Statistical reasoning in dependent $p$-generalized elliptically contoured distributions and beyond. Subtitle: Testing scaling parameters, the role semi-inner products play, and simulating star-shaped distributed random vectors

First, likelihood ratio statistics for checking the hypothesis of equal variances of two-dimensional Gaussian vectors are derived both under the standard ( σ 2 1 , σ 2 2 , ) -parametrization and under the geometric (a, b,α)-parametrization where a2 and b2 are the variances of the principle components and α is an angle of rotation. Then, the likelihood ratio statistics for checking the hypothesis of equal scaling parameters of principle components of p-power exponentially distributed two-dimensional vectors are considered both under independence and under rotational or correlation type dependence. Moreover, the role semi-inner products play when establishing various likelihood equations is demonstrated. Finally, the dependent p-generalized polar method and the dependent p-generalized rejection-acceptance method for simulating star-shaped distributed vectors are presented.