikitin, Y., Asymptotic efficiency and local
optimality of tests based on two-sample U- and
V-statistics. J. Math.
Sci.
152 (2008)
921–927. Google Scholar
Y. Nikitin, Asymptotic Efficiency of
Nonparametric Tests. Cambridge University Press, New York (1995).
E.S. Pearson and H.O Hartley, Biometrika
Tables for Statisticians II. Cambridge University Press, New York (1972).
Sethuraman, J., On the probability of large
deviations of families of sample means. Ann. Math.
Stat.
35 (1964)
1304–1316. Google Scholar
Sethuraman, J., On the probability of large
deviations of the mean for random variables in D [ 0,1
] . Ann. Math. Stat.
36 (1965)
280–285. Google Scholar
Stephens, M.A., The goodness-of-fit statistic
V N: distribution and significance
points. Biometrika
52 (1965)
309–321. Google Scholar
Wieand, H.S., A condition under which the
Pitman and Bahadur approaches to efficiency coincide.
Ann. Stat.
4 (1976)
1003–1011. Google Scholar
Withers, C.S. and Nadarajah, S., Power of a class of
goodness-of-fit test I. ESAIM : PS
13 (2009)
283–300. Google Scholar