Yes, nice solution. A couple of points. First, the posterior distribution you state is specifically the one that results when you use the improper prior f(mu) = constant. Second, following through on your last paragraph, if the cost function is specifically c(mu, mhat) = |mu - muhat|, then the answer is obviously the median of the posterior distribution, which is the sample *mean*! So, this drives home the point that, under this Bayesian framework, the sample median is not necessary. What if we take a frequentist point of view though? Then things get more difficult. Is there a uniformly minimum-cost unbiased estimate for the median? If so, is it the sample mean, the sample median, or something else? -Joshua On 9/29/05, Eugene Salamin <gene_salamin@yahoo.com> wrote:
--- joshua sweetkind-singer <sweetkindsinger@gmail.com> wrote:
Here's an interesting question: suppose we have data X_1, ..., X_n drawn from a Gaussian distribution with unknown mean mu and known variance 1. We wish to estimate mu with a guess muhat. Virtually everyone uses the sample mean of the dataset as an estimate of mu, but note that mu is also the *median* of the distribution. Under what circumstances would we be justified in prefering the sample median of the data to estimate mu? Since the sample average is a sufficient statistic, the answer might be never, but I'm not sure. Might it be the case that that the sample median is preferable if we are using L1 loss, i.e., seeking to minimize E_mu |mu - muhat| ?
I would solve this problem using the Bayesean method. Then the posterior distribution for mu will be a Gaussian with mean equal to the sample mean, and variance 1/n. This is all you can know about mu on the basis of the given information. For this particular estimation problem, where we are given that the underlying distribution is a Gaussian with unit variance, I would have no need for the sample median.
Now then, if you must pick a number muhat, and make some decision on that basis, and there is a cost c(mutrue,muhat) for being wrong, then you can calculate the muhat that minimizes the expected cost, using the p(mu) derived above.
Gene
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