Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals
Abstract
:1. Introduction with Historical Notes and Motivation
2. Preliminaries and Auxiliary Results
- 1.
- There exists so, that
- 2.
- Moreover, for any such choice of there holds
3. Approximating CDF of Distribution
- 1.
- Then, there exists such that
- 2.
- For there holds
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions; John Wiley & Sons, Inc.: New York, NY, USA, 1995; Volume 2. [Google Scholar]
- Robert, C. Modified Bessel functions and their applications in probability and statistics. Stat. Probab. Lett. 1990, 9, 155–161. [Google Scholar] [CrossRef]
- Kamel, A.S.; Abdel-Samad, A.I. On the computation of non-central Chi-square distribution function. Commun. Stat. Simul. Comput. 1990, 19, 1279–1291. [Google Scholar] [CrossRef]
- András, S.; Baricz, Á. Properties of the probability density function of the non–central chi–squared distribution. J. Math. Anal. Appl. 2008, 346, 395–402. [Google Scholar] [CrossRef] [Green Version]
- Scharf, L.L. Statistical Signal Processing: Detection, Estimation, and Time Series Analysis; Addison–Wesley Publishing Co.: Boston, MA, USA, 1990. [Google Scholar]
- Lancaster, H.O. Forerunners of the Pearson χ2. Aust. J. Stat. 1966, 8, 117–126. [Google Scholar] [CrossRef]
- Bienaymé, I.J. Sur la probabilité des erreurs d’après la méthode des moindres carrés. Liouville’s J. Math. Pures Appl. 1852, 17, 33–78. [Google Scholar]
- Pearson, K. On a criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philos. Mag. 1900, 50, 157–175. [Google Scholar] [CrossRef] [Green Version]
- Helmert, F.R. Über die Berechnung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wehrer Beobachtungsfehler. Z. Math. Phys. 1875, 20, 300–303. [Google Scholar]
- Helmert, F.R. Über die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen. Z. Math. Phys. 1876, 21, 192–218. [Google Scholar]
- Kruskal, W.H. Helmert’s distribution. Am. Math. Mon. 1946, 53, 435–438. [Google Scholar] [CrossRef]
- Sheynin, O.B. Origin of the theory of errors. Nature 1966, 211, 1003–1004. [Google Scholar] [CrossRef]
- Plackett, R.L. Karl Pearson and the Chi-Squared Test. Int. Stat. Rev. 1983, 51, 59–72. [Google Scholar] [CrossRef]
- Kendall, M.G. Studies in the history of probability and statistics. XXVI. The work of Ernst Abbe. Biometrika 1971, 58, 369–373. [Google Scholar] [CrossRef]
- Abbe, E. Über die Gesetzmässigkeit in der Vertheilung der Fehler bei Beobachtungsreihen; Dissertation zur Erlangung der Venia Docendi bei den Phyilosophischen Fakultät in Jena; Verlag Frommann: Jena, Germany, 1863. [Google Scholar]
- Watson, G.N. A Treatise on the Theory of Bessel Functions; Cambridge University Press: London, UK, 1922. [Google Scholar]
- Abramowitz, M.; Stegun, I.A. (Eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Applied Mathematics Series 55; National Bureau of Standards: Washington, DC, USA, 1964; Reprinted by Dover Publications, New York, 1972.
