Estimating a Response Surface with an Uncertain Number of Parameters, Assuming Normal Errors

Abstract

Let Y ( x ) be independent normal random variables with mean f′ ( x )θ and variance σ², and partition the vectors f′ and θ′ into (f ₁′, f ₂′) and (θ₁′, θ₂′). Estimate f′θ by , where and are the BLUEs of θ and θ₂, is the BLUE of θ₁ assuming θ₂ = 0, σ² D is the covariance matrix of , and r is any bounded non-negative nondecreasing function. Among such estimators with given fixed MSE when θ₂ = 0, MSE is minimized for θ₂ near 0 by making r constant. Numerical comparisons are given for the quadratic regression example.