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Mathematics of Computation

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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.

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