Nadaraya-Watson estimator for sensor fusion

In a system of N sensors, the sensor S-j, j=1,2,..., N, outputs Y-(j) is an element of [0,1], according to an unknown probability density p(j)(Y-(j)parallel to X), corresponding to input X is an element of [0,1]. A training n-sample (X(1),Y-1),(X(2),Y-2),...,(X(n),Y-n) is given where Y-i=(Y-i((1)),Y-i((2)),..., Y-i((N))) such that Y-i((j)) is the output of S-j in response to input X(i). The problem is to estimate a fusion rule f:[0,1](N)-->[0,1], based on the sample, such that the expected square error I(f) =integral[X-f(Y)](2)p(Y parallel to X)p(X)dY((1)) dY((2))... dY((N))dX is minimized over a family of functions F with uniformly bounded modulus of smoothness, where Y=(Y-(1), Y-(2),..., Y-(N)). Let f* minimize I(.) over F; f* cannot be computed since the underlying densities are unknown. We estimate the sample size sufficient to ensure that Nadaraya-Watson estimator (f) over cap satisfies P[I((f) over cap)-I(f*)> epsilon]<delta for any epsilon>0 and delta, 0< delta<1. (C) 1997 Society of Photo-Optical Instrumentation Engineers.