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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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