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Nils Lid Hjort

Publications and source records attributed to Nils Lid Hjort.

At least 73 records · Page 4Linked to original sources
arXiv2026

Optimal inference via confidence distributions for two-by-two tables modelled as Poisson pairs: fixed and random effects

This paper presents methods for meta-analysis of $2 \times 2$ tables, both with and without allowing heterogeneity in the treatment effects. Meta-analysis is common in medical research, but most existing methods are unsuited for $2 \times 2$ tables with rare events. Usually the tables are modelled as pairs of binomial variables, but we will model them as Poisson pairs. The methods presented here are based on confidence distributions, and offer optimal inference for the treatment effect parameter. We also propose an optimal method for inference on the ratio between treatment effects, and illustrate our methods on a real dataset.

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arXiv2026

Minimum L2 and robust Kullback-Leibler estimation

This paper introduces two new robust methods for estimation of parameters in a given parametric family. The first method is that of `minimum weighted L2', effectively minimising an estimate of the integrated (and possibly weighted) squared error. The second is `robust Kullback-Leibler', consisting of minimising a robust version of the empirical Kullback-Leibler distance, and can be viewed as a general robust modification of the maximum likelihood procedure. This second method is also related to recent local likelihood ideas for semiparametric density estimation. The methods are described, influence functions are found, as are formulae for asymptotic variances. In particular large-sample efficiencies are computed under the home turf conditions of the underlying parametric model. The methods and formulae are illustrated for the normal model.

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arXiv2026

Hybrid combinations of parametric and empirical likelihoods

This paper develops a hybrid likelihood (HL) method based on a compromise between parametric and nonparametric likelihoods. Consider the setting of a parametric model for the distribution of an observation $Y$ with parameter $θ$. Suppose there is also an estimating function $m(\cdot,μ)$ identifying another parameter $μ$ via $E\,m(Y,μ)=0$, at the outset defined independently of the parametric model. To borrow strength from the parametric model while obtaining a degree of robustness from the empirical likelihood method, we formulate inference about $θ$ in terms of the hybrid likelihood function $H_n(θ)=L_n(θ)^{1-a}R_n(μ(θ))^a$. Here $a\in[0,1)$ represents the extent of the compromise, $L_n$ is the ordinary parametric likelihood for $θ$, $R_n$ is the empirical likelihood function, and $μ$ is considered through the lens of the parametric model. We establish asymptotic normality of the corresponding HL estimator and a version of the Wilks theorem. We also examine extensions of these results under misspecification of the parametric model, and propose methods for selecting the balance parameter $a$.

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arXiv2026

Dynamic likelihood hazard rate estimation

The best known methods for estimating hazard rate functions in survival analysis models are either purely parametric or purely nonparametric. The parametric ones are sometimes too biased while the nonparametric ones are sometimes too variable. In the present paper a certain semiparametric approach to hazard rate estimation, proposed in Hjort (1991), is developed further, aiming to combine parametric and nonparametric features. It uses a dynamic local likelihood approach to fit the locally most suitable member in a given parametric class of hazard rates, and amounts to a version of nonparametric parameter smoothing within the parametric class. Thus the parametric hazard rate estimate at time $s$ inserts a parameter estimate that also depends on $s$. We study bias and variance properties of the resulting estimator and methods for choosing the local smoothing parameter. It is shown that dynamic likelihood estimation often leads to better performance than the purely nonparametric methods, while also having capacity for not losing much to the parametric methods in cases where the model being smoothed is adequate.

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arXiv2026

Parametric or nonparametric: the FIC approach for stationary time series

We seek to narrow the gap between parametric and nonparametric modelling of stationary time series processes. The approach is inspired by recent advances in focused inference and model selection techniques. The paper generalises and extends recent work by developing a new version of the focused information criterion (FIC), directly comparing the performance of parametric time series models with a nonparametric alternative. For a pre-specified focused parameter, for which scrutiny is considered valuable, this is achieved by comparing the mean squared error of the model-based estimators of this quantity. In particular, this yields FIC formulae for covariances or correlations at specified lags, for the probability of reaching a threshold, etc. Suitable weighted average versions, the AFIC, also lead to model selection strategies for finding the best model for the purpose of estimating e.g.~a sequence of correlations.

