On The Conditional L₁-median and its estimation

Abstract

The conditional L-median μ(x) is defined as the minimizer over of

where F{|x) is the conditional distribution function. From a nonparametric estimate F(_n) of F, a natural estimate φ_n of φ is defined, which is proved to be uniformly strongly consistent. Under suitable assumptions, the existence and uniqueness of a minimizer μ_n(x) of φ_n ( x,x) are derived. The function φ_n(.) is proved to be (with probability 1) continuous for n large and the sequence (φ_n(.)) is shown to estimate consistently the function φ(.), uniformly on compact sets.

Keywords:

• Conditional L ₁rmedian
• Spatial median
• Nonparametric estimation
• Strong uniform consistency