Examples of Likelihoods and Comparison with Point Estimates and Large Sample Approximations

Abstract The exact likelihood functions are examined for several examples of data chosen from the literature. These are compared with the likelihoods arising from the large sample approximations and with point estimates that were actually used in the literature. It is concluded that large sample approximations (application of standard maximum likelihood theory) can be misleading for inferences and should be checked against the actual likelihood functions. Similarly point estimates can be misleading or uninformative and their properties such as bias, variance, etc. are relatively unimportant. Because of the availability of high speed computers, exact methods and asymptotic comparisons are now feasible and this should be reflected where possible in the theory and application of statistical inference.