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.


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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