Estimation of the moments of a quantized random variable
A significant part of the research on the problems of quantization of random variables is devoted to practical aspects of optimal quantization in the sense of filling in information. For these purposes, certain quantitative characteristics of quantized random variables are used, such as: mathemati...
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| Format: | Article |
| Language: | English |
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Saint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University)
2023-06-01
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| Series: | Naučno-tehničeskij Vestnik Informacionnyh Tehnologij, Mehaniki i Optiki |
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| Online Access: | https://ntv.ifmo.ru/file/article/22078.pdf |
| _version_ | 1797798481728897024 |
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| author | Mikhail I. Lomakin Alexander V. Dokukin |
| author_facet | Mikhail I. Lomakin Alexander V. Dokukin |
| author_sort | Mikhail I. Lomakin |
| collection | DOAJ |
| description | A significant part of the research on the problems of quantization of random variables is devoted to practical aspects
of optimal quantization in the sense of filling in information. For these purposes, certain quantitative characteristics of
quantized random variables are used, such as: mathematical expectation, variance and mean square deviation. At the
same time, to determine the quantitative characteristics of quantized random variables, as a rule, well-known parametric distributions are used: uniform, exponential, normal and others. In real situations, it is not possible to identify the initial
parametric distribution based on the available statistical information. In this paper, a nonparametric model is proposed
for determining such numerical characteristics of a quantized random variable as the highest initial moments. The
mathematical formalization of the problem of estimating the higher initial moments of a quantized random variable
in the conditions of incomplete data represented by small samples of a quantized random variable is performed in the
form of an optimization model of a certain integral of a piecewise continuous function satisfying certain conditions. The
final estimates of the highest initial moments of the quantized random variable are found as extreme (lower and upper)
estimates of a certain integral on a set of distribution functions with given moments equal to the sample moments of the
quantized random variable. A model of the higher initial moments of a quantized random variable is presented in the
form of a definite integral of a piecewise continuous function; in the general case, the problem of finding extreme (lower
and upper) estimates of the higher initial moments of a quantized random variable on a set of distribution functions with
given moments is solved. Examples of finding higher initial moments and optimal quantization of a random variable
are given. The obtained results can be used by specialists in evaluating and optimizing the quantization of various
information presented by random signals. |
| first_indexed | 2024-03-13T04:04:22Z |
| format | Article |
| id | doaj.art-8a60190e985341e0a2c58db5043acc26 |
| institution | Directory Open Access Journal |
| issn | 2226-1494 2500-0373 |
| language | English |
| last_indexed | 2024-03-13T04:04:22Z |
| publishDate | 2023-06-01 |
| publisher | Saint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University) |
| record_format | Article |
| series | Naučno-tehničeskij Vestnik Informacionnyh Tehnologij, Mehaniki i Optiki |
| spelling | doaj.art-8a60190e985341e0a2c58db5043acc262023-06-21T09:53:55ZengSaint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University)Naučno-tehničeskij Vestnik Informacionnyh Tehnologij, Mehaniki i Optiki2226-14942500-03732023-06-0123364665110.17586/2226-1494-2023-23-3-646-651Estimation of the moments of a quantized random variableMikhail I. Lomakin0https://orcid.org/0000-0003-4191-1348Alexander V. Dokukin1https://orcid.org/0000-0002-3342-8770D.Sc. (Engineering), D.Sc. (Economics), Chief Researcher, All-Russian Scietific Reserch Institute for Civil Defence and Emergencies of the EMERCOM of Russia, Moscow, 121352, Russian Federation, sc 7003352786D.Sc. (Economics), Chief Researcher, All- Russian Scietific Reserch Institute for Civil Defence and Emergencies of the EMERCOM of Russia, Moscow, 121352, Russian Federation, sc 56161047500A significant part of the research on the problems of quantization of random variables is devoted to practical aspects of optimal quantization in the sense of filling in information. For these purposes, certain quantitative characteristics of quantized random variables are used, such as: mathematical expectation, variance and mean square deviation. At the same time, to determine the quantitative characteristics of quantized random variables, as a rule, well-known parametric distributions are used: uniform, exponential, normal and others. In real situations, it is not possible to identify the initial parametric distribution based on the available statistical information. In this paper, a nonparametric model is proposed for determining such numerical characteristics of a quantized random variable as the highest initial moments. The mathematical formalization of the problem of estimating the higher initial moments of a quantized random variable in the conditions of incomplete data represented by small samples of a quantized random variable is performed in the form of an optimization model of a certain integral of a piecewise continuous function satisfying certain conditions. The final estimates of the highest initial moments of the quantized random variable are found as extreme (lower and upper) estimates of a certain integral on a set of distribution functions with given moments equal to the sample moments of the quantized random variable. A model of the higher initial moments of a quantized random variable is presented in the form of a definite integral of a piecewise continuous function; in the general case, the problem of finding extreme (lower and upper) estimates of the higher initial moments of a quantized random variable on a set of distribution functions with given moments is solved. Examples of finding higher initial moments and optimal quantization of a random variable are given. The obtained results can be used by specialists in evaluating and optimizing the quantization of various information presented by random signals.https://ntv.ifmo.ru/file/article/22078.pdfrandom variablesamplequantumprobabilitymomentsdistribution function |
| spellingShingle | Mikhail I. Lomakin Alexander V. Dokukin Estimation of the moments of a quantized random variable Naučno-tehničeskij Vestnik Informacionnyh Tehnologij, Mehaniki i Optiki random variable sample quantum probability moments distribution function |
| title | Estimation of the moments of a quantized random variable |
| title_full | Estimation of the moments of a quantized random variable |
| title_fullStr | Estimation of the moments of a quantized random variable |
| title_full_unstemmed | Estimation of the moments of a quantized random variable |
| title_short | Estimation of the moments of a quantized random variable |
| title_sort | estimation of the moments of a quantized random variable |
| topic | random variable sample quantum probability moments distribution function |
| url | https://ntv.ifmo.ru/file/article/22078.pdf |
| work_keys_str_mv | AT mikhaililomakin estimationofthemomentsofaquantizedrandomvariable AT alexandervdokukin estimationofthemomentsofaquantizedrandomvariable |