Inequalities for quantiles of the chi-square distribution

• Autorzy: Inglot T.
• Języki publikacji: EN

Abstrakty

EN

We obtain a new sharp lower estimate for tails of the central chi-square distribution. Using it we prove quite accurate lower bounds for the chi-square quantiles covering the case of increasing number of degrees of freedom and simultaneously tending to zero tail probabilities. In the case of small tail probabilities we also provide upper bounds for these quantiles which are close enough to the lower ones. As a byproduct we propose a simple approximation formula which is easy to calculate for the chi-square quantiles. It is expressed explicitly in terms of tail probabilities and a number of degrees of freedom.

Słowa kluczowe

EN

chi-square quantiles; normal quantiles; lower and upper bounds; tails of chi-square distribution; Wilson–Hilferty formula.

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2010
• Tom: Vol. 30, Fasc. 2
• Strony: 339--351
• Opis fizyczny: Bibliogr. 10 poz., tab.

Twórcy

autor

Inglot T.

• Institute of Mathematics and Informatics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliografia

• [1] Y. Baraud, S. Huet and B. Laurent, Adaptive tests of linear hypotheses by model selection, Ann. Statist. 31 (2003), pp. 225-251.
• [2] C. W. Brain and J. Mi, On some properties of the quantiles of the chi-square distribution and their applications to interval estimation, Comm. Statist. Theory Methods 30 (2001), pp. 1851-1867.
• [3] R. A. Fisher, Statistical Methods for Research Workers, Oliver and Boyd, Edinburgh 1925.
• [4] T. Inglot and T. Ledwina, Asymptotic optimality of a new adaptive test in regression model, Ann. Inst. H. Poincaré 42 (2006), pp. 579-590.
• [5] C. Ittrich, D. Krause and W.-D. Richter, Probabilities and large quantiles of noncentral chi-square distributions, Statistics 34 (2000), pp. 53-101.
• [6] N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. 1, Wiley, 1994.
• [7] N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. 2, Wiley, 1995.
• [8] B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection, Ann. Statist. 28 (2000), pp. 1302-1338.
• [9] E. B. Wilson and M. M. Hilferty, The distribution of chi-square, Proc. Nat. Acad. Sci. USA 17 (1931), pp. 684-688.
• [10] J. H. Zar, Approximations for the percentage points of the chi-square distribution, Appl. Statist. 27 (1978), pp. 280-290.