Nils Lid Hjort

• Hjort, Nils Lid & Stoltenberg, Emil Aas (2026). Probability Proofs for Stirling (and More): The Ubiquitous Role of (Formula Presented.). American Statistician. ISSN 0003-1305. 80(3), p. 405–412. doi: 10.1080/00031305.2025.2603256. Full text in Research Archive Show summary

  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, with arguments which can be worked through by Master and PhD level students (and above). 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. A section detailing historical aspects and development is included, from Wallis Citation1656 and de Moivre and Stirling 1730 to Laplace 1778, etc.

• Simensen, Oskar Høgberg; Christensen, Dennis & Hjort, Nils Lid (2026). Random irregular histograms. Computational Statistics & Data Analysis. ISSN 0167-9473. 220. doi: 10.1016/j.csda.2026.108367. Full text in Research Archive
• Christensen, Dennis; Stoltenberg, Emil Aas & Hjort, Nils Lid (2026). Asymptotic properties of adaptive designs. Stochastic Processes and their Applications. ISSN 0304-4149. 196. doi: 10.1016/j.spa.2026.104922. Full text in Research Archive Show summary

  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 this paper we present new asymptotic theory for such processes of dependent regression data. This includes stand-alone results for adaptive covariate processes, which, in particular, enable us to develop new likelihood theory for inference on the regression model parameters. Existing 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.

• Dæhlen, Ingrid & Hjort, Nils Lid (2025). Model robust hybrid likelihood. Journal of Statistical Planning and Inference. ISSN 0378-3758. 241. doi: 10.1016/j.jspi.2025.106327. Full text in Research Archive Show summary

  This 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 hybrid likelihood function is sorted out, also outside of the parametric model conditions. We prove a profiling result as well as limiting behaviour of the maximizer of the hybrid likelihood function. 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.

• Dæhlen, Ingrid; Hjort, Nils Lid & Haff, Ingrid Hobæk (2023). Accurate bias estimation with applications to focused model selection. Scandinavian Journal of Statistics. ISSN 0303-6898. p. 1–36. doi: 10.1111/sjos.12696. Full text in Research Archive Show summary

  Abstract We derive approximations to the bias and squared bias with errors of order where is the sample size. Our results hold for a large class of estimators, including quantiles, transformations of unbiased estimators, maximum likelihood estimators in (possibly) incorrectly specified models, and functions thereof. Furthermore, we use the approximations to derive estimators of the mean squared error (MSE) which are correct to order . Since the variance of many estimators is of order , this level of precision is needed for the MSE estimator to properly take the variance into account. We also formulate a new focused information criterion (FIC) for model selection based on the estimators of the squared bias. Lastly, we illustrate the methods on data containing the number of battle deaths in all major inter‐state wars between 1823 and the present day. The application illustrates the potentially large impact of using a less‐accurate estimator of the squared bias.

• Aastveit, Marthe Elisabeth; Cunen, Celine Marie Løken & Hjort, Nils Lid (2023). A new framework for semi-Markovian parametric multi-state models with interval censoring. Statistical Methods in Medical Research. ISSN 0962-2802. 32(6), p. 1053–1246. doi: 10.1177/09622802231160550. Full text in Research Archive Show summary

  There are few computational and methodological tools available for the analysis of general multi-state models with interval censoring. Here, we propose a general framework for parametric inference with interval censored multi-state data. Our framework can accommodate any parametric model for the transition times, and covariates may be included in various ways. We present a general method for constructing the likelihood, which we have implemented in a ready-to-use R package, smms, available on GitHub. The R package also computes the required high-dimensional integrals in an efficient manner. Further, we explore connections between our modelling framework and existing approaches: our models fall under the class of semi-Markovian multi-state models, but with a different, and sparser parameterisation than what is often seen. We illustrate our framework through a dataset monitoring heart transplant patients. Finally, we investigate the effect of some forms of misspecification of the model assumptions through simulations.

• Cunen, Celine; Walløe, Lars; Konishi, Kenji & Hjort, Nils Lid (2021). Decline in body condition in the Antarctic minke whale (Balaenoptera bonaerensis) in the Southern Ocean during the 1990s. Polar Biology. ISSN 0722-4060. 44, p. 259–273. doi: 10.1007/s00300-020-02783-3. Full text in Research Archive Show summary

  Abstract Changes in the body condition of Antarctic minke whales ( Balaenoptera bonaerensis ) have been investigated in a number of studies, but remain contested. Here we provide a new analysis of body condition measurements, with particularly careful attention to the statistical model building and to model selection issues. We analyse body condition data for a large number (4704) of minke whales caught between 1987 and 2005. The data consist of five different variables related to body condition (fat weight, blubber thickness and girth) and a number of temporal, spatial and biological covariates. The body condition variables are analysed using linear mixed-effects models, for which we provide sound biological motivation. Further, we conduct model selection with the focused information criterion (FIC), reflecting the fact that we have a clearly specified research question, which leads us to a clear focus parameter of particular interest. We find that there has been a substantial decline in body condition over the study period (the net declines are estimated to 10% for fat weight, 7% for blubber thickness and 3% for the girth). Interestingly, there seems to be some differences in body condition trends between males and females and in different regions of the Antarctic. The decline in body condition could indicate major changes in the Antarctic ecosystem, in particular, increased competition from some larger krill-eating whale species.

