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### Robust Tests

Consider the location model (2) with known . We can therefore assume that the cumulative distribution function of our observations F satisfies for some fixed distribution function F0 and location parameter . We want to test the null hypothesis versus a one- or two-sided alternative. If Tn is the estimate defined by the equation (3) it follows that under H0 the statistic

is asymptotically normal. We can use a test of the form: reject H0 when En > C (or when if the alternative hypothesis is two-sided). The problem with this approach is that there is no known formula for the asymptotic variance of En under H0 when the scale has to be estimated and is asymmetric. For the same location model with known scale, other proposed test statistics (Ronchetti, 1979, and Sen, 1982) are of the form

where satisfies . Then, under the null hypothesis, we have

and we can proceed as in the previous case. The same criticism with respect to the of estimation of under asymmetric distributions applies here. For the linear model (8), Ronchetti (1982) introduced the class of -tests. They are defined as follows. Let be a function such that , , and is differentiable for all . Let be the vector of regression parameters, and let with and . Consider the null hypothesis . Let be the residuals as defined in (9). The test statistic is

where , are the GM-estimates calculated with (see section (2.1.2)) for the reduced and full models respectively. This test statistic can also be thought as the robust equivalent to the classical F test. To see the relationship note that instead of minimizing the residual sum of squares, and satisfy

where is the reduced parameter space. This test assumes that the scale parameter is known. As before, under symmetric distributions, the simultaneous estimation of does not change the asymptotic results. No suggestion is made on how to proceed when the distribution of the errors is not symmetric. The asymptotic distribution of the statistic Sn2 under the null hypothesis is that of a linear combination of random variables,

where Nj are independent standard normal random variables; are the q positive eigenvalues of the matrix

M-111 is the inverse of the upper part of the matrix

 (15)

Q is defined as

 (16)

and . Other proposals to test the same null hypothesis are the aligned GM tests (see Markatou and Hettmansperger, 1990). They are defined as follows. Assume that is known and hence, without loss of generality, let . Consider the statistic

where is the reduced model GM estimator, with and ,

and Q and M are defined in (16) and (15). Markatou, Stahel and Ronchetti (1991) study two other types of tests (see also Markatou and He, 1994). The Wald-type tests are based on statistics of the form

where is a robust estimate of and V22 is the sub matrix of , an estimator of the asymptotic covariance matrix. This matrix will depend on the design. The scores (Rao) type test is based on the statistic

where

and

where the matrices Mij correspond to the same partition as before. All these tests have good local robustness behavior (see the papers cited above). To study their global behavior when the fraction of contamination is relatively large He, Simpson and Portnoy (1990) introduced the concept of breakdown-point for tests. Denote and the null and alternative hypotheses respectively. The power breakdown function of a test statistic T is (see (1) for the definition of )

similarly, the level breakdown function is

The interpretation of these definitions is that for each , is the minimum proportion of outliers that will cause the null hypothesis to be confounded with the alternative . On the other hand, is the smallest amount of contamination that will result in a failure to reject H0 when the parameter is . Markatou and He (1994) studied some proposals to obtain simultaneously high breakdown points and good local behavior. Note that with this definition of breakdown point, if the boundaries of and intersect the resulting breakdown point is zero. This is due to the fact that if the asymptotic bias is positive, for any proportion of outliers there exists a contamination that will drive the estimator towards the wrong region. This last remark can be restated in terms of confidence intervals. If we construct a classical type confidence interval centered at a robust estimate Tn

its asymptotic level will be zero, because as the confidence interval converges to the point which is typically different from the parameter of interest .

Next: Robust Confidence Intervals and Up: Robust Inference Previous: Robust Inference
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2000-05-29