Minimum Bias Estimation and Experimental Design for Response Surfaces

A note on minimum bias estimation in response surfaces

“…Draper (1959, 1963) presented the average mean squared error (AMSE) criterion integrated over the experimental region as an evaluation criterion, involving variance and bias. Later, Karson et al (1969) introduced an alternative criterion of response surface estimators to AMSE, termed minimum bias estimation (MBE). Next, we will briefly recapitulate these two methodologies.…”

“…The MBE approach is introduced in the next section. Karson et al (1969) proposed an alternative estimation procedure to Box and Draper's approach, the minimization of APV can be achieved for a particular design selected by the experimenter under condition that ASB is first minimized. That is, the ASB is minimized not by design, instead by estimation.…”

“…For this type of design, the standard least-squares estimator of b 1 minimizes the ASB. So, we can say the Box and Draper's all-bias design is a special case of the minimum bias design in Karson et al (1969).…”

“…The AMSE criterion is the one that considers the magnitude of the prediction variance as well as provides some protection against model inadequacy of the fitted response, normally an underestimated model mismatch. Within the realm of AMSE, Karson et al (1969) presented an alternative approach of estimating a response surface called ''minimum bias estimation (MBE)''. As opposed to the standard least-squares estimation (LSE) method used in Box and Draper's approach, MBE concentrates on the minimization of the bias resulting from some higher-order terms omitted by the fitted response through the estimation of parameters.…”

“…Provided the ''estimability'' condition is met, the minimum bias estimator can also achieve minimum average variance of the fitted response for any fixed designs. However, Karson et al (1969) method is compared merely with the ''all-bias'' case addressed in Box and Draper (1959). To further the discussion of MBE, we will compare the Box and Draper's and Karson et al's approaches by means of the single-variable design problem appearing in the literature, and minimizing the sum of the prediction error and the bias error (i.e., AMSE value) will be opted as our performance measure.…”