• Hjort, Nils Lid & Stoltenberg, Emil Aas (2021). The partly parametric and partly nonparametric additive risk model. Lifetime Data Analysis. ISSN 1380-7870. doi: 10.1007/s10985-021-09535-3. Full text in Research Archive Show summary

  Aalen’s linear hazard rate regression model is a useful and increasingly popular alternative to Cox’ multiplicative hazard rate model. It postulates that an individual has hazard rate function h(s)=z1α1(s)+⋯+zrαr(s) in terms of his covariate values z1,…,zr. These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions αj(s) are the parameters of the model and are estimated from data. This is traditionally accomplished in a fully nonparametric way. This paper develops methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified. Large-sample results are reached inside this partly parametric, partly nonparametric framework, which also enables us to assess the goodness of fit of the model’s parametric components. In addition, these results are used to pinpoint how much precision is gained, using the parametric-nonparametric model, over the standard nonparametric method. A real-data application is included, along with a brief simulation study.

• Cunen, Celine & Hjort, Nils Lid (2021). Combining information across diverse sources: The II-CC-FF paradigm. Scandinavian Journal of Statistics. ISSN 0303-6898. p. 1–32. doi: 10.1111/sjos.12530. Full text in Research Archive Show summary

  We introduce and develop a general paradigm for combining information across diverse data sources. In broad terms, suppose 𝜑 is a parameter of interest, built up via components 𝜓1,…,𝜓𝑘 from data sources 1, … , k. The proposed scheme has three steps. First, the independent inspection (II) step amounts to investigating each separate data source, translating statistical information to a confidence distribution (CD) 𝐶𝑗(𝜓𝑗) for the relevant focus parameter 𝜓𝑗 associated with data source j. Second, confidence conversion (CC) techniques are used to translate the CDs to confidence log-likelihood functions. Finally, the focused fusion (FF) step uses relevant and context-driven techniques to construct a confidence distribution for the primary focus parameter 𝜑=𝜑(𝜓1,…,𝜓𝑘), acting on the combined confidence log-likelihood. In traditional setups, the II-CC-FF strategy amounts to versions of meta-analysis, and turns out to be competitive against state-of-the-art methods. Its potential lies in applications to harder problems, however. Illustrations are presented, related to actual applications.

• Cunen, Celine Marie Løken; Hjort, Nils Lid & Nygård, Håvard Mokleiv (2020). Statistical sightings of better angels: Analysing the distribution of battle-deaths in interstate conflict over time. Journal of Peace Research. ISSN 0022-3433. 57(2), p. 221–234. doi: 10.1177/0022343319896843. Full text in Research Archive Show summary

  Have great wars become less violent over time, and is there something we might identify as the long peace? We investigate statistical versions of such questions, by examining the number of battle-deaths in the Correlates of War dataset, with 95 interstate wars from 1816 to 2007. Previous research has found this series of wars to be stationary, with no apparent change over time. We develop a framework to find and assess a change-point in this battle-deaths series. Our change-point methodology takes into consideration the power law distribution of the data, models the full battle-deaths distribution, as opposed to focusing merely on the extreme tail, and evaluates the uncertainty in the estimation. Using this framework, we find evidence that the series has not been as stationary as past research has indicated. Our statistical sightings of better angels indicate that 1950 represents the most likely change-point in the battle-deaths series – the point in time where the battle-deaths distribution might have changed for the better.

• Cunen, Celine; Walløe, Lars & Hjort, Nils Lid (2020). Focused model selection for linear mixed models with an application to whale ecology. Annals of Applied Statistics. ISSN 1932-6157. 14(2). doi: 10.1214/20-AOAS1331. Full text in Research Archive Show summary

  A central point of disagreement, in certain long-standing discussions about a particular whaling dataset in the Scientific Committee of the International Whaling Commission, has directly involved model selection issues for linear mixed effect models. The biological question under discussion is associated with a clearly defined parameter of primary interest, a focus parameter, which makes model selection with the Focused Information Criterion (FIC) more appropriate than other selection methods. Since the existing FIC methodology has not covered the case of linear mixed effects models, this article sets up the required framework and develops the necessary formulae for the relevant FIC. Our new criterion requires the asymptotic distribution of estimators derived for a given candidate linear mixed model but with behaviour examined under a wider linear mixed model. These results, needed here to build our FIC, also have independent interest.

