Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case
In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous...
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MDPI AG
2022-12-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/24/4712 |
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| author | Vladimir Bening Victor Korolev |
| author_facet | Vladimir Bening Victor Korolev |
| author_sort | Vladimir Bening |
| collection | DOAJ |
| description | In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the 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 Poisson 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>. This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic <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 sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution. |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T16:08:24Z |
| publishDate | 2022-12-01 |
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| spelling | doaj.art-be0d4afd842a47ce83cd498fec1eb5ed2023-11-24T16:28:26ZengMDPI AGMathematics2227-73902022-12-011024471210.3390/math10244712Comparing Compound Poisson Distributions by Deficiency: Continuous-Time CaseVladimir Bening0Victor Korolev1Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, RussiaFaculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, RussiaIn the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the 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 Poisson 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>. This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic <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 sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution.https://www.mdpi.com/2227-7390/10/24/4712limit theoremcompound Poisson distributionPoisson random sumasymptotic expansionasymptotic deficiencykurtosis |
| spellingShingle | Vladimir Bening Victor Korolev Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case Mathematics limit theorem compound Poisson distribution Poisson random sum asymptotic expansion asymptotic deficiency kurtosis |
| title | Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case |
| title_full | Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case |
| title_fullStr | Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case |
| title_full_unstemmed | Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case |
| title_short | Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case |
| title_sort | comparing compound poisson distributions by deficiency continuous time case |
| topic | limit theorem compound Poisson distribution Poisson random sum asymptotic expansion asymptotic deficiency kurtosis |
| url | https://www.mdpi.com/2227-7390/10/24/4712 |
| work_keys_str_mv | AT vladimirbening comparingcompoundpoissondistributionsbydeficiencycontinuoustimecase AT victorkorolev comparingcompoundpoissondistributionsbydeficiencycontinuoustimecase |