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arXiv2026

Focused Relative Risk Information Criterion for Variable Selection in Linear Regression

This paper motivates and develops a novel and focused approach to variable selection in linear regression models. For estimating the regression mean $μ=\E\,(Y\midd x_0)$, for the covariate vector of a given individual, there is a list of competing estimators, say $\hattμ_S$ for each submodel $S$. Exact expressions are found for the relative mean squared error risks, when compared to the widest model available, say $\mse_S/\mse_\wide$. The theory of confidence distributions is used for accurate assessments of these relative risks. This leads to certain Focused Relative Risk Information Criterion scores, and associated FRIC plots and FRIC tables, as well as to Confidence plots to exhibit the confidence the data give in the submodels. The machinery is extended to handle many focus parameters at the same time, with appropriate averaged FRIC scores. The particular case where all available covariate vectors have equal importance yields a new overall criterion for variable selection, balancing complexity and fit in a natural fashion. A connection to the Mallows criterion is demonstrated, leading also to natural modifications of the latter. The FRIC and AFRIC strategies are illustrated for real data.

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arXiv2026

Confidence Distributions for FIC scores

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 endeavours. 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 fruitful weights determined by the relative sizes of the median- and quantile-FIC scores. And Mrs. Jones is pregnant.

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arXiv2026

Towards Semiparametric Bandwidth Selectors for Kernel Density Estimators

There is an intense and partly recent literature focussing on the problem of selecting the bandwidth parameter for kernel density estimators. Available methods are largely `very nonparametric', in the sense of not requiring any knowledge about the underlying density, or `very parametric', like the normality-based reference rule. This report aims at widening the scope towards the inclusion of many semiparametric bandwidth selectors, via Hermite type expansions aroundthe normal distribution. The resulting bandwidths may be seen as carrying out suitable corrections on the normal reference rule, requiring a low number of extra coefficients to be estimated from data. The present report introduces and discusses some basic ideas and develops the necessary initial theory, but modestly chooses to stop short of giving precise recommendations for specific procedures among the many possible constructions. This will require some further analysis, numerical work, and some simulation-based exploration.

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arXiv2026

The stochastic view used in climate sciences: (some) perspectives from (some of) mathematical statistics

Climate statistics is of course a very broad field, along with the many connections and impacts for yet other areas, with a history as long as mankind has been recording temperatures, describing drastic weather events, etc. The important work of Klaus Hasselmann, with crucial contributions to the field, along with various other connected strands of work,is being reviewed and discussed in other chapters. The aim of the present chapter is to point to a few statistical methodology themes of relevance for and joint interest with climate statistics. These themes, presented from a statistical methods perspective, include (i) more careful modelling and model selection strategies for meteorological type time series; (ii) methods for prediction, not only for future values of a time series, but for assessing when a trend might be crossing a barrier, along with relevant measures of uncertainty for these; (iii) climatic influence on marine biology; (iv) monitoring processes to assess whether and then to what extent models and their parameters have stayed reasonably constant over time; (v) combination of outputs from different information sources; and (vi) analysing probabilities and their uncertainties related to extreme events.

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arXiv2026

Anyone for chess? Analysing chess ratings above high thresholds

Suppose some cleverness score parameter is sufficiently interesting to be defined and then measured, perhaps for different strata of specialists or for the broader population. Such phenomena could have Gaussian distributions, when it comes to all players in a stratum, but when interest focuses on the very tails, for the top few percent, those above certain high thresholds, different models are called for, along with the need to analyse such based on the listed top scores only. In this note I develop such models and tools, and apply them to the top-100 and above 2100 points lists for regular chess ratings, for the currently active 14671 men and 753 women, as given by the FIDE, January 2026. It is argued that even when two or more distributions have close to identical expected values, or medians, even smaller differences in variance may explain gaps for the few very best ones.

