From Theorems 6 and 7, the following conclusions can be drawn:
The maximum likelihood estimator of is asymptotically unbiased: .
Asymptotically, , which, by a multi-parameter version of the Cramér-Rao theorem, is optimal for unbiased estimators.
If , then , so is a positive definite, symmetric matrix with elements ; hence, is the variance of . The quantity is the standard error of and .
We have argued that it is natural to base confidence regions for on the basis of including those values for which the likelihood is greatest (or, equivalently, the deviance is smallest). Thus, we should choose regions of the form
for some value of .
To obtain a region which is a confidence region, should be chosen so that .
The shape of contours of equal deviance near the MLE for large sample sizes can be studied by looking at the Taylor series expansion:
Therefore the boundary of the confidence region is given by such that . This is the equation of an ellipse, which is the shape we found in our contour plots of the log-likelihood function.
We can now obtain such a value approximately using the fact that asymptotically : choose such that . For example, if , then we obtain the following table:
drop from | ||
---|---|---|
1 | 3.84 | 1.92 |
2 | 5.99 | 3.00 |
3 | 7.81 | 3.90 |
4 | 9.49 | 4.75 |
5 | 11.07 | 5.53 |
6 | 12.59 | 6.25 |
Consequently, confidence regions for correspond to contours of the likelihood surface, and the appropriate contour for a specified degree of confidence can be obtained from the corresponding distribution.
For example, if (a bivariate likelihood), a 95% confidence region would be
i.e. the region is defined by parameter pairs within 3 units of the likelihood at the MLE (as shown in row 2 of the table).
Hence if we draw a likelihood contour plot with contours representing unit steps down from the maximum, a 95% confidence region would be described by the first (innermost) 3 contours (see Figure Figure 4.1 (Link)).