CONCOMITANT SCALE ESTIMATION IN REGRESSION PROBLEMS WITH INCREASING DIMENSION

We obtain Bahadur representations for the semi‐interquartile range and the median deviation when these estimators are based on the residuals from a linear regression model with increasing dimension. These representations yield a variety of central limit theorems and conditions under which the two estimators are equivalent. In particular, the representations justify the use of the estimators as concomitant scale estimators in general scale equivariant M‐estimation of a regression parameter when the dimension of the parameter increases with the sample size.