A new estimation procedure for response surface models

Several attempts have been made to find an estimator of a response which will have a smaller integrated mean square error than existing procedures. In this work another such attempt is made by introducing a shrinkage procedure.

Suppose the true functional relationship between a response η and ρ independent variables is

η = β₀ + Σ_i=1^pβ_ix_i + Σ_i=1^pβ_iix_i² + Σ_i=1^p-1Σ_j=1^p-1β_ijx_ix_j. i < j

We fit a model

ŷ(x) = β̂̂₀ + Σ_i=1^pk̂_iβ̂_ix_i .

We show that the k_i which minimize

Σ_i=1^pE(k_iβ̂_i - β_i)²

are of the form

k_i = (μ₂Nβ_i²)/(σ² + μ₂Nβ_i²)

We propose estimating k_i by k̂_i where

k̂_i = (μ₂Nβ̂_i²)/(σ̂² + μ₂Nβ̂̂_i²)

and where

β̂̂ = (X₁′X₁)⁻¹X₁′y

and

σ̂² = (y′y - β̂′X′y)/(N-p)

are the usual least squares estimators.

A A The distribution of k̂_i is derived and the probability that k̂_i is closer to the optimal k_i than a k using upper bounds on the parameters is computed. Also an expression for the integrated mean square error of the proposed procedure is found. Various comparisons among least squares estimation minimum variance and minimum bias designs and optimal least squares and shrink.age estimation are made for the one, two, and three variable cases.