Sufficient statistics for the Pareto distribution parameter
https://doi.org/10.32362/2500-316X-2021-9-3-88-97
Abstract
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.
About the Authors
I. S. Pulkin
MIREA – Russian Technological University
Russian Federation
Russian Federation
Igor S. Pulkin, Cand. Sci. (Phys.–Math.), Associate Professor, Higher Mathematics Department, Institute of Cybernetics
78, Vernadskogo pr., Moscow, 119454
A. V. Tatarintsev
MIREA – Russian Technological University
Russian Federation
Russian Federation
Andrey V. Tatarintsev, Cand. Sci. (Phys.–Math.), Associate Professor, Department of Mathematics, Institute of Physics and Technology
Scopus Autor ID: 57221996001, 7004076246
78, Vernadskogo pr., Moscow, 119454
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1. Graphs of the probability density of the distribution of a random variable Y with n = 10 (red) and n = 12 (green), α = 2.5 | |
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The study aimed to estimate the Pareto distribution parameters, first of all, an indicator of this distribution for a given sample. It is proved that the product of the sample elements, as well as any function of this product, in particular, the geometric mean, is a sufficient statistic for the Pareto distribution parameter. 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.
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For citations:
Pulkin I.S., Tatarintsev A.V. Sufficient statistics for the Pareto distribution parameter. Russian Technological Journal. 2021;9(3):88-97. (In Russ.) https://doi.org/10.32362/2500-316X-2021-9-3-88-97
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