For X randomly sampled from a Poisson(lambda) population, the method of moment estimate of lambda is the sample mean:
lambda_hat = X_bar
Now, say we are interested in the variance of lambda_hat (sample size = n)
Var(lambda_hat) = Var(X_bar) = Var(X) / n
Since we know the true Var(X) = lambda, we can just plug in the estimated lambda_hat into the above to get the estimated variance of lambda_hat
i.e. Var_hat(lambda_hat) = lambda_hat / n
Sub S^2 = sample variance into Var(X), i.e. Var_hat(lambda_hat) = S^2 / n
I personally don't understand why approach 2 can be correct, as effectively you're estimating lambda twice in the same procedure, once with X_bar and a second time with S^2, but that's what's written in my lecture notes.
Can someone provide some insights into the two approaches? Thank you very much!