  Cross validation is widely used for selecting tuning parameters in
regularization methods, but it is computationally intensive in general. To
lessen its computational burden, approximation schemes such as generalized
approximate cross validation (GACV) are often employed. However, such
approximations may not work well when non-smooth loss functions are involved.
As a case in point, approximate cross validation schemes for penalized quantile
regression do not work well for extreme quantiles. In this paper, we propose a
new algorithm to compute the leave-one-out cross validation scores exactly for
quantile regression with ridge penalty through a case-weight adjusted solution
path. Resorting to the homotopy technique in optimization, we introduce a case
weight for each individual data point as a continuous embedding parameter and
decrease the weight gradually from one to zero to link the estimators based on
the full data and those with a case deleted. This allows us to design a
solution path algorithm to compute all leave-one-out estimators very
efficiently from the full-data solution. We show that the case-weight adjusted
solution path is piecewise linear in the weight parameter, and using the
solution path, we examine case influences comprehensively and observe that
different modes of case influences emerge, depending on the specified
quantiles, data dimensions and penalty parameter.
