Limiting Behavior of Extrema of Certain Conditional Sample Functions

has the conditional distribution G(xjy) (unknown). We investigate the behavior of the extrema of sample functions of the form (x; G nk (xjy)), where G nk (xjy) is the k nearest neighbor (k-NN) estimator of G(xjy). Specii-cally, we rst study the properties of a multiparameter process Wn = k 1=2 (G nk (xjy) ?G(xjy)), (x; y) 2 R + C, where C is a compact set in R q. Then under some regular conditions we show that there exists a Gaussian process W (:); such that fk 1=2 (sup x (x; G nk (xjy)) ? sup x (x; G(xjy))); ; y 2 Cg convergences weakly to W. The above problem is motivated by the study of the strength of a bundle of parallel laments in reliability theory when there are concomitant variates.