Bias in P-value of MM-type estimators or Cochrans Q Penalized Regression There are a number of linear regression methods designed to limit the influence of outliers on estimates:


*

*For example, Cochrans Q Penalised regression as described in [1] will do an initial linear regression, then downweighting data points which are outlying from this initial regression.

*Or, MM-type estimators will run the regression twice calculating the residual of datapoints to the regression line then subsequently eliminating outliers. Running this procedure twice.
Is it true that:
Given, this two step procedure, the P-values derived from these estimates are biased since the model is observing the data twice.
One could repeatedly run an MM-type estimator and progressively constrain to find a significant regression line in any dataset purely by repeated exclusion of 'outlying' datapoints.
[1] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6756542/
 A: *

*Penalized regression does not do an initial regression and then downweight points "outlying" from this initial regression. $L_1$ and $L_2$-penalized regressions do not weight observations at all (de facto). In fact, $L_1$-penalized regression (LASSO) is more prone to bias d/t outlier because it tends to select regressors based on large coefficient values. The same is not true of ridge which tends to attenuate effects ($L_2$ penalty).

*The ordinary least squares regression coefficient is a method of moment estimator. $$ \mathbf{X}^T (Y - \mathbf{X}^T\beta) = 0$$ gives rise to the OLS parameters. Although the method is so broad, some of the specialized outlier-robust methods may have estimating equation formulations as well. By virtue of being a root finding algorithm, it does not "run" twice, but a solution can be obtained using Newton-Raphson or a similar approach.

*Data are never observed "twice" even in iterative estimation routines. Data are observed once by virtue of their being data.

*Automated or manual outlier deletion leads to biased and inefficient estimates.

*If data are wrong they should be deleted or, better, fixed regardless of whether they are outliers.

*The only routine which comes close to the two-step processes you are trying to describe would be a trimmed OLS where a proportion of the extreme residuals are excluded from analysis and the OLS is recalculated from the central subset.

*A minimax approach to the regression routine that would "downweight" residuals would be to minimize the following objective function:
$$ \beta = \text{arg min} \rho (Y-\mathbf{X} \beta$$
where $\rho$ is a convex loss function. Choosing $\rho(r) = r^2$ gives OLS, but you can, say, choose $\rho(r) = r^2/(r^2+1)$ where the loss is locally quadratic near $r=0$ but converges to a constant 1 in the tails. 

