Again, suppose we have a random sample X1,..., Xn from 1/Ï f[(x - 9)/Ï), a location- scale pdf, but we are now interested in estimating

Again, suppose we have a random sample X1,..., Xn from 1/σ f[(x - 9)/σ), a location- scale pdf, but we are now interested in estimating σ2. We can consider three groups of transformations:
G1 = (ga, c(x): - ˆž 0},
where ga, c(x1,... ,xn) = (cx1 + a,..., cxn + a);
G2 = {ga(x): - ˆž where ga(x1,... ,xn) = (x1 + a,... ,xn + a); and
G3 = (gc(x): c > 0},
where gc(x1,... ,xn) = (cx1,... ,cxn).
a. Show that estimators of σ2 of the form kS2, where k is a positive constant and S2 is the sample variance, are invariant with respect to G2 and equivariant with respect to the other two groups.
b. Show that the larger class of estimators of σ2 of the form

where Ï•(x) is a function, are equivariant with respect to G3 but not with respect to either G1 or G2, unless Ï•(x) is a constant (Brewster and Zidek 1974).
Consideration of estimators of this form led Stein (1964) and Brewster and Zidek (1974) to find improved estimators of variance (see Lehmann and Casella 1998, Section 3.3).

W(X1,..,Xn) = 6