Petunin Yu. I.
Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 147-163
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.
Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1217–1231
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.
Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1098–1105
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.
Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 980–988
From the geometric point of view, we consider the problem of construction of a minimum-area ellipse containing a given convex polygon. For an arbitrary triangle, we obtain an equation for the boundary of the minimum-area ellipse in explicit form. For a quadrangle, the problem of construction of a minimumarea ellipse is connected with the solution of a cubic equation. For an arbitrary polygon, we prove that if the boundary of the minimum-area ellipse has exactly three common points with the polygon, then this ellipse is the minimum-area ellipse for the triangle obtained.
Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1286–1290
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.
Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 506–515
We study the structure of the critical function of an optimal statistical criterion for testing an arbitrary finite set of simple alternative hypotheses.
Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 378–388
We propose a method for constructing statistical criteria. It can be used for testing an arbitrary finite set of simple alternative hypotheses. A concept of an optimal statistical criterion is introduced, special cases of which are the Bayesian criterion and the minimax criterion. It is proved that any optimal statistical criterion can be constructed on the basis of the likelihood ratio.
Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 779-786
Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 518–528
Ukr. Mat. Zh. - 1989. - 41, № 11. - pp. 1512–1521
Ukr. Mat. Zh. - 1988. - 40, № 6. - pp. 799–803
Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 87 – 93
Characterization of the Hilbert space L2 (Ω, U, μ) in terms of the additivity of the measure of dispersion
Ukr. Mat. Zh. - 1984. - 36, № 6. - pp. 683 – 688
Ukr. Mat. Zh. - 1982. - 34, № 3. - pp. 374—377
Ukr. Mat. Zh. - 1977. - 29, № 3. - pp. 344–350
Linear regression parameter estimation in the presence of constraints on linear regression coefficients
Ukr. Mat. Zh. - 1976. - 28, № 4. - pp. 463–472
Ukr. Mat. Zh. - 1976. - 28, № 2. - pp. 237–243
On the structure of a ?-algebra of borel sets and the convergence of certain stochastic series in Banach spaces
Ukr. Mat. Zh. - 1975. - 27, № 4. - pp. 435–442
Ukr. Mat. Zh. - 1974. - 26, № 1. - pp. 102–106
Asymptotic behavior of variance of best unbiased linear estimate of unknown mean of stationary random process obtained by uniform division of observation interval
Ukr. Mat. Zh. - 1973. - 25, № 2. - pp. 214—227
Ukr. Mat. Zh. - 1971. - 23, № 2. - pp. 157–167