- Mihoc, G.; Craiu, V. Treatise on Mathematical Statistics, Sampling and Estimation; With an English Table of Contents; Editura Academiei Republicii Socialiste România: Bucureşti, Romania, 1976; Volume I. (In Romanian) [Google Scholar]
- Fisher, R.A. The general sampling distribution of the multiple correlation coefficient. Proc. R. Soc. Lond. 1928, 121, 654–673. [Google Scholar]
- Tang, P.C. The power function of the analysis of variance tests with tables and illustrations of their use. Stat. Res. Mem. Lond. 1938, 2, 126–149. [Google Scholar]
- Patnaik, P.B. The non-central χ2– and the F–distributions and their applications. Biometrika 1949, 36, 202–234. [Google Scholar] [CrossRef]
- Pearson, E.S. Note on an approximation to the distribution of non-central χ2. Biometrika 1959, 46, 202–232. [Google Scholar] [CrossRef]
- Sankaran, M. Approximations to the non-central chi-square distribution. Biometrika 1963, 50, 199–204. [Google Scholar] [CrossRef]
- Temme, N.M. Asymptotic and numerical aspects of the non-central chi-square distribution. Comput. Math. Appl. 1993, 25, 55–63. [Google Scholar] [CrossRef] [Green Version]
- Jankov Maširević, D. On new formulas for the cumulative distribution function of the non-central chi-square distribution. Mediterr. J. Math. 2017, 14, 66. [Google Scholar] [CrossRef]
- András, S.; Baricz, Á.; Sun, Y. The generalized Marcum Q–function: An orthogonal polynomial approach. Acta Univ. Sapientiae Math. 2011, 3, 60–76. [Google Scholar]
- Brychkov, Y.A. On some properties of the Marcum Q function. Integral Transform. Spec. Funct. 2012, 23, 177–182. [Google Scholar] [CrossRef]
- Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. (Eds.) NIST Handbook of Mathematical Functions; NIST and Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Agrest, M.M.; Maksimov, M.S. Theory of Incomplete Cylindrical Functions and Their Applications; Springer: New York, NY, USA, 1971. [Google Scholar]
- Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I. Special Functions; Integrals and Series; Gordon and Breach Science Publishers: New York, NY, USA, 1986; Volume 2. [Google Scholar]
- Hobson, E.W. On the second mean-value theorem of the integral calculus. Trans. Am. Math. Soc. 1908, 2, 14–23. [Google Scholar] [CrossRef]
- Du Bois-Reymond, P. Über die allgemeinen Eigenschaften der Klasse von Doppelintegralen, zu welcher das Fouriersche Doppelintegral gehört. J. Reine Angew. Math. 1868, 69, 65–108. [Google Scholar]
- Schwind, W.J.; Ji, J.; Koditschek, D.E. A physically motivated further note on the mean-value theorem for integrals. Am. Math. Mon. 1999, 106, 559–564. [Google Scholar] [CrossRef]
- Bao-Lin, Z. A note on the mean-value theorem for integrals. Am. Math. Mon. 1997, 104, 561–562. [Google Scholar] [CrossRef]
- Jacobson, B. On the mean-value theorem for integrals. Am. Math. Mon. 1982, 89, 300–301. [Google Scholar] [CrossRef]
- Polezzi, M. On the weighted mean value theorem for integrals. Internat. J. Math. Ed. Sci. Technol. 2006, 37, 868–870. [Google Scholar] [CrossRef]
- Matsumoto, T. Hiroshi Okamura. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 1950, 26, 1–3. [Google Scholar] [CrossRef]
- Okamura, H. On the second mean-value theorem of integral. In Mathematics; Kyoto Mathematical Society: Kyoto, Japan, 1947; Volume 1. (In Japanese) [Google Scholar]
- Baricz, Á.; Pogány, T.K. On a sum of modified Bessel functions. Mediterr. J. Math. 2014, 11, 349–360. [Google Scholar] [CrossRef] [Green Version]
- Wituła, R.; Hetmaniok, E.; Słota, D. A stronger version of the second mean value theorem for integrals. Comput. Math. Appl. 2016, 64, 1612–1615. [Google Scholar] [CrossRef] [Green Version]
- Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, 6th ed.; Jeffrey, A., Zwillinger, D., Eds.; Academic Press: New York, NY, USA, 2000. [Google Scholar]
- Pratt, W.K. Partial differentials of Marcum’s Q function. Proc. IEEE 1968, 56, 1220–1221. [Google Scholar] [CrossRef]
- Jones, A.L. An extension of an inequality involving modified Bessel functions. J. Math. Phys. 1968, 47, 220–221. [Google Scholar] [CrossRef]
- Cochran, J.A. The monotonicity of modified Bessel functions with respect to their order. J. Math. Phys. 1967, 46, 220–222. [Google Scholar] [CrossRef]
- Nåsell, I. Inequalities for modified Bessel functions. Math. Comput. 1974, 28, 253–256. [Google Scholar] [CrossRef]
- Soni, R.P. On an inequality for modified Bessel functions. J. Math. Phys. 1965, 44, 406–407. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Baricz, Á.; Jankov Maširević, D.; Pogány, T.K. Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals. Mathematics 2021, 9, 129. https://doi.org/10.3390/math9020129
Baricz Á, Jankov Maširević D, Pogány TK. Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals. Mathematics. 2021; 9(2):129. https://doi.org/10.3390/math9020129
Chicago/Turabian StyleBaricz, Árpád, Dragana Jankov Maširević, and Tibor K. Pogány. 2021. "Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals" Mathematics 9, no. 2: 129. https://doi.org/10.3390/math9020129
APA StyleBaricz, Á., Jankov Maširević, D., & Pogány, T. K. (2021). Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals. Mathematics, 9(2), 129. https://doi.org/10.3390/math9020129