Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case
In the paper, we consider a new approach to the comparison of the distributions of sums of random variables. Unlike preceding works, for this purpose we use the notion of deficiency that is well known in mathematical statistics. This approach is used, first, to determine the distribution of a separa...
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| Format: | Article |
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MDPI AG
2022-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/3/454 |
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| author | Vladimir E. Bening Victor Y. Korolev |
| author_facet | Vladimir E. Bening Victor Y. Korolev |
| author_sort | Vladimir E. Bening |
| collection | DOAJ |
| description | In the paper, we consider a new approach to the comparison of the distributions of sums of random variables. Unlike preceding works, for this purpose we use the notion of deficiency that is well known in mathematical statistics. This approach is used, first, to determine the distribution of a separate random variable in the sum that provides the least possible number of summands guaranteeing the prescribed value of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>-quantile of the normalized sum for a given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and second, to determine the distribution of a separate random variable in the sum that provides the least possible number of summands guaranteeing the prescribed value of the probability for the normalized sum to fall into a given interval. Both problems are solved under the condition that possible distributions of random summands possess coinciding three first moments. In both settings the best distribution delivers the smallest number of summands. Along with distributions of a non-random number of summands, we consider the case of random summation and introduce an analog of deficiency which can be used to compare the distributions of sums with random and non-random number of summands. The main mathematical tools used in the paper are asymptotic expansions for the distributions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>-valued functions of random vectors, in particular, normalized sums of independent identically distributed r.v.s and their quantiles. Along with the general case, main attention is paid to the situation where the summarized random variables are independent and identically distributed. The approach under consideration is applied to determination of the distribution of insurance payments providing the least insurance portfolio size under prescribed Value-at-Risk or non-ruin probability. |
| first_indexed | 2024-03-09T23:32:19Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T23:32:19Z |
| publishDate | 2022-01-01 |
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| spelling | doaj.art-99712c705c564273a17988605ea5e4622023-11-23T17:07:46ZengMDPI AGMathematics2227-73902022-01-0110345410.3390/math10030454Comparing Distributions of Sums of Random Variables by Deficiency: Discrete CaseVladimir E. Bening0Victor Y. Korolev1Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991 Moscow, RussiaFaculty of Computational Mathematics and Cybernetics, Moscow State University, 119991 Moscow, RussiaIn the paper, we consider a new approach to the comparison of the distributions of sums of random variables. Unlike preceding works, for this purpose we use the notion of deficiency that is well known in mathematical statistics. This approach is used, first, to determine the distribution of a separate random variable in the sum that provides the least possible number of summands guaranteeing the prescribed value of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>-quantile of the normalized sum for a given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and second, to determine the distribution of a separate random variable in the sum that provides the least possible number of summands guaranteeing the prescribed value of the probability for the normalized sum to fall into a given interval. Both problems are solved under the condition that possible distributions of random summands possess coinciding three first moments. In both settings the best distribution delivers the smallest number of summands. Along with distributions of a non-random number of summands, we consider the case of random summation and introduce an analog of deficiency which can be used to compare the distributions of sums with random and non-random number of summands. The main mathematical tools used in the paper are asymptotic expansions for the distributions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>-valued functions of random vectors, in particular, normalized sums of independent identically distributed r.v.s and their quantiles. Along with the general case, main attention is paid to the situation where the summarized random variables are independent and identically distributed. The approach under consideration is applied to determination of the distribution of insurance payments providing the least insurance portfolio size under prescribed Value-at-Risk or non-ruin probability.https://www.mdpi.com/2227-7390/10/3/454limit theoremsum of independent random variablesrandom sumasymptotic expansionasymptotic deficiencykurtosis |
| spellingShingle | Vladimir E. Bening Victor Y. Korolev Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case Mathematics limit theorem sum of independent random variables random sum asymptotic expansion asymptotic deficiency kurtosis |
| title | Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case |
| title_full | Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case |
| title_fullStr | Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case |
| title_full_unstemmed | Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case |
| title_short | Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case |
| title_sort | comparing distributions of sums of random variables by deficiency discrete case |
| topic | limit theorem sum of independent random variables random sum asymptotic expansion asymptotic deficiency kurtosis |
| url | https://www.mdpi.com/2227-7390/10/3/454 |
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