|EDIT GOMBAY, Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada|
|Correcting some limit theorems about the likelihood ratio|
To put the new results into perspective, first, we review some aspects of the history of the likelihood ratio since its definition by Neyman and Pearson in 1928.
Strong approximation of the maximum likelihood ratio statistic by a diffusion process under the null hypothesis is given. This allows one to develop statistics using different weight functions. Sequential tests will be defined and the precision of the approximation is examined.
The asymptotic distribution of the likelihood ratio under noncontiguous alternatives is shown to be normal for the exponential family of distributions. The rate of convergence of the parameters to the hypothetical value is specified where the asymptotic noncentral chi-square distribution no longer holds. It is only a little slower than O(n-1/2). The result provides compact power approximation formulae and is shown to work reasonably well even for moderate sample sizes.
In conclusion, we briefly consider the consequences of these theorems in sequential testing theory.