Yu. I. Petunin

Minimum-Area ellipse containing a finite set of points. II

We continue the investigation of the problem of construction of a minimum-area ellipse for a given convex polygon (this problem is solved for a rectangle and a trapezoid). For an arbitrary polygon, we prove that, in the case where the boundary of the minimum-area ellipse has exactly four or five common points with the polygon, this ellipse is the minimum-area ellipse for the quadrangles and pentagons formed by these common points.

A Nonparametric Test for the Equivalence of Populations Based on a Measure of Proximity of Samples

We propose a new measure of proximity of samples based on confidence limits for the bulk of a population constructed using order statistics. For this measure of proximity, we compute approximate confidence limits corresponding to a given significance level in the cases where the null hypothesis on the equality of hypothetical distribution functions may or may not be true. We compare this measure of proximity with the Kolmogorov–Smirnov and Wilcoxon statistics for samples from various populations. On the basis of the proposed measure of proximity, we construct a statistical test for testing the hypothesis on the equality of hypothetical distribution functions.

On the concept of generalized solution of operator equations in banach spaces

We introduce a new concept of generalized solution of operator equations with closed linear operator in a Banach space as an element of the completion of the space in certain locally convex topology. We prove a theorem on the existence and uniqueness of a generalized solution and give examples of finding the generalized solution for infinite systems of the linear algebraic equations.

A generalization of Bernoulli's model occurring in order statistics II

The frequency of occurrences of events in a modified Bernoulli model under the assumption Fx(u) ≡ Fy(u) is studied. A homogeneity test for two samples based on properties of this frequency and its properties are investigated.

Theory of quadratic estimates of variance

A class of quadratic estimates is constructed for the second-order moment and variance of a random variable. These estimates are found on the basis of sample values obtained by simple sampling. The best quadratic estimates are found for the second-order moment and variance in the case of known mathematical expectation. The exactness of biased and unbiased estimates of variance is investigated in the case of unknown mathematical expectation.

Testing hypotheses by using optimal statistical criteria. II

We study the structure of the critical function of an optimal statistical criterion for testing an arbitrary finite set of simple alternative hypotheses.

Chebyshev inequalities in symmetric spaces

The characterization (by means of inequalities) of some special Banach spaces is investigated.