We study natural links between various types of consistency: usual consistency, strong consistency, uniform consistency, and pointwise consistency. On the base of these results, we provide both sufficient conditions and necessary conditions for the existence of various types of consistent tests for a wide spectrum of problems of hypothesis testing which arise in statistics: on a probability measure of an independent sample, on a mean measure of a Poisson process, on a solution of an ill-posed linear problem in a Gaussian noise, on a solution of the deconvolution problem, for signal detection in a Gaussian white noise. In the last three cases, the necessary and sufficient conditions coincide.
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References
M. Arcones, “The large deviation principle for stochastic processes. II,” Teor. Veroyatn. Primen., 48, 19–44 (2004).
R. G. Bahadur and R. G. Savage, “The nonexistence of certain statistical procedures in nonparametric problems,” Ann. Math. Statist., 27, 1115–1122 (1956).
N. Balakrishnan, M. S. Nikulin, and V. Voinov, Chi-squared Goodness of Fit Tests With Applications, Elsevier, Oxford (2013).
A. R. Barron, “Uniformly powerful goodness of fit tests,” Ann. Statist., 17, 107–124 (1989).
A. Berger, “On uniformly consistent tests,” Ann. Math. Statist., 18, 289–293 (1989).
V. I. Bogachev, Gaussian Measures [in Russian], Moscow (1997).
V. I. Bogachev, Measure Theory, Springer, New York (2000).
M. V. Burnashev, “On the minimax solution of inaccurately known signal in a white Gaussian noise. Background,” Teor. Veroyatn. Primen., 24, 107-119 (1979).
L. Comminges and A. S. Dalalyan, “Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression,” Electronic Journal of Statistics, 7, 146–190 (2013).
T. M. Cover, “On determining irrationality of the mean of a random variable,” Ann. Statist., 1, 862–871 (1973).
A. Dembo and Y. Peres, “A topological criterion for hypothesis testing,” Ann. Statist., 22, 106–117 (1994).
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston (1993).
L. Devroye and G. Lugosi, “Almost sure classification of densities,” J. Nonpar. Statist., 14, 675–698 (2002).
L. L. Donoho, “One-sided inference about functionals of a density,” Ann. Statist., 16, 1390–1420 (1988).
N. Dunford and J. T. Schwartz, Linear Operators. Part I, Interscience Publishers, New York (1958).
M. S. Ermakov, “Large deviations of empirical measures and statistical tests,” Zap. Nauchn. Semin. POMI, 207, 37–59 (1993).
M. S. Ermakov, “On distinguishability of two nonparametric sets of hypotheses,” Statist. Probab. Letters., 48, 275–282 (2000).
M. S. Ermakov, “Nonparametric signal detection with small type I and type II error probabilities,” Statistical Inference for Stochastic Processes, 14, 1–19 (2011).
P. Ganssler, “Compactness and sequential compactness on the space of measures,” Z.Wahrsch.Verw. Gebiete, 17, 124–146 (1971).
P. Groeneboom, J. Oosterhoff, and F. H. Ruymgaart, “Large deviation theorems for empirical probability measures,” Ann. Probab., 7, 553–586 (1979).
W. Hoeffding and J. Wolfowitz, “Distinguishability of sets of distributions,” Ann. Math. Stat., 29, 700–718 (1958).
J. L. Horowitz and V. S. Spokoiny, “An adaptive, rate optimal test of a parametric model against a nonparametric alternative,” Econometrica, 69, 599–631 (2001).
I. A. Ibragimov and R. Z. Khasminskii, “On the estimation of infinitely dimensional parameter in Gaussian white noise,” Dokl. Akad. Nauk USSR, 236, 1053–1055 (1977).
Yu. I. Ingster, “Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III,” Mathematical Methods of Statistics, 2, 85–114, 171–189, 249–268 (1993).
Yu. I. Ingster and Yu. A. Kutoyants, “Nonparametric hypothesis testing for intensity of the Poisson process,” Mathematical Methods of Statistics, 16, 218–246 (2007).
Yu. I. Ingster and I. A. Suslina, “Nonparametric goodness-of-fit testing under Gaussian models,” Lect. Notes Statist., 169, Springer (2002).
A. Janssen, “Global power function of goodness of fit tests,” Ann. Statist., 28, 239–253 (2000).
C. Kraft, “Some conditions for consistency and uniform consistency of statistical procedures,” Univ. Californ. Publ. Stat., 2, 125–142 (1955).
S. R. Kulkarni and O. Zeitouni, “A general classification rule for probability measures,” Ann. Statist. 23, 1393–1407 (1995).
L. Le Cam, “Convergence of estimates under dimensionality restrictions,” Ann. Statist., 1, 38–53 (1973).
L. Le Cam and L. Schwartz, “A necessary and sufficient conditions for the existence of consistent estimates,” Ann. Math. Statist., 31, 140–150 (1960).
E. L. Lehmann and J. P. Romano, Testing Statistical Hypothesis, Springer-Verlag, New York (2005).
A. B. Nobel, “Hypothesis testing for families of ergodic processes,” Bernoulli, 12, 251–269 (2006).
L. Schwartz, “On Bayes procedures,” Z. Wahrsch. Verw. Gebiete, 4, 10–26 (1965).
J. Pfanzagl, “On the existence of consistent estimates and tests,” Z.Wahrsch.Verw. Gebiete, 10, 43–62 (1968).
A. Shapiro, “On concepts of directional differentiability,” J. Optimization Theory and Appl., 66, 477–487 (1990).
A. W. van der Vaart, Asymptotic Statistics, Cambridge Univ. Press, Cambridge (1998).
K. Yosida, Functional Analysis, Springer, Berlin (1968).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 442, 2015, pp. 48–74.
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Ermakov, M. On Consistent Hypothesis Testing. J Math Sci 225, 751–769 (2017). https://doi.org/10.1007/s10958-017-3491-4
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DOI: https://doi.org/10.1007/s10958-017-3491-4