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arXiv2026

Six-Minute Man Sander Eitrem 5:58.52 -- first man below the 6:00.00 barrier

In Calgary, November 2005, Chad Hedrick was the first to skate the 5,000 m below 6:10. His world record time 6:09.68 was then beaten a week later, in Salt Lake City, by Sven Kramer's 6:08.78. Further top races and world records followed over the ensuing seasons; up to and including the 2024-2025 season, a total of 126 races have been below 6:10, with Nils van der Poel's 2021 world record being 6:01.56. The appropriately hyped-up canonical question for the friends and followers and aficionados of speedskating has then been when (and by whom we for the first time would witness a below 6:00.00 race. In this note I first use extreme value statistics modelling to assess the state of affairs, as per the end of the 2024-2025 season, with predictions and probabilities for the 2025-2026 season. Under natural modelling assumptions the probability of seeing a new world record during this new season is shown to be about ten percent. We were indeed excited but in reality merely modestly surprised that a race better than van der Poel's record was clocked, by Timothy Loubineaud, in Salt Lake City, November 14, 2025. But Six-Minute Man Sander Eitrem's outstanding 5:58.52 in Inzell, on January 24, 2026, is truly beamonesquely shocking. I also use the modelling machinery to analyse the post-Eitrem situation, and suggest answers to the question of how fast the 5,000 m ever can be skated.

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arXiv2025

Random irregular histograms

We propose a new method of histogram construction, providing a fully Bayesian approach to irregular histograms. Our procedure applies Bayesian model selection to a piecewise constant model of the underlying distribution, resulting in a method that selects both the number of bins as well as their location based on the data in a fully automatic fashion. We show that the histogram estimate is consistent with respect to the Hellinger metric under mild regularity conditions, and that it attains a convergence rate equal to the minimax rate (up to a logarithmic factor) for Hölder continuous densities. Simulation studies indicate that the new method performs comparably to other histogram procedures, both for minimizing the estimation error and for identifying modes. A software implementation is included as supplementary material.

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arXiv2025

Asymptotic properties of adaptive designs through differentiability in quadratic mean

There exist multiple regression applications in engineering, industry and medicine where the outcomes follow an adaptive experimental design in which the next measurement depends on the previous observations, so that the observations are not conditionally independent given the covariates. In the existing literature on such adaptive designs, results asserting asymptotic normality of the maximum likelihood estimator require regularity conditions involving the second or third derivatives of the log-likelihood. Here we instead extend the theory of differentiability in quadratic mean (DQM) to the setting of adaptive designs, which requires strictly fewer regularity assumptions than the classical theory. In doing so, we discover a new DQM assumption, which we call summable differentiability in quadratic mean (S-DQM). As applications, we first verify asymptotic normality for two classical adaptive designs, namely the Bruceton 'up-and-down' design and the Robbins-Monro design. Next, we consider a more complicated problem, namely a Markovian version of the Langlie design.

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arXiv2024

Probability Proofs for Stirling (and More): the Ubiquitous Role of $\mathbf{\sqrt{2π}}$

The Stirling approximation formula for $n!$ dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and also to approximations of marginal distributions in Bayesian setups. Certain formulae emerge by working through particular instances, some independently verifiable but others perhaps not. A particular case yielding new formulae is that of summing independent uniforms, related to the Irwin--Hall distribution. Yet further proofs of the Stirling flow from examining aspects of limiting normality of the sample median of uniforms, and from these again we find a proof for the Wallis product formula for $π$.

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arXiv2024

Model robust hybrid likelihood

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.

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arXiv2019

Multivariate Estimation of Poisson Parameters

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.

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arXiv2011

Asymptotics for minimisers of convex processes

By means of two simple convexity arguments we are able to develop a general method for proving consistency and asymptotic normality of estimators that are defined by minimisation of convex criterion functions. This method is then applied to a fair range of different statistical estimation problems, including Cox regression, logistic and Poisson regression, least absolute deviation regression outside model conditions, and pseudo-likelihood estimation for Markov chains. Our paper has two aims. The first is to exposit the method itself, which in many cases, under reasonable regularity conditions, leads to new proofs that are simpler than the traditional proofs. Our second aim is to exploit the method to its limits for logistic regression and Cox regression, where we seek asymptotic results under as weak regularity conditions as possible. For Cox regression in particular we are able to weaken previously published regularity conditions substantially.

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