For parametric sets of alternatives the efficiency of nonparametric tests has been studied quite thoroughly. If sets of alternatives are contained in a certain class of functions of given smoothness, the distinguishability of these sets by nonparametric tests is also fairly well explored. 

In talk, we explore more general setups. New results are presented and a review of recent results is provided, answering two questions: 

Under what necessary and sufficient conditions can nonparametric sets of alter-natives be distinguishable based on some nonparametric tests? Here we allow sets of alternatives to approach the hypothesis [1, 2]. 

What necessary and sufficient conditions should nonparametric sets of alternatives satisfy if they are distinguished by one of the most wide spread nonparametric tests? The results are obtained for sets of alternatives defined both in terms of distribution functions and densities [3]. 

References 

[1] Ermakov, M.S. On Consistent Hypothesis Testing. Journal of Mathematical Sciences. 225(5), (2017), 751-769. 

[2] Ermakov, M.S. On uniformly consistent tests. https://arxiv.org/abs/2303.00680 

[3] Ermakov, M S On Uniform consistency of nonparametric tests I. Journal of Mathematical Sciences, 258 (2021), 802-837.

I, Mikhail Sergeyevich Ermakov, was born on 12/25/1952. In December 1975 I graduated from the Faculty of Mathematics and Mechanics of Leningrad State University with a degree in mathematics. Since January 1976 I have been a trainee researcher at the Laboratory of Statistical Methods of the Leningrad Branch of the Mathematical Institute, Academy of Sciences of the USSR. Since January 1978. I am a junior researcher there. In 1979. I defended my PhD thesis. At the Mathematical Institute, I, together with Professors I.A.Ibragimov (POMI) and R.Z.Khasminsky (Institute of Information Transmission Problems of the USSR Academy of Sciences), explored the problem ”when sequential estimation provides a significant gain compared to estimation with a fixed sample size” [1]. This problem was set by academician Yu.V.Linnik.

From September 1981 to September 1992, I worked at the applied institutes of the USSR Academy of Sciences and Leningrad State University. Basically, I was engaged in solving applied problems of probability theory and mathematical statistics. In particular, for solving applied problems in the geophysics in 1990, I received a diploma of a senior researcher in the specialty ”geophysics”. 

At the same time, I continued my research on mathematical statistics. I have constructed asymptotically minimax estimators in the deconvolution problem with Gaussian stationary noise [2, 3], if a priori information is given that solution belongs to a ball in Sobolev space. I found asymptotically minimax tests for the problem of detecting a signal in Gaussian white noise when the set of alternatives is an ellipsoid in L2 with a cut-out ”small ball”[4]. These results are obtained for the strong asymptotics of the convergence rate of L2-norms of estimators and probabilities of type I and type II errors, respectively. 

In 1992 I defended my doctoral dissertation in probability theory and mathematical statistics in St.Peterburg Branch of the Mathematical Institute, Academy of Sciences. The research has been devoted asymptotic problems of statistics.

From September 1992 to the present I have been working a leading researcher at the Laboratory of Reliability Theory Methods at the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences. Below main scientific results for subsequent years are provided. 

1. Extension to the zone of moderate deviation probabilities of Hajek-Le Cam lower bound in statistical estimation. Similar results were obtained for Pitman efficiency in hypothesis testing. These results were obtained for both strong and logarithmic asymptotics [5, 6]. 

2. Proof of the asymptotic minimax properties of classical nonparametric tests when the set of alternatives are provided in terms of the distance method [7, 8] 

3. The necessary and sufficient conditions were established for the uniform consistency of sets of hypothesis and alternatives for the most widespread setups of hypothesis testing (hypothesis testing on a density, intensity of Poisson process, signal detection, ill-posed problems with Gaussian noise and so on) [9, 10]. Here we supposed that the sets of hypotheses and alternatives are consistent if there is tests for which they are consistent. 

4. The necessary and sufficient conditions for the uniform consistency of sets of alternatives are established for the most widespread nonparametric tests. Conditions were provided for sets of alternatives defined both in terms of distribution functions and densities [8]. 

From September 1996 to the present I have been a professor at the Department of Statistical Modeling of the Faculty of Mathematics and Mechanics of St. Petersburg State University. I lecture on intermediate and advanced statistics for bachelors, masters and postgraduate students.