Robust Tests

Consider the location model (2) with known $\sigma$. We can therefore assume that the cumulative distribution function of our observations F satisfies $F \left( x \right) = F_0 \left( x - \mu \right)$ for some fixed distribution function F₀ and location parameter $\mu$. We want to test the null hypothesis $H_0: \ \mu=\mu_0$ versus a one- or two-sided alternative. If T_n is the estimate defined by the equation (3) it follows that under H₀ the statistic

$$E_n = \sqrt{n} \left( T_n - \mu_0 \right)$$

is asymptotically normal. We can use a test of the form: reject H₀ when E_n > C (or when $\left\vert E_n \right\vert > C$ if the alternative hypothesis is two-sided). The problem with this approach is that there is no known formula for the asymptotic variance of E_n under H₀ when the scale has to be estimated and $F_{\mu_0}$ is asymmetric. For the same location model with known scale, other proposed test statistics (Ronchetti, 1979, and Sen, 1982) are of the form

$$E_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n\psi \left( X_i - \mu_0 \right)$$

where $\psi$ satisfies $\int \psi \left( u - \mu_0 \right) \ dF_{\mu_0} = 0$. Then, under the null hypothesis, we have

$$E_n \longrightarrow N \left( 0 , \int \psi^2 \left( u - \mu_0 \right) \ dF_{\mu_0} \right)$$

and we can proceed as in the previous case. The same criticism with respect to the of estimation of $\sigma$ under asymmetric distributions applies here. For the linear model (8), Ronchetti (1982) introduced the class of $\tau$-tests. They are defined as follows. Let $\tau \left( \cdot, \cdot \right): {\cal R}^p \times {\cal R} \rightarrow {\cal R}$ be a function such that $\tau \left( {\bf x}, r \right) \ge 0$, $\tau \left({\bf x}, 0 \right) = 0$, and $\tau \left({\bf x}, \cdot \right)$ is differentiable for all ${\bf x}\in {\cal R}^p$. Let $\theta \in {\mbox{I}\!\mbox{R}}^p$ be the vector of regression parameters, and let $\theta = \left( \theta_1, \theta_2 \right)$ with $\theta_1 \in {\mbox{I}\!\mbox{R}}^{p-q}$ and $\theta_2 \in {\mbox{I}\!\mbox{R}}^q$. Consider the null hypothesis $H_0: \ \theta_2 = 0$. Let $r_i \left( \beta \right)$ be the residuals as defined in (9). The test statistic is

$$S^2_n = \frac{2}{q} \frac{1}{n} \sum_{i=1}^n\left\{ \tau \le... ..., r_i \left( T_{\Omega n} \right) / \sigma \right) \right\}$$

where $T_{\omega n}$, $T_{\Omega n}$ are the GM-estimates calculated with $\phi \left( {\bf x}, r \right) = \partial \tau \left({\bf x}, r \right) / \partial r$ (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, $T_{\omega n}$ and $T_{\Omega n}$ satisfy

$$T_{\omega n} = \arg \min_{\beta \in \Theta_\omega} \sum_{i=... ...\left( {\bf x}_i, r_i \left( \beta \right) / \sigma \right)$$

where $\Theta_\omega$ is the reduced parameter space. This test assumes that the scale parameter $\sigma$ is known. As before, under symmetric distributions, the simultaneous estimation of $\sigma$ 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 S_n² under the null hypothesis is that of a linear combination of $\chi^2$ random variables,

$$\sum_{j=p-q+1}^p \lambda_j N_j^2$$

where N_j are independent standard normal random variables; $\lambda_{p-q+1} \ge \lambda_{p-q+2} \ge \ldots \lambda_p > 0$ are the q positive eigenvalues of the matrix

$$Q \left[ M^{-1} - \left( \begin{array}{cc} M^{-1}_{11} & 0 \\ 0 & 0 \end{array} \right) \right],$$

