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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

Approach 1

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

Approach 2

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!

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