Exact extreme value, product, and ratio distributions under non-standard assumptions

The exact distributions of many functions of random vectors are derived in the literature mainly for the case of a Gaussian vector distribution or under the assumption that the vector follows a spherical or an elliptically contoured distribution. Numerous standard statistical applications are given for these cases. Deriving analogous results, if the sample distribution comes from a large family of probability laws, needs to make use of new analytical tools from the area of exact distribution theory. The present paper provides the application of such tools suitable for deriving the exact cumulative distribution functions and density functions of extreme values, products, and ratios in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$l_{2,p}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>l</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>p</mi> </mrow> </msub> </math> </EquationSource> </InlineEquation>-symmetrically distributed populations. Accompanying simulation studies are presented in cases of power-exponentially distributed populations and for different sample sizes. As an application, well-known results on the increasing failure rate properties of extremes from Gaussian samples are extended to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>p</mi> </math> </EquationSource> </InlineEquation>-power exponential sample distributions. Copyright Springer-Verlag Berlin Heidelberg 2015