On the conditional L1-median and its estimation

The conditional L-1-median mu(x) is defined as the minimizer over alpha epsilon R-d of phi(alpha, x) = integral(Rd) (parallel toy - alphaparallel to - parallel toyparallel toF(dy\x), where F((.)\x) is the conditional distribution function. From a nonparametric estimate F-n of F, a natural estimate phi(n) of phi is defined which is proved to be uniformly strongly consistent. Under suitable assumptions, the existence and uniqueness of a minimizer mu(n)(x) of phi(n)(alpha, x) are derived. The function mu(n)((.)) is proved to be (with probability 1) continuous for n large and the sequence (mu(n)((.))) is shown to estimate consistently the function p(.), uniformly on compact sets.