Let \(X\) be a random variable and let \(\mathrm{E}(X)=\theta eq 0\) and \(\mathrm{V}(X)=\) \(\tau^{2}>0\).

Let \(b\) be a constant such that \(0

(a) Show that \(\mathrm{E}(b X) eq \theta\); that is, the shrunken estimator \(b X\) is not unbiased for \(\theta\).

(b) Show that \(\mathrm{V}(b X)<\tau^{2}\).

(c) Now "shrink" \(X\) toward the constant \(c\); that is, let \(Y=c+b(X-c)\). Show that \(\mathrm{V}(Y)<\tau^{2}\).