Three enigmatic examples and inference from likelihood

Statistics has many inference procedures for examining a model with data to obtain information concerning the value of a parameter of interest. If these give different results for the same model and data, one can reasonably want a satisfactory explanation. Over the last eighty years, three very simple examples have appeared intermittently in the literature, often with contradictory or misleading results; these enigmatic examples come from Cox, Behrens, and Box & Cox. The procedures in some generality begin with an observed likelihood function, which is known to provide just first order accuracy unless there is additional information that calibrates the parameter. In particular, default Bayes analysis seeks such calibration in the form of a model‐based prior; such a prior with second order accuracy is examined for the Behrens problem, but none seems available for the Box and Cox problem. Alternatively, higher‐order likelihood theory obtains such information by examining likelihood at and near the data and achieves third order accuracy. We examine both Bayesian and frequentist procedures in the context of the three enigmatic examples; simulations support the indicated accuracies. The Canadian Journal of Statistics © 2009 Statistical Society of Canada