Open AccessSufficient statistics for the Pareto distribution parameter
Author(s) -
I. S. Pulkin,
Andrey Tatarintsev
Publication year - 2021
Publication title -
rossijskij tehnologičeskij žurnal/russian technological journal
Language(s) - English
Resource type - Journals
eISSN - 2782-3210
pISSN - 2500-316X
DOI - 10.32362/2500-316x-2021-9-3-88-97
Subject(s) - mathematics , statistic , probability density function , geometric distribution , pareto distribution , pareto interpolation , differential entropy , exponential family , statistics , lomax distribution , sampling distribution , pareto principle , random variable , principle of maximum entropy , probability distribution , maximum entropy probability distribution , generalized pareto distribution , extreme value theory
The task of estimating the parameters of the Pareto distribution, first of all, of an indicator of this distribution for a given sample, is relevant. This article establishes that for this estimate, it is sufficient to know the product of the sample elements. It is proved that this product is a sufficient statistic for the Pareto distribution parameter. On the basis of the maximum likelihood method the distribution degree indicator is estimated. It is proved that this estimate is biased, and a formula eliminating the bias is justified. For the product of the sample elements considered as a random variable the distribution function and probability density are found; mathematical expectation, higher moments, and differential entropy are calculated. The corresponding graphs are built. In addition, it is noted that any function of this product is a sufficient statistic, in particular, the geometric mean. For the geometric mean also considered as a random variable, the distribution function, probability density, and the mathematical expectation are found; the higher moments, and the differential entropy are also calculated, and the corresponding graphs are plotted. In addition, it is proved that the geometric mean of the sample is a more convenient sufficient statistic from a practical point of view than the product of the sample elements. Also, on the basis of the Rao–Blackwell–Kolmogorov theorem, effective estimates of the Pareto distribution parameter are constructed. In conclusion, as an example, the technique developed here is applied to the exponential distribution. In this case, both the sum and the arithmetic mean of the sample can be used as sufficient statistics.