Abstract:
We study the lower bounds of efficiency for the moderate deviation probabilities
of tests and estimators. These bounds cover both the logarithmic
and strong asymptotics.
For the problems of hypothesis testing we propose a natural representation for
the lower bounds of type I
and type II error probabilities in terms of inverse function of
the standard normal distribution. The lower bounds for the
moderate deviation probabilities of estimators are deduced easily from
the corresponding bounds in hypothesis testing.
Keywords:
large deviations, moderate deviations, efficiency, Bahadur efficiency, Chernoff efficiency.
This publication is cited in the following 11 articles:
Theory Probab. Appl., 69:1 (2024), 150–160
M. S. Ermakov, “Lokalnaya asimptoticheskaya normalnost logarifma otnosheniya pravdopodobiya v zone veroyatnostei umerennykh uklonenii”, Veroyatnost i statistika. 34, Posvyaschaetsya yubileyu Andreya Nikolaevicha BORODINA, Zap. nauchn. sem. POMI, 525, POMI, SPb., 2023, 71–85
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Inglot T., “Intermediate Efficiency of Tests Under Heavy-Tailed Alternatives”, Prob. Math. Stat.., 40:2 (2020), 331–348
Inglot T. Ledwina T. Cmiel B., “Intermediate Efficiency in Nonparametric Testing Problems With An Application to Some Weighted Statistics”, ESAIM-Prob. Stat., 23 (2019), 697–738
Ji P. Nussbaum M., “Sharp Minimax Adaptation Over Sobolev Ellipsoids in Nonparametric Testing”, Electron. J. Stat., 11:2 (2017), 4515–4562
M. S. Ermakov, “On asymptotically efficient statistical inference on a signal parameter”, J. Math. Sci. (N. Y.), 206:2 (2015), 159–170
Ermakov M., “The Sharp Lower Bound of Asymptotic Efficiency of Estimators in the Zone of Moderate Deviation Probabilities”, Electron. J. Stat., 6 (2012), 2150–2184
Mikhail Ermakov, “Nonparametric signal detection with small type I and type II error probabilities”, Stat Inference Stoch Process, 14:1 (2011), 1
M. S. Ermakov, “Nonparametric Hypothesis Testing with Small Type I or Type II Error Probabilities”, Problems Inform. Transmission, 44:2 (2008), 119–137
M. S. Ermakov, “On semiparametric inference in moderate deviation zone”, J. Math. Sci. (N. Y.), 152:6 (2008), 869–874