Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables where \(X_{n}\) has a \(\operatorname{Gamma}(\alpha, \beta)\) distribution for all \(n \in \mathbb{N}\).

a. Compute one- and two- term Edgeworth expansions for the density of \(n^{1 / 2} \sigma^{-1}\left(\bar{X}_{n}-\muight)\) where in this case \(\mu=\alpha \beta\) and \(\sigma^{2}=\alpha \beta^{2}\). What effect do the values of \(\alpha\) and \(\beta\) have on the accuracy of the expansion? Is it possible to eliminate either the first or second term through a specific choice of \(\alpha\) and \(\beta\) ?

b. Compute one- and two-term Edgeworth expansions for the distribution function of \(n^{1 / 2} \sigma^{-1}\left(\bar{X}_{n}-\muight)\).

c. Compute one- and two-term Cornish-Fisher expansions for the quantile function of \(n^{1 / 2} \sigma^{-1}\left(\bar{X}_{n}-\muight)\).