Approximation of multivariate distribution functions.(English)Zbl 1195.62008

Let $$X$$ be a $$p$$-dimensional random vector with known distribution function $$F( x)$$ and density function $$f(x)$$. Let $$Y$$ be a $$p$$-dimensional random vector with unknown distribution function $$F_{Y}(x)$$. The main result of the paper is an explicitly expressed approximation of the unknown distribution function $$F_{Y}(x)$$ using the known distribution function $$F(x)$$ by means of Taylor expansions. The paper contains necessary results of matrix algebra. A new operation, the matrix integral, is introduced, studied, and applied. An application of the main result to the case when $$X$$ has a bivariate normal distribution is presented. For simulated bivariate data the goodness of the last mentioned estimated approximation is established. The paper also contains many useful formulas.

MSC:

 62E17 Approximations to statistical distributions (nonasymptotic) 62H10 Multivariate distribution of statistics 62E20 Asymptotic distribution theory in statistics 26B12 Calculus of vector functions
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