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

Corwin L. Atwood
Journal of the American Statistical Association
Vol. 70, No. 351 (Sep., 1975), pp. 613-617
DOI: 10.2307/2285942
Stable URL: http://www.jstor.org/stable/2285942
Page Count: 5
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## 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 (f1', f2') and (θ1',θ2'). Estimate f'θ by $\lbrack 1 - r(\mathbf{\hat\theta}_2' D^{-1}\mathbf{\hat\theta}_2)\rbrack\mathbf{f}_1'\mathbf{\hat\theta}_1 + r(\mathbf{\hat\theta}_2' D^{-1}\mathbf{\hat\theta}_2)\mathbf{f}'\mathbf{\hat\theta}$, where $\mathbf{\hat \theta}$ and $\mathbf{\hat\theta}_2$ are the BLUEs of θ and $\mathbf{\theta}_2, \mathbf{\hat\theta}_1$ is the BLUE of θ1 assuming θ2 = 0, σ2D is the covariance matrix of $\mathbf{\hat\theta}_2$, and r is any bounded nonnegative 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.

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