An asymptotic expansion for the upper percentage points of the $\chi ^{2}$-distribution
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- by Henry E. Fettis PDF
- Math. Comp. 33 (1979), 1059-1064 Request permission
Abstract:
An asymptotic development is given for estimating the value of the variable $\chi$ for which the ${\chi ^2}$-distribution \[ Q({\chi ^2},v) = \frac {1}{{\Gamma (v/2)}}\int _{{\chi ^2}/2}^\infty {t^{v/2 - 1}}{e^{ - t}}dt\] assumes a preassigned value $\alpha$, in the region where the quantity $\eta = - \ln [\Gamma (v/2)\alpha ]$ satisfies \[ \eta > > \ln \eta .\] This development generalizes a similar one given by Blair and coauthors [2] for the case $v = 1$. It is also shown how the estimates thus obtained may be used in conjunction with various iterative schemes to give more accurate values.References
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M. ABRAMOWITZ & I. STEGUN, Editors, Handbook of Mathematical Functions, with Formulas, Graphs and Tables, Dover, New York, 1966.
- J. M. Blair, C. A. Edwards, and J. H. Johnson, Rational Chebyshev approximations for the inverse of the error function, Math. Comp. 30 (1976), no. 136, 827–830. MR 421040, DOI 10.1090/S0025-5718-1976-0421040-7
- Henry E. Fettis, A stable algorithm for computing the inverse error function in the “tail-end” region, Math. Comp. 28 (1974), 585–587. MR 341812, DOI 10.1090/S0025-5718-1974-0341812-5 W. GANDER, "A machine independent algorithm for computing percentage points of the ${\chi ^2}$-distribution," Z. Angew. Math. Phys., v. 28, 1977, pp. 1133-1136.
- J. R. Philip, The function inverfc $\theta$, Austral. J. Phys. 13 (1960), 13–20. MR 118857, DOI 10.1071/PH600013
- Anthony Strecok, On the calculation of the inverse of the error function, Math. Comp. 22 (1968), 144–158. MR 223070, DOI 10.1090/S0025-5718-1968-0223070-2 M. ZYCZKOWSKI, "Potenzieren von verallgemeinerten Potenzreihen mit beliebigen Exponent," Z. Angew. Math. Phys., v. 12, 1961, pp. 572-576.
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 1059-1064
- MSC: Primary 62E20
- DOI: https://doi.org/10.1090/S0025-5718-1979-0528059-9
- MathSciNet review: 528059