Nadaraya-Watson estimator for sensor fusion

In a system of N sensors, the sensor S_j, j=1,2, . . . , N, outputs Y^(j) ∈ [0,1], according to an unknown probability density p_j(Y^(j)|X), corresponding to input X ∈ [0,1]. A training n-sample (X₁, Y₁),(X₂, Y₂), . . . , (X_n, Y_n) is given where Y_i=(Y⁽¹⁾_i, Y⁽²⁾_i, . . . , Y^(N)_i) such that Y^(j)_i 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) = ∫[X-f(Y)]²p(Y|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 f̂ satisfies P[I(f̂) - I(f*) > ε] < δ for any ε>0 and δ, 0<δ<1.