• Cunen, Celine; Hjort, Nils Lid & Schweder, Tore (2020). Confidence in confidence distributions! Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. ISSN 1364-5021. doi: 10.1098/rspa.2019.0781. Full text in Research Archive Show summary

  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. A475, 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.

• Cunen, Celine & Hjort, Nils Lid (2020). Confidence Distributions for FIC Scores. Econometrics. 8(27). doi: 10.3390/econometrics8030027. Full text in Research Archive Show summary

  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.

• Stoltenberg, Emil Aas & Hjort, Nils Lid (2020). Models and inference for on–off data via clipped Ornstein–Uhlenbeck processes. Scandinavian Journal of Statistics. ISSN 0303-6898. doi: 10.1111/sjos.12472. Full text in Research Archive Show summary

  We introduce a model for recurrent event data subject to left‐, right‐, and intermittent‐censoring. The observations consist of binary sequences (along with covariates) for each individual under study. These sequences are modeled as generated by latent Ornstein–Uhlenbeck processes being above or below certain thresholds. Features of the latent process and the thresholds are taken as functions of covariates, allowing the researcher to distinguish factors that have an effect on the frailty, from those that have an effect on the variability, of the observational unit. Inference is achieved by a quasi‐likelihood approach, for which consistency and asymptotic normality is established. An advantage of our model is that particularities regarding the censoring need not be taken actively into account, and that it is well suited for situations where the individuals under study are irregularly and asynchronously observed. The motivation for our model came from a dataset pertaining to the incidence of diarrhoea among Brazilian children growing up under rather harsh conditions. We analyze these data with our model and contrast the results with an intensity‐based counting process analysis of the same data.

• Claeskens, Gerda; Cunen, Celine Marie Løken & Hjort, Nils Lid (2019). Model Selection via Focused Information Criteria for Complex Data in Ecology and Evolution. Frontiers in Ecology and Evolution. 7. doi: 10.3389/fevo.2019.00415. Full text in Research Archive Show summary

  Datasets encountered when examining deeper issues in ecology and evolution are often complex. This calls for careful strategies for both model building, model selection, and model averaging. Our paper aims at motivating, exhibiting, and further developing focused model selection criteria. In contexts involving precisely formulated interest parameters, these versions of FIC, the focused information criterion, typically lead to better final precision for the most salient estimates, confidence intervals, etc. as compared to estimators obtained from other selection methods. Our methods are illustrated with real case studies in ecology; one related to bird species abundance and another to the decline in body condition for the Antarctic minke whale.

• Walløe, Lars; Hjort, Nils Lid & Thoresen, Marianne (2019). Why results from Bayesian Statistical analyses of clinical trials with a strong prior and small sample sixes may be misleading. Acta Paediatrica. ISSN 0803-5253. p. 1190–1191. doi: 10.1111/apa.14800. Full text in Research Archive
• Stoltenberg, Emil Aas & Hjort, Nils Lid (2019). Multivariate estimation of Poisson parameters. Journal of Multivariate Analysis. ISSN 0047-259X. 175, p. 1–19. doi: 10.1016/j.jmva.2019.104545. Full text in Research Archive Show summary

  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 also 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.

• Ko, Vinnie; Hjort, Nils Lid & Haff, Ingrid Hobæk (2019). Focused information criteria for copulae. Scandinavian Journal of Statistics. ISSN 0303-6898. 46, p. 1117–1140. doi: 10.1111/sjos.12387. Full text in Research Archive Show summary

  In this paper, we extend the focused information criterion (FIC) to copula models. Copulas are often used for applications where the joint tail behavior of the variables is of particular interest, and selecting a copula that captures this well is then essential. Traditional model selection methods such as the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) aim at finding the overall best‐fitting model, which is not necessarily the one best suited for the application at hand. The FIC, on the other hand, evaluates and ranks candidate models based on the precision of their point estimates of a context‐given focus parameter. This could be any quantity of particular interest, for example, the mean, a correlation, conditional probabilities, or measures of tail dependence. We derive FIC formulae for the maximum likelihood estimator, the two‐stage maximum likelihood estimator, and the so‐called pseudo‐maximum‐likelihood (PML) estimator combined with parametric margins. Furthermore, we confirm the validity of the AIC formula for the PML estimator combined with parametric margins. To study the numerical behavior of FIC, we have carried out a simulation study, and we have also analyzed a multivariate data set pertaining to abalones. The results from the study show that the FIC successfully ranks candidate models in terms of their performance, defined as how well they estimate the focus parameter. In terms of estimation precision, FIC clearly outperforms AIC, especially when the focus parameter relates to only a specific part of the model, such as the conditional upper‐tail probability.

