Best unbiased predictor

We next describe the first steps in the derivation of the best linear unbiased predictor (BLUP) of Y x) at an untried input vector x (see, for example, Sacks et al 1989). Similar steps are used in Section 4 to estimate the effects of one, two, or more input variables. It is then apparent how to adapt results and computational methods for predicting Y(x) to the problem of estimating such effects. 

Following the random-function model (1), consider the prediction of T(x ) by f (.v) = a x)y, that is, a linear combination of the n values of the output variable observed in the experiment. The best linear unbiased predictor is obtained by minimizing the mean squared error of the linear predictor or approximator, Y(x). The mean squared error, MSIi K(x), is... 

This constraint is also sometimes motivated by unbiasedness, that is, from E[T(a )] = E[T(a )] for all /3. Thus, the best linear unbiased predictor, or optimal value of a(x), results from the following optimization problem,... 

Derivation of the Best Linear Unbiased Predictor of an Effect... 

The best linear unbiased predictor (BLUP) of Ye(xe) in (18) follows from the properties of Ze xe) in (17). Clearly, Ze(xe), like Z(x), has expectation zero. Its variance, however, differs from one effect to another ... 

Also under OLS assumptions, the regression parameter estimates have a number of optimal properties. First, 0 is an unbiased estimator for 0. Second, the standard error of the estimates are at a minimum, i.e., the standard error of the estimates will be larger than the OLS estimates given any other assumptions. Third, assuming the errors to be normally distributed, the OLS estimates are also the maximum likelihood (ML) estimates for 0 (see below). It is often stated that the OLS parameter estimates are BLUE (Best Linear Unbiased Predictors) in the sense that best means minimum variance. Fourth, OLS estimates are consistent, which in simple terms means that as the sample size increases the standard error of the estimate decreases and the bias of the parameter estimates themselves decreases. 

Figure 8,2 Methodology of grapevine selection in Portugal. EBLUPs empirical best linear unbiased predictors.

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