Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\) with mean \(\theta\) and finite variance \(\sigma^{2}\). Suppose now that we are interested in estimating \(g(\theta)=c \theta^{3}\) using the estimator \(g\left(\bar{X}_{n}ight)=c \bar{X}_{n}^{3}\) where \(c\) is a known real constant.

a. Find the bias and variance of \(g\left(\bar{X}_{n}ight)\) as an estimator of \(g(\theta)\) directly.

b. Find asymptotic expressions for the bias and variance of \(g\left(\bar{X}_{n}ight)\) as an estimator of \(g(\theta)\) using Theorem 10.1. Compare this result to the exact expressions derived above.

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Theorem 10.1. Let {X}1 be a sequence of independent and identically distributed random variables from a distribution F with mean 0 and variance 02. Suppose that f has a finite fourth moment and let g be a function that has at least four derivatives. 1. If the fourth derivative of g is bounded, then as n. -1 E[g(Xn)] = g(0) + nog (0) + O(n), 2. If the fourth derivative of g is also bounded, then as n. V[g(Xn)] = ng'(0)] + O(n),