Suppose for an i.i.d. sample of size \(n\), the disease status, \(d_{i}\), is MCAR with the probability of each \(d_{i}\) being observed given by \(\pi=0.75\).

(a) Show that \(\frac{1}{n} \sum_{d_{i} \text { observed }} \frac{d_{i}}{\pi}\) is a consistent estimate of population prevalence, \(\operatorname{Pr}\left(d_{i}=1\right)\) (IPW estimate with known probabilities).

(b) Show that \(\frac{1}{n} \sum_{d_{i} \text { observed }} \frac{d_{i}}{\pi}=\frac{\sum_{d_{i} \text { observed }} d_{i}}{\sum_{d_{i} \text { observed }} 1}\) is a consistent estimate of the population prevalence, where \(\widehat{\pi}=\frac{\sum_{d_{i} \text { observed } 1}^{1}}{n}\) is an estimate of \(\pi\) (IPW estimate with estimated probabilities).

(c) Compare the variances of the two estimates in (a) and (b) to confirm that the use of estimated probability \(\pi\) improves efficiency, provided that the model for \(\pi\) is correct.