ON SOME PROPERTIES OF THE QUANTILES OF THE CHI-SQUARE DISTRIBUTION AND THEIR APPLICATIONS TO INTERVAL ESTIMATION

Abstract

Three theorems are presented which describe certain mathematical properties of the quantiles of the central chi-square distribution with ν degrees of freedom. The first two theorems deal with certain inequalities of the quantiles with respect to , and , for a certain range of tail probabilities, while the third theorem deals with the monotonicity of the ratio of the quantiles to ν, as a function of ν, also for a certain range of tail probabilities. Applications of these theorems are illustrated in a few examples by showing analytically certain properties of interval estimators which are either not often thought of or are taken for granted more by intuition than by formal mathematical reasoning. One of the interval estimator properties that is discussed deals with whether or not an interval always contains the corresponding maximum likelihood estimator, and another property deals with whether or not a larger sample size is better than a smaller one, on the average. In either case, nontrivial illustrations are dealt with analytically.

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Keywords:

• Chi-square distribution
• Confidence bounds
• Interval estimation
• Quantile properties

ACKNOWLEDGMENT

The authors thank Dr. Nandini Kanan for the invitation extended to one of us to participate and present a paper in the statistical conference organized in honor of Professor C. R. Rao's 80th birthday. The authors also thank the anonymous referee for his/her kind comments and suggestions.

Notes

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