Statistics

If a vector of variates is transformed into another vector, what is the resulting distribution? This chapter details the main methods of making such transformations and obtaining the new distribution, density, and characteristic functions. Although the proof of the transformation theorem for densities is not usually given in statistics textbooks, we use the shortcut of conditioning to give a statistical proof. Applications of the three methods range from simple transformation (including convolutions) to products and ratios. General transformations are also studied. These include rotations of vectors (and their Jacobian), transformations from uniform variates to others (useful for generating simulated samples in Monte-Carlo studies) via the probability integral transformation (PIT), exponential tilting, and others. Extreme-value distributions are studied, to complement the study of sample averages. The chapter concludes by revisiting the copula, which transforms the marginals into a joint distribution.