9.21 PROPERTIES OF LS ESTIMATORS

In this section, we present some properties of the weighted LS estimator in (9.315) using matrix a in (9.311).

9.21.1 Minimum

Substituting the WLS estimator into (9.313) gives

(9.347)

where a symmetric weighting W = WT has been assumed. Substituting the LS parameters causes the third term in the expansion to cancel, and the last expression has reinserted after no further simplification is possible. A plot of the LS cost function for Example 9.44 with W = I is shown in Figure 9.17(a). It has a convex quadratic form, and the minimum is close to . Note that is not an ensemble quantity; it is a function of the samples so that the plot would change with different outcomes for X and Y. However, for large N, the basic shape and location of the minimum are as shown in the figure.

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