1. a) Show that any squared-error consistent is asymptotically unbiased and b) Show that any
squared consistent is consistent in the sense of Definition 5.7.1
Solution
a)An estimator ?^of ? isunbiasedif E(?^? ?) = ? for all ?. Thebiasis represented as bias?^(?)=
E(?^? ?) - ?.
Let ?^nbe an estimator of ? based on a sample size of n. The estimator isasymptotically
unbiasedif limn??(E(?^n? ?)) = ? for all ?.
An estimator isconsistentorweakly consistentif, for all ? > 0 and any ?,
limn??(Pr(??^n- ?? > ?). To show that an estimator is weakly consistent, it is sufficient to show
that the estimator is asymptotically unbiased and that limn??(Var(?^n)) = 0.
Themean squared error (MSE)of an estimator is MSE?^(?) = E((?^- ?)2?? ).
The following equation is true: MSE?^(?) = Var(?^? ?) + (bias?^(?))2.
b)