Reliabilities of χ2- and F tests in Gauss-Markov model

Abstract

Power and level breakdown points are used in robust statistics to determine the global reliability of a test. A method for calculating the finite-sample versions of these for tests such as chi(2) Or F is given here when observations have the Gauss-Markov model. Outliers are compared with the generating model using a coordinate transformation and simulated samples. Outliers are considered in two groups, namely, doubtful and clear. The reliability of a chi(2)- or F test changes with the magnitude, number, and kind of outliers (i.e., random outliers or influential outliers), and also with the level of significance, alpha. The chi(2)- and F tests are generally insensitive to both doubtful and clear outliers. The chi(2) test can reliably reject the null hypothesis at alpha = 0.05, when the observations contain multiple clear random outliers, only if the magnitude of one of them is at least greater than 5 sigma. Also, the F test can reliably reject the null hypothesis at alpha = 0.05, when the observations contain multiple clear random outliers, only if the magnitude of one of them is at least greater than 6 sigma. The greater the magnitude and number of clear outliers that are contained in the observations, the more successful is the chi(2)- or F test in rejecting the null hypothesis. The reliabilities of the chi(2)- and F tests decrease as the number of unknowns increases. For this case, the power and level finite-sample breakdown points of the chi(2)- and F tests are estimated here to be approximately 1/2.