• Ko, Vinnie & Hjort, Nils Lid (2019). Copula information criterion for model selection with two-stage maximum likelihood estimation. Econometrics and Statistics. ISSN 2452-3062. 12, p. 167–180. doi: 10.1016/j.ecosta.2019.01.001. Full text in Research Archive Show summary

  In parametric copula setups, where both the margins and copula have parametric forms, two-stage maximum likelihood estimation, often referred to as inference functions for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Exploiting the existing model robust inference of two-stage maximum likelihood estimation, a copula information criterion (CIC) for model selection is developed. In a nutshell, CIC aims for the model that minimizes the Kullback–Leibler divergence from the real data generating mechanism. CIC does not assume that the chosen parametric model captures this true model, unlike what is assumed for AIC. In this sense, CIC is analogous to the Takeuchi Information Criterion (TIC), which is defined for the full maximum likelihood. If the additional assumption that a candidate model is correctly specified is made, then CIC for that model simplifies to AIC. Additionally, CIC can easily be extended to the conditional copula setup where covariates are parametrically linked to the copula model. As a numerical illustration, simulation studies were performed to show that the better model according to CIC also has better prediction performance in general. The result also shows that the bias correction term of CIC penalizes the misspecified model more heavily. This bias correction term has a strong positive relationship with the prediction performance of the model. So, a model with bad prediction performance is being penalized more by CIC. Although this behavior of the bias correction part is an important conceptual advance of CIC, this is not sufficient to make CIC outperform AIC in practice. This is because each misspecified model has the bias correction term and they grow at different speeds, depending on the model. The difference between CIC and AIC becomes minimal as sample size grows because the log-likelihood part outgrows the bias correction part.

• Ko, Vinnie & Hjort, Nils Lid (2019). Model robust inference with two-stage maximum likelihood estimation for copulas. Journal of Multivariate Analysis. ISSN 0047-259X. 171, p. 362–381. doi: 10.1016/j.jmva.2019.01.004. Full text in Research Archive Show summary

  This article is concerned with inference in parametric copula setups, where both the marginals and the copula have parametric forms. For such models, two-stage maximum likelihood estimation, often referred to as inference function for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Previous studies of the two-stage maximum likelihood estimator have largely been based on the assumption that the chosen parametric model captures the true model that generated data. We study the impact of dropping this true model assumption, both theoretically and numerically. We first show that the two-stage maximum likelihood estimator is consistent for a well-defined least false parameter value, different from the analogous least false parameter associated with the full maximum likelihood procedure. Then we demonstrate limiting normality of the full vector of estimators, with concise matrix notation for the variance matrices involved. Along with consistent estimators for these, we have built a model-robust machinery for inference in parametric copula models. The special case where the parametric model is assumed to hold corresponds to situations studied earlier in the literature, with simpler formulas for variance matrices. As a numerical illustration, we perform a set of simulations. We also analyze five-dimensional Norwegian precipitation data. We find that the variance of the copula parameter estimate can both increase and decrease, by dropping the true model assumption. In addition, we observe that the two-stage maximum likelihood estimator is still highly efficient when the true model assumption is dropped and thus the model robust asymptotic variance formulas are used. Additionally, we discover that using highly misspecified models can lead to situations where the asymptotic variance of the two-stage maximum likelihood estimator is lower than that of full maximum likelihood estimator. Our results are also used to analyze the mean squared error properties for both the full and the two-stage maximum likelihood estimators of any focus parameter.

• Walløe, Lars; Hjort, Nils Lid & Thoresen, Marianne (2019). Start cooling as soon as possible. Acta Paediatrica. ISSN 0803-5253. 108(4), p. 771–771. doi: 10.1111/apa.14710. Full text in Research Archive
• Jullum, Martin & Hjort, Nils Lid (2018). What price semiparametric Cox regression? Lifetime Data Analysis. ISSN 1380-7870. 25(3), p. 406–438. doi: 10.1007/s10985-018-9450-7. Full text in Research Archive Show summary

  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.

• Cunen, Celine Marie Løken; Hermansen, Gudmund Horn & Hjort, Nils Lid (2018). Confidence distributions for change-points and regime shifts. Journal of Statistical Planning and Inference. ISSN 0378-3758. 195, p. 14–34. doi: 10.1016/j.jspi.2017.09.009. Full text in Research Archive Show summary

  Suppose observations y 1 ,...,y n stem from a parametric model f ( y,θ ), with the parameter taking one value θ L for y 1 ,...,y τ and another value θ R for y τ +1 ,...,y n . This article provides and examines two different general strategies for not merely estimating the break point τ but also to complement such an estimate with full confidence distributions, both for the change-point τ and for associated measures of differences between the two levels of θ . The first idea worked with involves testing homogeneity for the two segments to the left and the right of a candidate change-point value at a fine-tuned level of significance. Carrying out such a scheme requires having a goodness-of-fit test for constancy of the θ parameter over a segment of indices, and we also develop classes of such tests. These also have some independent interest. The second general method uses the log-likelihood function, profiled over the other parameters, and we show how this may lead to confidence inference for τ . Our methods are illustrated for four real data stories, with these meeting different types of challenges.

