Multiple Case High Leverage Diagnosis in Regression Quantiles

Regression Quantiles (RQs) (see Koenker and Bassett, 1978) can be found as optimal solutions to a Linear Programming (LP) problem. Also, these optimal solutions correspond to specific elemental regressions (ERs). On the other hand, single case ordinary least squares (OLS) leverage statistics can be expressed as weighted averages of ER ones. Using this three-tier relationship amongst RQs, ERs, and OLS leverage statistics some relationships between single case leverage statistics and ER ones are explored and deduced. We build upon these results and propose a multiple-case RQ weighted predictive leverage statistic, TJ. We do this using an ER view of the well-known leverage relationship, ∑i=1nhi=p$\sum\nolimits_{i = 1}^n {h_i = } p $ , by summing the ER weighted predictive leverage statistics over all ERs (RQs included) instead of over observations, i.e., ∑J=1KTJ=p$\sum\nolimits_{J = 1}^K {T_J } = p $ . As an ad-hoc cut-off value of this statistic we make use of the analog of the Hoaglin and Welsch (1978) one, i.e., high leverage points have hi>2p2pnn$h_i > {{2p} \mathord{\left/ {\vphantom {{2p} n}} \right. \kern-\nulldelimiterspace} n} $ . So in the RQ weighted predictive leverage scenario, the cut-off value becomes 2p2pKK${{2p} \mathord{\left/ {\vphantom {{2p} K}} \right. \kern-\nulldelimiterspace} K} $ , where K is the total number of ERs. We then apply this RQ high leverage diagnostic to well-known data sets in the literature. The cut-off value used generally seems too small. Some proposals of cut-off values based on some analytical bounds and a simulation study are therefore given and shown to be reasonable.