Furthermore, assuming "sample-with-replacement," we can apply the large sample theory 5 (Lehmann and Casella 1998, Theorem 6.3.10), which says thatâ MLE is asymptotically unbiased and converges in distribution to a Normal with mean a and variance 1 I(a) , where I(a), the expected Fisher Information, is where we evaluate E(a For "sampling-without-replacement," we correct the asymptotic variance 1 I(a) by multiplying by the finite population correction factor 1 − D s D : Comparing (17) with (29), we know that Var (â MLE |D s ) < Var (â MF |D s ), and the difference could be substantial.
when we know the margins, we ought to use them.