• Hjort, Nils Lid; McKeague, Ian & Keilegom, Ingrid Van (2018). Hybrid combinations of parametric and empirical likelihoods. Statistica sinica. ISSN 1017-0405. 28(4), p. 2389–2407. doi: 10.5705/ss.202017.0291. Full text in Research Archive
• Hjort, Nils Lid & Sørensen, Øystein (2018). Rikdom og pengenes verdi i Andeby. Donald Duck & Co. Full text in Research Archive
• Walløe, Lars; Hjort, Nils Lid & Thoresen, Marianne (2018). Major concerns about late hypothermia study. Acta Paediatrica. ISSN 0803-5253. doi: 10.1111/apa.14640. Full text in Research Archive
• Hellton, Kristoffer Herland & Hjort, Nils Lid (2018). Fridge: Focused fine‐tuning of ridge regression for personalized predictions. Statistics in Medicine. ISSN 0277-6715. 37(8), p. 1290–1303. doi: 10.1002/sim.7576. Full text in Research Archive Show summary

  Statistical prediction methods typically require some form of fine‐tuning of tuning parameter(s), with K‐fold cross‐validation as the canonical procedure. For ridge regression, there exist numerous procedures, but common for all, including cross‐validation, is that one single parameter is chosen for all future predictions. We propose instead to calculate a unique tuning parameter for each individual for which we wish to predict an outcome. This generates an individualized prediction by focusing on the vector of covariates of a specific individual. The focused ridge—fridge—procedure is introduced with a two‐part contribution: First we define an oracle tuning parameter minimizing the mean squared prediction error of a specific covariate vector, and then we propose to estimate this tuning parameter by using plug‐in estimates of the regression coefficients and error variance parameter. The procedure is extended to logistic ridge regression by using parametric bootstrap. For high‐dimensional data, we propose to use ridge regression with cross‐validation as the plug‐in estimate, and simulations show that fridge gives smaller average prediction error than ridge with cross‐validation for both simulated and real data. We illustrate the new concept for both linear and logistic regression models in two applications of personalized medicine: predicting individual risk and treatment response based on gene expression data. The method is implemented in the R package fridge.

• Jullum, Martin & Hjort, Nils Lid (2017). Parametric or nonparametric: The FIC approach. Statistica sinica. ISSN 1017-0405. 27(3), p. 951–981. doi: 10.5705/ss.202015.0364. Full text in Research Archive Show summary

  Should one rely on a parametric or nonparametric model when analysing a given data set? This classic question cannot be answered by traditional model selection criteria like AIC and BIC, since a nonparametric model has no likelihood. The purpose of the present paper is to develop a focused information criterion (FIC) for comparing general non-nested parametric models with a nonparametric alternative. It relies in part on the notion of a focus parameter, a population quantity of particular interest in the statistical analysis. The FIC compares and ranks candidate models based on estimated precision of the different model-based estimators for the focus parameter. It has earlier been developed for several classes of problems, but mainly involving parametric models. The new FIC, including also nonparametrics, is novel also in the mathematical context, being derived without the local neighbourhood asymptotics underlying previous versions of FIC. Certain average-weighted versions, called AFIC, allowing several focus parameters to be considered simultaneously, are also developed. We concentrate on the standard i.i.d. setting and certain direct extensions thereof, but also sketch further generalisations to other types of data. Theoretical and simulation-based results demonstrate desirable properties and satisfactory performance.

• Hjort, Nils Lid & Schweder, Tore (2017). Confidence distributions and related themes. Journal of Statistical Planning and Inference. ISSN 0378-3758. 195, p. 1–13. doi: 10.1016/j.jspi.2017.09.017. Full text in Research Archive
• Glad, Ingrid Kristine & Hjort, Nils Lid (2016). Model uncertainty first, not afterwards. Statistical Science. ISSN 0883-4237. 31(4), p. 490–494. doi: 10.1214/16-STS559. Full text in Research Archive Show summary

  Watson and Holmes propose ways of investigating robustness of statistical decisions by examining certain neighbourhoods around a posterior distribution. This may partly amount to ad hoc modelling of extra uncertainty. Instead of creating neighbourhoods around the posterior a posteriori, we argue that it might be more fruitful to model a layer of extra uncertainty first, in the model building process, and then allow the data to determine how big the resulting neighbourhoods ought to be. We develop and briefly illustrate a general strategy along such lines.

