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) Numbered Display Equation

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.

FIGURE 9.17 LS cost function for Example 9.44. (a) Nonsingular AT A. (b) Singular ...

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