The Calculation of the Density and Distribution Functions of Strictly Stable Laws
Integral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/5/775 |
| _version_ | 1827717074045108224 |
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| author | Viacheslav Saenko |
| author_facet | Viacheslav Saenko |
| author_sort | Viacheslav Saenko |
| collection | DOAJ |
| description | Integral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function of standard strictly stable laws through a definite integral. Using the methods of numerical integration, the obtained integral representations allow us to calculate the probability density and distribution function of a strictly stable law for a wide range of admissible values of parameters <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>α</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. A number of cases were given when numerical algorithms had difficulty in calculating the density. Formulas were given to calculate the density and distribution function with an arbitrary value of the scale parameter <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula>. |
| first_indexed | 2024-03-10T19:53:40Z |
| format | Article |
| id | doaj.art-83f0cc71e70a4bd1b2539f2226ad4122 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T19:53:40Z |
| publishDate | 2020-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-83f0cc71e70a4bd1b2539f2226ad41222023-11-20T00:09:03ZengMDPI AGMathematics2227-73902020-05-018577510.3390/math8050775The Calculation of the Density and Distribution Functions of Strictly Stable LawsViacheslav Saenko0Department of Theoretical Physics, Ulyanovsk State University, 432970 Ulyanovsk, RussiaIntegral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function of standard strictly stable laws through a definite integral. Using the methods of numerical integration, the obtained integral representations allow us to calculate the probability density and distribution function of a strictly stable law for a wide range of admissible values of parameters <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>α</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. A number of cases were given when numerical algorithms had difficulty in calculating the density. Formulas were given to calculate the density and distribution function with an arbitrary value of the scale parameter <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/8/5/775stable distributionprobability density functiondistribution function |
| spellingShingle | Viacheslav Saenko The Calculation of the Density and Distribution Functions of Strictly Stable Laws Mathematics stable distribution probability density function distribution function |
| title | The Calculation of the Density and Distribution Functions of Strictly Stable Laws |
| title_full | The Calculation of the Density and Distribution Functions of Strictly Stable Laws |
| title_fullStr | The Calculation of the Density and Distribution Functions of Strictly Stable Laws |
| title_full_unstemmed | The Calculation of the Density and Distribution Functions of Strictly Stable Laws |
| title_short | The Calculation of the Density and Distribution Functions of Strictly Stable Laws |
| title_sort | calculation of the density and distribution functions of strictly stable laws |
| topic | stable distribution probability density function distribution function |
| url | https://www.mdpi.com/2227-7390/8/5/775 |
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