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• Schweder, Tore & Hjort, Nils Lid (2016). Confidence, Likelihood, Probability. Statistical Inference with Confidence Distributions. Cambridge University Press. ISBN 9780521861601. 511 p. Full text in Research Archive

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• Christensen, Dennis; Stoltenberg, Emil Aas & Hjort, Nils Lid (2024). Theory for adaptive designs in regression. arXiv.org. ISSN 2331-8422. doi: 10.48550/arXiv.2312.13387. Full text in Research Archive
• Hjort, Nils Lid & Cunen, Celine Marie Løken (2023). Independent Inspection, Confidence Conversion, Focused Fusion: Combining (Very) Different Information Sources. Full text in Research Archive Show summary

  Invited talk at the Nordic-Baltic Biometric Conference, Göteborg, June 2023. I presented the II-CC-FF paradigm, joint work with Celine Cunen.

• Hjort, Nils Lid (2023). Oljeeventyret -- i Norge og i Andeby. Donald Duck & Co. 9(Norsk Donald (bokserie)), p. 4–9. Full text in Research Archive Show summary

  "Oljeeventyret - i Norge og i Andeby", invitert forord i bind 9 i bokserien "Donald på norsk", utgitt av Egmont, 2023. Cristin har ikke pre-lagret denne bokserien, eller forlaget, så hun tillater meg ikke å registrere dette som populærvitenskapelig bidrag i en bok, og det blir irriterende nok stående, inntil videre, som bidrag i Donald Duck & Co., altså bladet, eller tidsskriftet.

• Hjort, Nils Lid (2023). How Many Were Killed in Guatemala, 1978 to 1996? Counting the Uncounted. Full text in Research Archive Show summary

  The task of estimating N000, the unseen individuals.

• Hjort, Nils Lid (2023). The Window of Change. Full text in Research Archive Show summary

  The Window of Change, talk at the ChangeTrend Workshop, organised via CAS and its programme Stability and Change, August 2022 to June 2023, led by Hegre and Hjort.

• Hjort, Nils Lid (2023). Ongoing Themes and Activities for the Stability and Change Project Group. Full text in Research Archive Show summary

  Håvard Hegre and Nils Lid Hjort presented ongoing themes and activities, to the general CAS audience, in their project Stability and Change.

• Hjort, Nils Lid (2023). How many were killed in Guatemala, 1978-1996? A FIC for N. Full text in Research Archive Show summary

  Talk in the section on model selection at NordStat 2023, Göteborg. The slides are available at the FocuStat site. The work relates to the Stability and Change programme at CAS, August 2022 to June 2023, led by Hjort and Hegre.

• Hjort, Nils Lid (2023). Does Polity Score Influence Size of War and Probability of Conflict Escalation? Full text in Research Archive Show summary

  From the DemoIndex Workshop, organised via the CAS programme Stability and Change, August 2022 to June 2023, led by Hjort and Hegre.

• Hjort, Nils Lid (2023). Bayes, Confidence Distributions, and US Pre-War Economy. Full text in Research Archive Show summary

  Invited talk at the NSF Oslo årsmøte, May 2023. I disagree with Sims, the winner of the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, regarding his methods and conclusions in his Nobel Prize acceptance lecture. We had pizza afterwards.

• Christensen, Dennis; Unneberg, Erik; Høyheim, Eirik; Jensen, Tomas Lunde & Hjort, Nils Lid (2023). Improved measurements of impact sensitivities of energetic materials. Full text in Research Archive Show summary

  Accurate estimation of impact sensitivity is crucial for safe production, handling, storage and transport of energetic materials. Indeed, molecular characteristics will affect sensitivity, and for solid materials, factors like particle size, lattice defects and morphology also play a role and make reproducibility difficult. As various synthesis and recrystallisation methods may lead to differences in crystal properties, it is important to determine the impact sensitivity whenever an energetic material is prepared. Of particular interest is the median ℎ50, namely the impact energy level at which there is a probability of 50% of an explosion occurring. This value has been shown to correlate with quantum chemical properties of the energetic material in question, providing insight into the underlying causes which govern sensitivity. However, in practical applications, it may be more important to estimate extreme values like the 99% quantile ℎ99. In addition to providing point estimates, we would like to derive confidence intervals to address their uncertainty. In this work, we cover the most common methods for constructing such confidence intervals (the delta method, Fieller’s theorem and the likelihood-ratio test) and compare their performance on sensitivity data via simulations. Our experiments indicate that Fieller’s theorem is the superior method, and we therefore use it to construct confidence intervals for ℎ50 and ℎ99 for cyclotetramethylene- tetranitramine (HMX), using new data. Based on our results, we formulate recommendations for researchers measuring sensitivities of synthesised molecules.

