Regresja wektorów losowych dla wielowymiarowego rozkładu t-Studenta
REGRESSION OF RANDOM VECTORS FOR MULTIVARIATE t-STUDENT DISTRIBUTION
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The paper examines the concept of „conditional expected value”, which is of great importance in modern finance. The considerations are carried out on a five dimensional random vector with multivariate t-Student distribution. In the first part we construct a distribution of its coordinates in a 2:3 ratio (i.e., the vectors are two-and three-dimensional, respectively) in order to find an effective two-dimensional vector regression function in relation to the three-dimensional vector. To that end, the probability density distribution of the boundary three-dimensional vector is determined (by calculating the appropriate double integral), and then the conditional probability density distribution of two-dimensional vector was used to produce the three-dimensional vector. The second part of the paper discusses the reasoning presented in the first part and then generalises it for a random vector of any size that will remain applicable provided that it is a multi-dimensional random vectors with t-Student distribution. The results (the general form of the regression function) are illustrated with a specific quantitative example that maintains a „hyperplane” regression.
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