### Asymptotic approach

As above the likelihood is

 (34)

so the log-likelihood is

 (35)

where is a constant. Differentiating and equating to zero gives

 (36)

so that

 (37)

and a second differentiation gives

 (38)

giving, in this case, and .

A histogram of 1000 samples from is shown in figure 3. While not a good density estimator, a histogram is useful for the gross comparison of two samples needed here. Note that this histogram appears to come from a distribution with a smaller standard deviation than that shown in figure 1; the predictive sample is a safer estimate as it allows for the uncertainty in the point density estimation.

danny 2009-07-23