# Article

 Title: A modification of the Hartung-Knapp confidence interval on the variance component in two-variance-component models (English) Author: Arendacká, Barbora Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 43 Issue: 4 Year: 2007 Pages: 471-480 Summary lang: English . Category: math . Summary: We consider a construction of approximate confidence intervals on the variance component $\sigma ^2_1$ in mixed linear models with two variance components with non-zero degrees of freedom for error. An approximate interval that seems to perform well in such a case, except that it is rather conservative for large $\sigma ^2_1/\sigma ^2,$ was considered by Hartung and Knapp in [hk]. The expression for its asymptotic coverage when $\sigma ^2_1/\sigma ^2\rightarrow \infty$ suggests a modification of this interval that preserves some nice properties of the original and that is, in addition, exact when $\sigma ^2_1/\sigma ^2\rightarrow \infty .$ It turns out that this modification is an interval suggested by El-Bassiouni in [eb]. We comment on its properties that were not emphasized in the original paper [eb], but which support use of the procedure. Also a small simulation study is provided. (English) Keyword: variance components Keyword: approximate confidence intervals Keyword: mixed linear model MSC: 62F25 MSC: 62J10 idZBL: Zbl 1134.62018 idMR: MR2377925 . Date available: 2009-09-24T20:25:47Z Last updated: 2012-06-06 Stable URL: http://hdl.handle.net/10338.dmlcz/135789 . Reference: [1] Arendacká A.: Approximate confidence intervals on the variance component in a general case of a two-component model.In: Proc. ROBUST 2006 (J. Antoch and G. Dohnal, eds.), Union of the Czech Mathematicians and Physicists, Prague 2006, pp. 9–17 Reference: [2] Billingsley P.: Convergence of Probability Measures.Wiley, New York 1968 Zbl 0944.60003, MR 0233396 Reference: [3] Boardman T. J.: Confidence intervals for variance components – a comparative Monte Carlo study.Biometrics 30 (1974), 251–262 Zbl 0286.62055 Reference: [4] Burdick R. K., Graybill F. A.: Confidence Intervals on Variance Components.Marcel Dekker, New York 1992 Zbl 0755.62055, MR 1192783 Reference: [5] El-Bassiouni M. Y.: Short confidence intervals for variance components.Comm. Statist. Theory Methods 23 (1994), 7, 1951–1933 Zbl 0825.62194, MR 1281896 Reference: [6] Hartung J., Knapp G.: Confidence intervals for the between group variance in the unbalanced one-way random effects model of analysis of variance.J. Statist. Comput. Simulation 65 (2000), 4, 311–323 Zbl 0966.62044, MR 1847242 Reference: [7] Park D. J., Burdick R. K.: Performance of confidence intervals in regression models with unbalanced one-fold nested error structures.Comm. Statist. Simulation Computation 32 (2003), 3, 717–732 Zbl 1081.62540, MR 1998237 Reference: [8] Seely J., El-Bassiouni Y.: Applying Wald’s variance component test.Ann. Statist. 11 (1983), 1, 197–201 Zbl 0516.62028, MR 0684876 Reference: [9] Tate R. F., Klett G. W.: Optimal confidence intervals for the variance of a normal distribution.J. Amer. Statist. Assoc. 54 (1959), 287, 674–682 Zbl 0096.12801, MR 0107926 Reference: [10] Thomas J. D., Hultquist R. A.: Interval estimation for the unbalanced case of the one-way random effects model.Ann. Statist. 6 (1978), 3, 582–587 Zbl 0386.62057, MR 0484702 Reference: [11] Tukey J. W.: Components in regression.Biometrics 7 (1951), 1, 33–69 Reference: [12] Wald A.: A note on the analysis of variance with unequal class frequencies.Ann. Math. Statist. 11 (1940), 96–100 MR 0001502 Reference: [13] Wald A.: A note on regression analysis.Ann. Math. Statist. 18 (1947), 4, 586–589 Zbl 0029.30703, MR 0023498 Reference: [14] Williams J. S.: A confidence interval for variance components.Biometrika 49 (1962), 1/2, 278–281 Zbl 0138.13101, MR 0144424 .

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