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Mikhail Ermakov

Publications and source records attributed to Mikhail Ermakov.

17 recordsLinked to original sources
arXiv2026

Parametric Statistical Inference in the Zone of Moderate Deviation Probabilities

A parametric theory of statistical inference is developed for the moderate deviation probability zone. The new approach to the proofs is based on the Taylor series expansion of the logarithm of the likelihood ratio based on the Hellinger distance. The Large Deviation Principle in the moderate deviation probability zone is proven for Bayesian estimators and maximum likelihood estimators. A uniform approximation of the logarithm of the likelihood ratio and Theorem on concentration of the posterior Bayesian measure are also established for the zone of moderate deviation probabilities.

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arXiv2025

Bahadur asymptotic efficiency in the zone of moderate deviation probabilities

For a sequence of independent identically distributed random variables having a distribution function with an unknown parameter from a set $\Theta \subset \mathbf{R}^d$, we prove an analogue of the lower bound of Bahadur asymptotic efficiency for the zone of moderate deviation probabilities. The assumptions coincide with assumptions conditions under which the locally asymptotically minimax lower bound of Hajek-Le Cam was proved. The lower bound for local Bahadur asymptotic efficiency is a special case of this lower bound.

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arXiv2023

On uniformly consistent tests

Necessary and sufficient conditions of uniform consistency are explored. A hypothesis is simple. Nonparametric sets of alternatives are bounded convex sets in $\mathbb{L}_p$, $p >1$ with "small" balls deleted. The "small" balls have the center at the point of hypothesis and radii of balls tend to zero as sample size increases. For problem of hypothesis testing on a density, we show that, for the sets of alternatives, there are uniformly consistent tests for some sequence of radii of the balls, if and only if, convex set is relatively compact. The results are established for problem of hypothesis testing on a density, for signal detection in Gaussian white noise, for linear ill-posed problems with random Gaussian noise and so on.

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arXiv2022

Nonparametric signal detection with small values of type I and type II error probabilities

We consider problem of signal detection in Gaussian white noise. Test statistics are linear combinations of squares of estimators of Fourier coefficients or $\mathbb{L}_2$-norms of kernel estimators. We point out necessary and sufficient conditions when nonparametric sets of alternatives have a given rate of exponential decay for type II error probabilities.

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arXiv2021

Chi-squared test for hypothesis testing of homogeneity

We provide necessary and sufficient conditions of uniform consistency of nonparametric sets of alternatives of chi-squared test for testing of hypothesis of homogeneity. The number of cells of chi-squared test increases with sample size growth. Nonparametric sets of alternatives can be defined both in terms of densities and distribution functions.

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arXiv2020

On uniform consistency of nonparametric tests I

We point out necessary and sufficient conditions of uniform consistency of nonparametric sets of alternatives for widespread nonparametric tests. Nonparametric sets of alternatives can be defined both in terms of distribution function and in terms of density (or signals in the problem of signal detection in Gaussian white noise). In this part of paper such conditions are provided for $χ^2-$tests with increasing number of cells, Cramer-von Mises tests, tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients.

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arXiv2020

On uniform consistency of nonparametric tests II

For Kolmogorov test we find natural conditions of uniform consistency of sets of alternatives approaching to hypothesis. Sets of alternatives can be defined both in terms of distribution functions and in terms of densities.

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arXiv2019

On consistency and inconsistency of nonparametric tests

For $χ^2-$tests with increasing number of cells, Cramer-von Mises tests, tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients, we find necessary and sufficient conditions of consistency and inconsistency for sequences of alternatives having a given rate of convergence to hypothesis in $\mathbb{L}_2$-norm. We provide transparent interpretations of these conditions allowing to understand the structure of such consistent sequences. For problem of signal detection in Gaussian white noise we show that, if set of alternatives is bounded closed center-symmetric convex set $U$ with deleted "small" $\mathbb{L}_2$ -- ball, then compactness of set $U$ is necessary condition for existence of consistent tests.

