I'll keep on adding problems for which I can't find any solutions.
1) We want to estimate the mean of the normal distribution $N(\mu^*,1)$ given a sample from the distribution. An obvious choice is to take the sample mean. In this case the estimator, say $\overline{\theta}$, is unbiased i.e. $E[\overline{\theta}] = \mu^*$ and variance of estimate is $\frac{1}{2n}$. The question is, what is the mean and variance of the estimate based on sample median. It seems that the estimate should be unbiased, but i don't know how to show it. No idea about the variance.
1) We want to estimate the mean of the normal distribution $N(\mu^*,1)$ given a sample from the distribution. An obvious choice is to take the sample mean. In this case the estimator, say $\overline{\theta}$, is unbiased i.e. $E[\overline{\theta}] = \mu^*$ and variance of estimate is $\frac{1}{2n}$. The question is, what is the mean and variance of the estimate based on sample median. It seems that the estimate should be unbiased, but i don't know how to show it. No idea about the variance.
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