Exponential and related probability distributions on symmetric matrices. (English) Zbl 1490.60038

Hassairi, Abdelhamid; Roula, Amel

Stat. Probab. Lett. 187, Article ID 109499, 10 p. (2022).

Summary: Pursuing the study initiated in [A. Hassairi and A. Roula, Stat. Probab. Lett. 145, 37–42 (2019; Zbl 1414.62174)], we show in the present paper that the reliability function of a probability distribution on the cone \(\varOmega\) of positive definite symmetric matrices characterizes the distribution without any invariance condition. We also show that the characterization of the exponential probability distribution on \(\varOmega\) by a memoryless property holds without assuming an invariance condition. We then study the connection between the exponential distribution on \(\varOmega\) and the uniform distribution on a bounded interval of \(\varOmega\). A notion of matrix Pareto distribution is introduced, and it is shown that this distribution possesses the long tail property.

MSC:

60E05	Probability distributions: general theory
60B11	Probability theory on linear topological spaces
62E15	Exact distribution theory in statistics

Keywords:

exponential distribution; uniform distribution; Pareto distribution; Helgason-Fourier transform; reliability function

Citations:

Zbl 1414.62174

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