Consider the following method of estimating lambda for a Poisson distribution.

Observe that p₀ = P(X=0) = e^(-lambda)

Letting Y denote the number the number of zeros from i.i.d. sample of size n, lambda might be estimated by

lambda_hat = -log(Y/n)

Use the method of propagation of error to obtain approximate expressions for the variance and the bias of this estimate. Compare the variance of this estimate to the variance of the mle, computing relative efficiencies for various values of lambda. Note that Y ~ bin( n , p₀).