Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals
The cumulative distribution function of the non-central chi-square distribution <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>χ</mi><mi>n</mi><mrow><mo>′</mo><mn>2</mn></mrow>&l...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-01-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/2/129 |
| _version_ | 1797414150418202624 |
|---|---|
| author | Árpád Baricz Dragana Jankov Maširević Tibor K. Pogány |
| author_facet | Árpád Baricz Dragana Jankov Maširević Tibor K. Pogány |
| author_sort | Árpád Baricz |
| collection | DOAJ |
| description | The cumulative distribution function of the non-central chi-square distribution <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>χ</mi><mi>n</mi><mrow><mo>′</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <i>n</i> degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin. |
| first_indexed | 2024-03-09T05:28:57Z |
| format | Article |
| id | doaj.art-0bff98fafc3d4faca7ff1c56bf11a375 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T05:28:57Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-0bff98fafc3d4faca7ff1c56bf11a3752023-12-03T12:34:19ZengMDPI AGMathematics2227-73902021-01-019212910.3390/math9020129Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for IntegralsÁrpád Baricz0Dragana Jankov Maširević1Tibor K. Pogány2Department of Economics, Babeş-Bolyai University, 400591 Cluj-Napoca, RomaniaDepartment of Mathematics, University of Osijek, Trg Lj. Gaja 6, 31000 Osijek, CroatiaInstitute of Applied Mathematics, Óbuda University, Bécsi út 96/b, 1034 Budapest, HungaryThe cumulative distribution function of the non-central chi-square distribution <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>χ</mi><mi>n</mi><mrow><mo>′</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <i>n</i> degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin.https://www.mdpi.com/2227-7390/9/2/129non-central <i>χ</i>2 distributionsecond mean-value theorem for definite integralsmodified Bessel function of the first kindMarcum <i>Q</i>–functionlower incomplete gamma function |
| spellingShingle | Árpád Baricz Dragana Jankov Maširević Tibor K. Pogány Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals Mathematics non-central <i>χ</i>2 distribution second mean-value theorem for definite integrals modified Bessel function of the first kind Marcum <i>Q</i>–function lower incomplete gamma function |
| title | Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals |
| title_full | Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals |
| title_fullStr | Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals |
| title_full_unstemmed | Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals |
| title_short | Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals |
| title_sort | approximation of cdf of non central chi square distribution by mean value theorems for integrals |
| topic | non-central <i>χ</i>2 distribution second mean-value theorem for definite integrals modified Bessel function of the first kind Marcum <i>Q</i>–function lower incomplete gamma function |
| url | https://www.mdpi.com/2227-7390/9/2/129 |
| work_keys_str_mv | AT arpadbaricz approximationofcdfofnoncentralchisquaredistributionbymeanvaluetheoremsforintegrals AT draganajankovmasirevic approximationofcdfofnoncentralchisquaredistributionbymeanvaluetheoremsforintegrals AT tiborkpogany approximationofcdfofnoncentralchisquaredistributionbymeanvaluetheoremsforintegrals |