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Approximations for F-tests which are ratios of sums of squares of independent variables with a model close to the normal

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Abstract

The F (a,b)-distribution is very frequently used in Statistics to compute significance levels when the individual observations follow a normal distribution. In this paper, we obtain good analytic approximations for the p-value and critical value of tests in which the test statistic is a ratio of sums of squares of independent and identically distributed random variables (i.e., F-tests under a normal) when the underlying distribution is close to but different from the normal. The class of distributions for the approximations to be valid is delimited with a “breakdown condition.” With these approximations, we can study, for example, the robustness of validity of this kind of tests and corroborate the thought that they have robustness of validity if only the second degree of freedom depends on a large n and that they do not have robustness of validity if both degrees of freedom depend on n and are similar. These robustness properties are displayed with the “Robustness of Validity Plot,” a diagram of the nominal level versus the actual level of a test. The simulations carried out confirm these conclusions.

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Authors and Affiliations

  1. Departamento de Estadística, Investigación Operativa y Cálculo Numérico, Facultad de Ciencias, U.N.E.D., Paseo Senda del Rey 9, 28040, Madrid, Spain

    Alfonso García-Pérez

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  1. Alfonso García-Pérez
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Correspondence to Alfonso García-Pérez.

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García-Pérez, A. Approximations for F-tests which are ratios of sums of squares of independent variables with a model close to the normal. TEST 17, 350–369 (2008). https://doi.org/10.1007/s11749-006-0036-4

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  • DOI: https://doi.org/10.1007/s11749-006-0036-4

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