• Hjort, Nils Lid (2022). Prediction With Uncertainty. Full text in Research Archive
• Hjort, Nils Lid (2022). Why, when, and how to disobey your government's instructions and save 15,000 lives. Full text in Research Archive Show summary

  Impromptu talk at the impromptu excursion, during the CAS Workshop Prediction With Uncertainty, to the workshop participants, fifty metres away from the workshop room, to the White Buses, in front of Wanda Hjort Heger painting.

• Hjort, Nils Lid (2022). From Processes to Models: a few sets of ideas. Full text in Research Archive
• Hjort, Nils Lid (2022). Processes With Steadily Rarer but Steadily Bigger Shocks. Full text in Research Archive Show summary

  Invited talk at the Bo Lindqvist emeritus celebration meeting.

• Hjort, Nils Lid (2022). Topics in Spatial Statistics (with Henning as my boss), invited talk at the Henning Omre Emeritus celebration. Full text in Research Archive
• Hjort, Nils Lid & Stoltenberg, Emil Aas (2022). The partly parametric and partly nonparametric additive risk model. Full text in Research Archive Show summary

  Invited talk at the Ørnulf Borgan emeritus celebration seminar, Nils Lid Hjort presenting work based on joint work with Emil Stoltenberg.

• Røislien, Jo & Hjort, Nils Lid (2021). Abels kjeller - kortstokkproblemet. [Radio]. NRK Radio - Abels tårn. Full text in Research Archive Show summary

  Aslak fant ett kort på gata. Så et til. Hvor mange må han finne før han har en komplett kortstokk? Dette tilsynelatende enkle problemet satt fast panelets statistiker da han prøvde seg på det, seint på kvelden før sendinga 18. juni. Men tårnets enestående lyttere kasta seg over problemet og fant riktig svar. I denne kjellerepisoden graver vi oss ned i dypet av problemet, diskuterer analytiske vs. simulerte løsninger, og ender opp med å bestemme rekkefølgen av Platons sokratiske dialoger med statistisk metode. Panelet: Statistikerene Jo Røislien og Nils Lid Hjort Lytterpanelet: Andreas Hafver, Pål Kjeldsen og Tor Inge Birkenes Lønmo

