Sufficient statistics for the Pareto distribution parameter
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 suff...
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| Language: | Russian |
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MIREA - Russian Technological University
2021-06-01
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| Series: | Российский технологический журнал |
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| Online Access: | https://www.rtj-mirea.ru/jour/article/view/331 |
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| author | I. S. Pulkin A. V. Tatarintsev |
| author_facet | I. S. Pulkin A. V. Tatarintsev |
| author_sort | I. S. Pulkin |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-4e27f2a845be4a86bfa1857bbae8c0d0 |
| institution | Kabale University |
| issn | 2500-316X |
| language | Russian |
| publishDate | 2021-06-01 |
| publisher | MIREA - Russian Technological University |
| record_format | Article |
| series | Российский технологический журнал |
| spelling | doaj-art-4e27f2a845be4a86bfa1857bbae8c0d02025-02-03T11:45:50ZrusMIREA - Russian Technological UniversityРоссийский технологический журнал2500-316X2021-06-0193889710.32362/2500-316X-2021-9-3-88-97265Sufficient statistics for the Pareto distribution parameterI. S. Pulkin0A. V. Tatarintsev1MIREA – Russian Technological UniversityMIREA – Russian Technological UniversityThe 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.https://www.rtj-mirea.ru/jour/article/view/331pareto distributionsufficient statisticseffective estimationdistribution functionmoments |
| spellingShingle | I. S. Pulkin A. V. Tatarintsev Sufficient statistics for the Pareto distribution parameter Российский технологический журнал pareto distribution sufficient statistics effective estimation distribution function moments |
| title | Sufficient statistics for the Pareto distribution parameter |
| title_full | Sufficient statistics for the Pareto distribution parameter |
| title_fullStr | Sufficient statistics for the Pareto distribution parameter |
| title_full_unstemmed | Sufficient statistics for the Pareto distribution parameter |
| title_short | Sufficient statistics for the Pareto distribution parameter |
| title_sort | sufficient statistics for the pareto distribution parameter |
| topic | pareto distribution sufficient statistics effective estimation distribution function moments |
| url | https://www.rtj-mirea.ru/jour/article/view/331 |
| work_keys_str_mv | AT ispulkin sufficientstatisticsfortheparetodistributionparameter AT avtatarintsev sufficientstatisticsfortheparetodistributionparameter |