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arXiv2018

On maxispaces of nonparametric tests

For the problems of nonparametric hypothesis testing we introduce the notion of maxisets and maxispace. We point out the maxisets of $χ^2-$tests, Cramer-von Mises tests, tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients. For these tests we show that, if sequence of alternatives having given rates of convergence to hypothesis is consistent, then each altehrnative can be broken down into the sum of two parts: a function belonging to maxiset and orthogonal function. Sequence of functions belonging to maxiset is consistent sequence of alternatives. We point out asymptotically minimax tests if sets of alternatives are maxiset with deleted "small" $\mathbb{L}_2$-balls.

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arXiv2017

On minimax nonparametric estimation of signal in Gaussian noise

For the problem of nonparametric estimation of signal in Gaussian noise we point out the strong asymptotically minimax estimators on maxisets for linear estimators (see \cite{ker93,rio}). It turns out that the order of rates of convergence of Pinsker estimator on this maxisets is worse than the order of rates of convergence for the class of linear estimators considered on this maxisets. We show that balls in Sobolev spaces are maxisets for Pinsker estimators.

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arXiv2017

On One Property of Tikhonov Regularization Algorithm

For linear inverse problem with Gaussian random noise we show that Tikhonov regularization algorithm is minimax in the class of linear estimators and is asymptotically minimax in the sense of sharp asymptotic in the class of all estimators. The results are valid if some a priori information on a Fourier coefficients of solution is provided. For trigonometric basis this a priori information implies that the solution belongs to a ball in Besov space $B^r_{2\infty}$.

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arXiv2015

On Consistent Hypothesis Testing

This paper explores conditions of existence of different types of consistent tests. New links of these types of consistency are also established. The existence of discernible (strong consistent) tests follows from the existence of pointwise consistent tests. If there are consistent tests, then the set of alternatives can be represented as countable union of nested subsets such that there are uniformly consistent tests for the hypothesis and each of this subset of alternatives. Implementing these results we explore both sufficient conditions and necessary conditions for existence of consistent, pointwise consistent, uniformly consistent and discernible tests for hypothesis testing on a probability measure of independent sample, on a mean measure of Poisson process, on a solution of linear ill-posed problems in Gaussian noise, on a solution of deconvolution problem and for the problem of signal detection in Gaussian white noise. In the last three cases the necessary conditions and sufficient conditions coincide.

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arXiv2015

On asymptotically efficient statistical inference on a signal parameter

We consider the problems of confidence estimation and hypothesis testing on a parameter of signal observed in Gaussian white noise. For these problems we point out lower bounds of asymptotic efficiency in the zone of moderate deviation probabilities. These lower bounds are versions of local asymptotic minimax Hajek-Le Cam lower bound in estimation and the lower bound for Pitman efficiency in hypothesis testing. The lower bounds were obtained for both logarithmic and sharp asymptotic of moderate deviation probabilities.

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arXiv2014

A moderate deviation principle for empirical bootstrap measure

We prove two Large deviations principles (LDP) in the zone of moderate deviation probabilities. First we establish LDP for the conditional distributions of moderate deviations of empirical bootstrap measures given empirical probability measures. Second we establish LDP for the joint distributions of empirical measure and bootstrap empirical measures. Using these LDPs, similar LDPs for statistical differentiable functionals can be established. The LDPs for moderate deviations of empirical quantile processes and empirical bootstrap copula function are provided as illustration of these results.

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arXiv2013

On distinguishability of hypotheses

We consider the problems of hypothesis testing on a probability measure of independent sample, on solution of ill-posed problem, on deconvolution problem and on Poisson mean measure. For all these setups necessary conditions and sufficient conditions are given for distinguishability of sets of hypothesis. In the case of hypothesis testing on a probability measure and on Poisson mean measure the results are given in terms of weak topology and topology of weak convergence on all Borel sets. The problem of discernibility of hypothesis is also studied. In other cases the necessary and sufficient conditions of distinguishability are given if the sets of hypotheses are bounded sets in $L_2$.

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