Consider two estimators for the probability, \(p\), that a tossed coin will land on "heads", based on a iid random sample of size \(n\) from whatever Bernoulli distribution governs the probability of observing "heads" and "tails". One estimator is the sample mean \(\bar{x}_{n}\), and the other is the estimator defined as \(t(X)=\sum_{i=1}^{n} X_{i} /(n+k)\), where \(k\) is some positive integer.

a. For each of the estimators, determine which of the following properties apply: unbiased, asymptotically unbiased, BLUE, and/or consistent?

b. Define asymptotic distributions for both estimators. On the basis of their asymptotic distributions, do you favor one estimator over the other?

c. Define the expected squared distances of the estimators from the unknown value of \(p\). Is there any validity to the statement that "for an appropriate choice of \(k, " t(X)\) will be superior to \(\bar{X}\) in terms of expected squared distance from \(p\) ? Explain.

d. Can you foresee any practical problems in using \(t(X)\) to generate estimates of the population mean?

e. Suggest a way of estimating when it might make sense to use \(t(X)\) in place of the sample mean for estimating \(p\).