43 Let \( Z_{1}, \ldots, Z_{n} \) be a random sample from \( N\left(0, \theta^{2}\right), \theta>0 \). Define \( X_{t}=\left|Z_{i}\right| \), and consider estimation of \( \theta \) and \( \theta^{2} \) on the basis of the random sample \( X_{1}, \ldots, X_{n} \).
(a) Find the UMVUE of \( \theta^{2} \) if such exists.
(b) Find an estimator of \( \theta^{2} \) that has uniformly smaller mean-squared error than the estimator that you found in part (a).
(c) Find the UMVUE of \( \theta \) if such exists.
(d) Find the Pitman estimator for the scale parameter \( \theta \).
(e) Does the estimator that you found in part (d) have uniformly smaller meansquared error than the estimator that you found in part (c)?