Consider the statistic in (2.14).

(a) Show that this statistic is asymptotically normal with the asymptotic variance given by

\[\operatorname{Var}_{a}\left(\sqrt{n}\left(\widehat{p}_{1}-\widehat{p}_{2}\right)\right)=\frac{n}{n_{1+}} p_{1}\left(1-p_{1}\right)+\frac{n}{n_{2+}} p_{2}\left(1-p_{2}\right) .\]

(b) By estimating \(p_{1}\) and \(p_{2}\) using \(\widehat{p}_{1}=\widehat{p}_{2}=\frac{n_{+1}}{n}\), confirm the asymptotic distribution in (2.15).