• Cunen, Celine & Hjort, Nils Lid (2021). Combining information across diverse studies: the II-CC-FF paradigm. Full text in Research Archive
• Nygård, Håvard Mokleiv; Cunen, Celine Marie Løken & Hjort, Nils Lid (2019). Blir verden egentlig fredeligere? Full text in Research Archive
• Cunen, Celine & Hjort, Nils Lid (2019). Gamma process models for survival and competing risks. Full text in Research Archive
• Cunen, Celine Marie Løken & Hjort, Nils Lid (2018). II-CC-FF: Combination of information beyond meta-analysis. Full text in Research Archive
• Cunen, Celine Marie Løken & Hjort, Nils Lid (2018). Statistical Sightings of Better Angels. Full text in Research Archive
• Cunen, Celine Marie Løken & Hjort, Nils Lid (2018). Whales, politics, and statisticians. Full text in Research Archive
• Ørstavik, Iver; Hjelseth, Arve & Hjort, Nils Lid (2018). Å fjerne allround på skøyter er som å endre reglene i fotball for å legge opp til flere mål. Full text in Research Archive
• Hjort, Nils Lid (2018). Towards a More Peaceful World [insert `!' or `?' here]. Full text in Research Archive
• Cunen, Celine Marie Løken & Hjort, Nils Lid (2018). Statistical Sightings of Better Angels (invited seminar). Full text in Research Archive
• Hjort, Nils Lid (2018). One Thousand is Unfair, Two Thousand is Fair. Full text in Research Archive
• Hjort, Nils Lid (2018). Survival and event history models via Gamma processes (VVV conference). Full text in Research Archive
• Hjort, Nils Lid (2018). Welcoming remarks & introduction to the FocuStat VVV themes. Full text in Research Archive
• Hjort, Nils Lid (2018). Prediction and Confidence (NordStat2018). Full text in Research Archive
• Hjort, Nils Lid (2018). Olympic unfairness -- analysing 2 x 1000 m speedskating races. FocuStat Combo Kitchen. Full text in Research Archive
• Walker, Sam-Erik & Hjort, Nils Lid (2018). Focused model selection and inference using robust estimators. Full text in Research Archive
• Jullum, Martin & Hjort, Nils Lid (2018). Parametric or nonparametric, that’s the question. Full text in Research Archive
• Walker, Sam-Erik & Hjort, Nils Lid (2017). Focused model selection and inference using robust estimators. Full text in Research Archive
• Walker, Sam-Erik & Hjort, Nils Lid (2017). Estimation and model selection by data-driven weighted likelihoods. Full text in Research Archive
• Walker, Sam-Erik & Hjort, Nils Lid (2017). Estimation and model selection by data-driven weighted likelihoods. Full text in Research Archive
• Walker, Sam-Erik & Hjort, Nils Lid (2017). Estimation and model selection via maximum weighted likelihoods. Full text in Research Archive
• Cunen, Celine Marie Løken & Hjort, Nils Lid (2017). Combining diverse information sources with the II-CC-FF paradigm: Independent Inspection, Confidence Conversion, Focused Fusion. Full text in Research Archive
• Cunen, Celine Marie Løken; Walløe, Lars & Hjort, Nils Lid (2017). Decline in energy storage in Antarctic Minke whales during the JARPA period: Assessment via the Focused Information Criterion (FIC). Full text in Research Archive
• Cunen, Celine Marie Løken & Hjort, Nils Lid (2017). Confidence distributions: A potential bridge for combining parametric and nonparametric analyses. Full text in Research Archive
• Cunen, Celine Marie Løken & Hjort, Nils Lid (2017). Combining diverse information sources with the II-CC-FF paradigm. Full text in Research Archive
• Cunen, Celine Marie Løken & Hjort, Nils Lid (2017). Combining diverse information sources with the II-CC- FF paradigm. Full text in Research Archive
• Hellton, Kristoffer Herland & Hjort, Nils Lid (2017). Fridge: focused fine-tuning of ridge regression for personalized prediction. Full text in Research Archive
• Hjort, Nils Lid (2017). Empirical Likelihood and Bayesian Nonparametrics. Full text in Research Archive
• Hjort, Nils Lid (2017). Confidence distributions for change-points and regime shifts (again). Full text in Research Archive
• Hjort, Nils Lid (2017). Building Bridges (at Bislett): general workshop themes. Full text in Research Archive
• Hjort, Nils Lid (2017). Minimum Dispair, Maximum Despair. Full text in Research Archive
• Hjort, Nils Lid (2017). A gentle introduction to Bayesian Nonparametrics. Full text in Research Archive
• Hjort, Nils Lid (2017). The semifinals factor for skiing fast in the finals. Full text in Research Archive
• Hjort, Nils Lid (2017). Model selection and model averaging (two hours). Full text in Research Archive
• Hjort, Nils Lid (2017). The hybrid likelihood. Full text in Research Archive
• Hjort, Nils Lid (2017). Data fusion with confidence curves: The II-CC-FF paradigm. Full text in Research Archive
• Hjort, Nils Lid (2017). Bayesian nonparametrics (two hours). Full text in Research Archive
• Hjort, Nils Lid (2017). All models are wrong, but some are more biologically plausible than other: Survival analysis models based on crossing times for cumulative damage processes. Full text in Research Archive
• Hjort, Nils Lid (2017). Confidence distributions and data fusion (two hours). Full text in Research Archive
• Hjort, Nils Lid (2016). The world's first novel, skiings days near Oslo, Kola temperatures and the Hjort time series. Full text in Research Archive
• Hjort, Nils Lid (2016). Statistikk, sannsynlighetsteori, sjansespill, samfunn, solidaritet. Full text in Research Archive
• Hjort, Nils Lid (2016). Confidence distributions for change-points and regime shifts. Full text in Research Archive
• Hjort, Nils Lid (2016). Confidence, Likelihood, Probability; Ten-hour short course. Full text in Research Archive
• Hjort, Nils Lid (2016). Post-Selection Distributions, Model Averaging, Bagging, Inference. Full text in Research Archive
• Hjort, Nils Lid (2016). Combining Information Across Diverse Sources via Confidence Distributions: the II-CC-FF Paradigm. Full text in Research Archive
• Hjort, Nils Lid (2016). The hybrid likelihood: Combining parametrics and nonparametrics. Full text in Research Archive
• Hjort, Nils Lid (2016). The hybrid likelihood: Bridging parametrics and nonparametrics. Full text in Research Archive
• Hjort, Nils Lid (2016). Confidence distributions and objective Bayes. Full text in Research Archive
• Walker, Sam-Erik & Hjort, Nils Lid (2016). Model selection and focused inference using robust estimators. Full text in Research Archive
• Walker, Sam-Erik & Hjort, Nils Lid (2016). Model selection and focused inference using robust estimators. Full text in Research Archive
• Tveten, Martin; Glad, Ingrid Kristine & Hjort, Nils Lid (2021). Scalable change and anomaly detection in cross-correlated data. Universitetet i Oslo. Full text in Research Archive
• Nygård, Håvard Mokleiv; Hjort, Nils Lid & Haug, Jens Kristoffer (2019). Focused Model Selection for Markov Chain Models, With an Application to Armed Conflict Data. Universitetet i Oslo. Full text in Research Archive
• Cunen, Celine; Walløe, Lars & Hjort, Nils Lid (2017). Decline in energy in Antarctic minke whales during the JARPA period. Assessment via the Focused Information Criterion (FIC). IWC_SC/67A/EM04. International Whaling Commission. Full text in Research Archive

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