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Biometrika 1976 63(1):185-190; doi:10.1093/biomet/63.1.185
© 1976 by Biometrika Trust
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On the extermal quotient from a gamma sample

ALAN JULIAN IZENMAN

Department of Statistics, Tel-Aviv University Israel

The extremad quotient, Q, is defined for nonnegative observations as the ratio of the largest observation in a sample to the smallest observation. This paper gives the exact distribution of this quotient based on a sample drawn from a gamma distribution. The statistic is scale-free, and its distribution depends only on the sample size n and on the shape parameter {eta} of the underlying gamma distribution. Applications include certain areas of industrial quality control and Hartley's maximum F ratio, Fmax for testing heterogeneity of variances in a multisample, equal degree of freedom situation. The kth moment of Q exists only if k < {eta} the first and second moments are evaluated for various values of n and {eta}. Comparisons are made with David's tables of the percentage points of Fmax.

Key Words: Extremal quotient • Gamma distribution • Maximum F ratio


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