M^-1₁₁ is the inverse of the upper $\left( p-q+1 \right) \times \left( p-q+1 \right)$ part of the matrix

$$M = E \left( \phi'\left({\bf x},r\right) {\bf x}{\bf x}' \right),$$ (15)

Q is defined as

$$Q = E \left( \phi^2 \left({\bf x},r\right) {\bf x}{\bf x}' \right)$$ (16)

and $\phi' = \partial \phi \left({\bf x}, r \right) / \partial r$. Other proposals to test the same null hypothesis $H_0: \ \theta_2 = 0$ are the aligned GM tests (see Markatou and Hettmansperger, 1990). They are defined as follows. Assume that $\sigma^2$ is known and hence, without loss of generality, let $\sigma=1$. Consider the statistic

$$W_n^2 = \left[ n^{-1/2} \ \sum_{i=1}^n \phi \left({\bf x}_i,... ...i, y_i - {\bf x}_i' \hat{\beta}_0 \right) {\bf x}^*_i \right]$$

where $\hat{\beta}_0$ is the reduced model GM estimator, ${\bf x}_i' = \left( {\bf x}_i^{** \ \prime}, {\bf x}_i^{* \ \prime} \right)$ with ${\bf x}_i^{**} \in {\cal R}^{p-q}$ and ${\bf x}_i^* \in {\cal R}^q$,

$$\hat{U}^{-1} = \hat{Q}_{22} - \hat{M}_{21} \hat{M}_{11}^{-1} ... ...{11}^{-1} \hat{Q}_{11}^{-1} \hat{M}_{11}^{-1} \hat{M}_{12}$$

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

$$T^2 = \hat{\beta}_2' V^{-1}_{22} \hat{\beta}_2$$

where $\hat{\beta}_2$ is a robust estimate of $\beta_2$ and V₂₂ is the $q \times q$ sub matrix of $\hat{V}$, an estimator of the asymptotic covariance matrix. This matrix will depend on the design. The scores (Rao) type test is based on the statistic

$$R^2 = Z_n' M_{\left(22.1\right)} V^{-1}_{22} M_{\left(22.1\right)} Z_n$$

where

$$Z_n = \frac1n \sum_{i=1}^n\phi \left({\bf x}_i, \frac{y_i - {\bf x}_i'T_{\omega n}}{ \hat{\sigma}} \right) {\bf x}_i^*$$

and

$$M_{\left(22.1\right)} = M_{22} - M_{21} M^{-1}_{11} M_{12}$$

where the matrices M_ij 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 $\Theta_0$ and $\Theta_1$ the null and alternative hypotheses respectively. The power breakdown function of a test statistic T is (see (1) for the definition of ${\cal Q}_\epsilon \left( F \right)$)

$$\epsilon^*_\theta \left( T \right) = \inf \left\{ \epsilon >... ...ht) \right) \mbox{ for some } \theta_0 \in \Theta_0 \right\}$$

similarly, the level breakdown function is

$$\epsilon^{**}_\theta \left( T \right) = \inf \left\{ \epsilo... ...ht) \right) \mbox{ for some } \theta_0 \in \Theta_0 \right\}$$

The interpretation of these definitions is that for each $\theta$, $\epsilon^*_\theta$ is the minimum proportion of outliers that will cause the null hypothesis to be confounded with the alternative $\theta$. On the other hand, $\epsilon^{**}_\theta$ is the smallest amount of contamination that will result in a failure to reject H₀ when the parameter is $\theta$. 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 $\Theta_0$ and $\Theta_1$ 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 T_n

$$\left( T_n - a \ \mbox{SE} / \sqrt{n}, T_n + a \ \mbox{SE} / \sqrt{n} \right)$$

its asymptotic level will be zero, because as $n \rightarrow \infty$ the confidence interval converges to the point $T_\infty$ which is typically different from the parameter of interest $\mu_0$.