A common way of getting confidence intervals for the mean of a normal random variable is to estimate the mean and variance by the method of moments and then compute a confidence interval based on those estimates. Since this is commonly used, I would assume that these are asymptotically valid and (in some sense) optimal. Is this so? If so, is there somewhere where I could read a proof of this?
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$\begingroup$ This paper by Lars Hansen establishes large sample properties of generalized method of moments estimators under weak stationarity and ergodicity, which nests the standard IID method of moments case. You can also find basic textbook treatments but I’ll suggest trying this on your own (the Taylor expansion method you use to get asymptotic normality for MLE should work here!) $\endgroup$– Yashaswi MohantyMar 5 at 19:52
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For a normal random variable, the moment-matching estimator (MME) for the mean is the maximum likelihood estimate (MLE). For the variance, the MME and the MLE differ just by the bias adjustment ( n/(n-1) ), so asymptotically they will coincide, and the MME will have the asymptotic properties of the MLE.
You can observe this in the Wikipedia article about the Normal distribution
The MLE has the properties you mention: consistency, efficiency, asymptotic normality, etc.
But the empirical mean of a sample coming from a normal distribution actually has a normal distribution, this is not true only asymptotically but for any sample size. You can understand this easily if you recall that the sum of two normal random variables is itself a normal random variable. The mean is a sum of normal random variables, multiplied by a scalar (1/n), so it also is a normal random variables. There should be a wealth of material for a step by step proof, here is one.
