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The paper is devoted to goodness of fit tests based on probability density estimates generated by kernel functions. The test statistic is considered in the form of maximum of the normalized deviation of the estimate from its expected value or a hypothesized distribution density function. A comparative Monte Carlo power study of the investigated criterion is provided. Simulation results show that the proposed test is a powerful competitor to the existing classical criteria testing goodness of fit against a specific type of alternative hypothesis. An analytical way for establishing the asymptotic distribution of the test statistic is proposed, using the theory of high excursions of close to Gaussian random processes and fields introduced by Rudzkis (1992, 2012). [ABSTRACT FROM AUTHOR]

Let { ξ( t) , t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E ξ( t) ≡ 0, D ξ( t) ≡ 1, and continuous trajectories defined on the m-dimensional interval $ T \subset {\mathbb{R}^m} $. The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{ −v( t) < ξ( t) < u( t) , t ∈ T}, when, for all t ∈ T, u( t) , v( t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions of empirical processes and fields, Sov. Math., Dokl., 45(1):226-228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e + Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R( t) = cov( ξ′( t) , ξ′( t)). [ABSTRACT FROM AUTHOR]

This paper discusses a soft sample clustering problem for multivariate independent random data satisfying the mixture model of the Gaussian distribution. The theory recommends to estimate the parameters of model by the maximum likelihood method and to use “plug‐in” approach for data clustering. Unfortunately, the calculation problem of the maximum likelihood estimate is not completely solved in multivariate case. This work proposes a new constructive a few stage procedure to solve this task. This procedure includes statistical distribution analysis of a large number of the univariate projections of observations, geometric clustering of a multivariate sample and application of EM algorithm. The results of the accuracy analysis of the proposed methods is made by means of Monte‐Carlo simulation. [ABSTRACT FROM AUTHOR]