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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.

Reviewer: Gejza Wimmer (Bratislava)

### MSC:

62E17 | Approximations to statistical distributions (nonasymptotic) |

62H10 | Multivariate distribution of statistics |

62E20 | Asymptotic distribution theory in statistics |

26B12 | Calculus of vector functions |

### References:

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