Nils Lid Hjort

The article concerns hybrid combinations of empirical and parametric likelihood functions. Combining the two allows classical parametric likelihood to be crucially modified via the nonparametric counterpart, making possible model misspecification less problematic. Limit theory for the maximum hybrid likelihood estimator is sorted out, also outside the parametric model conditions. Results include consistency of the estimated parameter in the parametric model towards a well-defined limit, as well as asymptotic normality after proper scaling and centring of the same quantity. Our results allow for the presence of plug-in parameters in the hybrid and empirical likelihood framework. Furthermore, the variance and mean squared error of these estimators are studied, with recipes for their estimation. The latter is used to define a focused information criterion, which can be used to choose how the parametric and empirical part of the hybrid combination should be balanced. This allows for hybrid models to be fitted in a context driven way, minimizing the estimated mean squared error for estimating any pre-specified quantity of interest.

When using the Focused Information Criterion (FIC) for assessing and ranking candidate models with respect to how well they do for a given estimation task, it is customary to produce a so-called FIC plot. This plot has the different point estimates along the y-axis and the root-FIC scores on the x-axis, these being the estimated root-mean-square scores. In this paper we address the estimation uncertainty involved in each of the points of such a FIC plot. This needs careful assessment of each of the estimators from the candidate models, taking also modelling bias into account, along with the relative precision of the associated estimated mean squared error quantities. We use confidence distributions for these tasks. This leads to fruitful CD–FIC plots, helping the statistician to judge to what extent the seemingly best models really are better than other models, etc. These efforts also lead to two further developments. The first is a new tool for model selection, which we call the quantile-FIC, which helps overcome certain difficulties associated with the usual FIC procedures, related to somewhat arbitrary schemes for handling estimated squared biases. A particular case is the median-FIC. The second development is to form model averaged estimators with weights determined by the relative sizes of the median- and quantile-FIC scores.

The recent article ‘Satellite conjunction analysis and the false confidence theorem’ (Balch et al . 2019, Satellite conjunction analysis and the false confidence theorem. Proc. R. Soc. A 475 , 20180565) points to certain difficulties with Bayesian analysis when used for models for satellite conjuntion and ensuing operative decisions. Here, we supplement these previous analyses and findings with further insights, uncovering what we perceive of as being the crucial points, explained in a prototype set-up where exact analysis is attainable. We also show that a different and frequentist method, involving confidence distributions, is free of the false confidence syndrome.

This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss function that penalises bad estimates of each of the parameters and the sum (or equivalently the mean) of the parameters is introduced. Under this loss function, a class of minimax estimators that uniformly dominate the maximum likelihood estimator is derived. Crucially, these methods have the property that for estimating a given component parameter, the full data vector is utilised. Estimators in this class can be fine-tuned to limit shrinkage away from the maximum likelihood estimator, thereby avoiding implausible estimates of the sum of the parameters. Further light is shed on this new class of estimators by showing that it can be derived by Bayesian and empirical Bayesian methods. In particular, we exhibit a generalisation of the Clevenson-Zidek estimator, and prove its admissibility. Moreover, a class of prior distributions for which the Bayes estimators uniformly dominate the maximum likelihood estimator under the new loss function is derived. A section is included involving weighted loss functions, notably also leading to a procedure improving uniformly on the maximum likelihood method in an infinite-dimensional setup. Importantly, some of our methods lead to constructions of new multivariate models for both rate parameters and count observations. Finally, estimators that shrink the usual estimators towards a data based point in the parameter space are derived and compared.

Cox’s proportional hazards regression model is the standard method for modelling censored life-time data with covariates. In its standard form, this method relies on a semiparametric proportional hazards structure, leaving the baseline unspecified. Naturally, specifying a parametric model also for the baseline hazard, leading to fully parametric Cox models, will be more efficient when the parametric model is correct, or close to correct. The aim of this paper is two-fold. (a) We compare parametric and semiparametric models in terms of their asymptotic relative efficiencies when estimating different quantities. We find that for some quantities the gain of restricting the model space is substantial, while it is negligible for others. (b) To deal with such selection in practice we develop certain focused and averaged focused information criteria (FIC and AFIC). These aim at selecting the most appropriate proportional hazards models for given purposes. Our methodology applies also to the simpler case without covariates, when comparing Kaplan–Meier and Nelson–Aalen estimators to parametric counterparts. Applications to real data are also provided, along with analyses of theoretical behavioural aspects of our methods.