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An asymptotic expansion for the upper percentage points of the $ \chi \sp{2}$-distribution

Author: Henry E. Fettis
Journal: Math. Comp. 33 (1979), 1059-1064
MSC: Primary 62E20
MathSciNet review: 528059
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Abstract: An asymptotic development is given for estimating the value of the variable $ \chi $ for which the $ {\chi ^2}$-distribution

$\displaystyle Q({\chi ^2},v) = \frac{1}{{\Gamma (v/2)}}\int _{{\chi ^2}/2}^\infty {t^{v/2 - 1}}{e^{ - t}}dt$

assumes a preassigned value $ \alpha $, in the region where the quantity $ \eta = - \ln [\Gamma (v/2)\alpha ]$ satisfies

$\displaystyle \eta > > \ln \eta .$

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 [Enhancements On Off] (What's this?)

  • [1] M. ABRAMOWITZ & I. STEGUN, Editors, Handbook of Mathematical Functions, with Formulas, Graphs and Tables, Dover, New York, 1966.
  • [2] J. M. BLAIR, C. A. EDWARDS & J. H. JOHNSON, "Rational Chebyshev approximations for the inverse of the error function," Math. Comp., v. 30, 1976, pp. 827-830. MR 0421040 (54:9047)
  • [3] H. E. FETTIS, "A stable algorithm for computing the inverse error function in the 'tail-end' region," Math. Comp., v. 28, 1975, pp. 585-587. MR 0341812 (49:6558)
  • [4] 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.
  • [5] J. R. PHILIP, "The function inverfc $ \theta $," Austral. J. Phys., v. 13, 1960, pp. 13-20. MR 0118857 (22:9626)
  • [6] A. J. STRECOK, "On the calculation of the inverse of the error function," Math. Comp., v. 22, 1968, pp. 144-158. MR 0223070 (36:6119)
  • [7] M. ZYCZKOWSKI, "Potenzieren von verallgemeinerten Potenzreihen mit beliebigen Exponent," Z. Angew. Math. Phys., v. 12, 1961, pp. 572-576.

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Keywords: Chi-square distribution, inverse incomplete gamma function, percentage points, asymptotic expansion
Article copyright: © Copyright 1979 American Mathematical Society

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