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ON LIMIT DISTRIBUTION OF MAXIMAL DEVIATION OF EMPIRICAL DISTRIBUTION DENSITY AND REGRESSION FUNCTION. II

THEORY OF PROBABILITY AND ITS APPLICATIONS 2012年第1期56卷 155-U1514页

作者： Muminov, M. S. Inst Math & Informat AN RUz Dept Theory Probabil & Math Stat Tashkent 100125 Uzbekistan

In this paper, for an unknown distribution function f(t), t is an element of R-nu, a random vector X is an element of R-nu, and a regression function r(t) = E(Y vertical bar X = t) of a random vector (X, Y), X is an element of R-nu, Y is an element of R-1, nonparametric kernel estimates f(n)(t) and r(n)(t) are constructed. It is proved that distribution of the maximal deviation of these estimators from the true distribution density f(t) and the regression function r(t) tend to the double exponential law as n -> infinity. With the aid of the constructed estimators we find a confidence region for f(t) and r(t), corresponding to the given confidence coefficient alpha (0 < alpha < 1), and construct a criterion for testing the hypothesis H-0 : f(t) = f(0)(t) (respectively, H'(0) : r(t) = r(0)(t)), where f(0)(t) is a given a priori distribution density, and r(0)(t) is a given function.

关键词： sequence of random vectors distribution density regression function nonparametric kernel estimates wiener process with multidimensional parameter local stationarity of random fields confidence region testing statistical hypothesis

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ON LIMIT DISTRIBUTION OF MAXIMAL DEVIATION OF EMPIRICAL DISTRIBUTION DENSITY AND REGRESSION FUNCTION. I

引用

THEORY OF PROBABILITY AND ITS APPLICATIONS 2011年第3期55卷 509-U516页

作者： Muminov, M. S. Inst Math & Informat AN RUz Dept Theory Probabil & Math Stat Tashkent 100125 Uzbekistan

In the paper, for unknown distribution density f(t), t is an element of R-nu, a random vector X is an element of R-nu, and regression function r(t) = E(Y vertical bar X = t) of a random vector (X, Y), X is an element of R-nu, Y is an element of R-1, nonparametric kernel estimates f(n)(t) and r(n)(t) are constructed. It is proved that the distribution of maximal deviation of these estimates from the true distribution density f(t) and regression function r(t) tends to the double exponential law for n -> infinity. With the help of the constructed estimates we found a confidence region for f(t) and r(t), corresponding to the given confidence coefficient alpha (0 < alpha < 1), and we constructed a criterion for testing the hypothesis H-0 : f(t) - f(0)(t) (respectively, H'(0) : r(t) = r(0)(t)), where f(0)(t) is a given a priori distribution density, r(0)(t) is a given function.

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