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