Geometric understanding of likelihood ratio statistics

It is well known that twice a log-likelihood ratio statistic follows asympt otically a chi-square distribution. The result is usually understood and pr oved via Taylor's expansions of likelihood functions and by assuming asympt otic normality of maximum likelihood estimators (MLEs). We obtain more gene ral results by using a different approach the Wilks type of results hold as long as likelihood contour sets are fan-shaped. The classical Wilks theore m corresponds to the situations in which the likelihood contour sets are el lipsoidal. This provides a geometric understanding and a useful extension o f the likelihood ratio theory. As a result, even if the MLEs are not asympt otically normal, the likelihood ratio statistics can still be asymptoticall y chi-square distributed. Our technical arguments are simple and easily und erstood.