arXiv subjects

Nils Lid Hjort

Publications and source records attributed to Nils Lid Hjort.

At least 55 records · Page 3Linked to original sources

arXiv2026

Second order asymptotics for the number of times an estimator is more than epsilon from its target value

Suppose $\{\widehatθ_n\colon n\ge1\}$ is a strongly consistent sequence of estimators for a parameter $θ$, where $\widehatθ_n$ is based on the first $n$ observations. Consider $Q_\varepsilon$, the number of times $|\widehatθ_n-θ|\ge\varepsilon$. In another paper (Hjort and Fenstad, 1992) we have shown that $\varepsilon^2 Q_\varepsilon$ has a limit distribution as $\varepsilon\rightarrow0$, depending only on $σ$, the standard deviation of the limit distribution for $\sqrt{n}(\widehatθ_n-θ)$, under natural regularity conditions. The present paper investigates some second order asymptotics for differences between $Q_\varepsilon$ variables. The limit of ${\rm E}(Q_{1,\varepsilon}-Q_{2,\varepsilon})$ is calculated in cases where ${\rm E} Q_{1,\varepsilon}/{\rm E} Q_{2,\varepsilon}$ goes to 1, leading to a notion of `asymptotic relative deficiency' in cases where the asymptotic relative efficiency is 1. This is used to distinguish between competing estimators with identical limit distributions. Thus using denominator $n-{1\over3}$ in the familiar formula for estimating a normal variance is better than both $n$ and $n-1$ and indeed all other choices, for example, in the sense of leading to the smallest possible expected number of $\varepsilon$ errors. Results of this type are found in a selection of familiar estimation problems, using limit results for expected differences, and are compared to corresponding asymptotic relative deficiency analysis in the sense of Hodges and Lehmann. Some second order distributional results are reached as well. It is shown how $\varepsilon$ times a $Q_\varepsilon$-difference tends to a variable which is related to some exponential distributions associated with Brownian motion, and that have recently been investigated by Hjort and Khasminskii (1993).

math.ST↗

arXiv2026

On the last time and the number of times an estimator is more than epsilon from its target value

Suppose $\widehatθ_n$ is a strongly consistent estimator for $θ_0$ in some i.i.d. situation. Let $N_\varepsilon$ and $Q_\varepsilon$ be respectively the last $n$ and the total number of $n$ for which $\widehatθ_n$ is at least $\varepsilon$ away from $θ_0$. The limit distributions for ${\varepsilon}^2 N_\varepsilon$ and ${\varepsilon}^2 Q_\varepsilon$ as $\varepsilon$ goes to zero are obtained under natural and weak conditions. The theory covers both parametric and nonparametric cases, multi-dimensional parameters, and general distance functions. Our results are of probabilistic interest, and, on the statistical side, suggest ways in which competing estimators can be compared. In particular several new optimality properties for the maximum likelihood estimator sequence in parametric families are established. Another use of our results is ways of constructing sequential fixed-volume or shrinking-volume confidence sets, as well as sequential tests with power 1. The paper also includes limit distribution results for the last $n$ and the number of $n$ for which the supremum distance $\|F_n-F\|\ge\varepsilon$, where $F_n$ is the empirical distribution function. Yet other results are reached for $\varepsilon^{5/2} N_\varepsilon$ and $\varepsilon^{5/2} Q_\varepsilon$ in the context of nonparametric density estimation, referring to the last time and the number of times where $|f_n(x) f(x)|\ge\varepsilon$. Finally it is shown that our results extend to several non-i.i.d. situations.

math.ST↗

arXiv2026

Philosophical foundations of statistics

The philosophical foundations of statistics involve issues in theoretical statistics, such as goals and methods to meet these goals, and interpretation of the meaning of inference using statistics. They are related to the philosophy of science and to the philosophy of probability. We review the core and partly interrelated themes and place them in context.

stat.OT↗

arXiv2026

The partly parametric and partly nonparametric additive risk model

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 $h(s)=z_1α_1(s)+\cdots+z_rα_r(s)$ in terms of his covariate values $z_1,\ldots,z_r$. These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions $α_j(s)$ 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.

stat.ME↗

arXiv2026

Focused Information Criteria for Semiparametric Linear Hazard Regression

The semiparametric linear hazard regression model introduced by McKeague and Sasieni (1994) is an extension of the linear hazard regression model developed by Aalen (1980). Methods of model selection for this type of model are still underdeveloped. In the process of fitting a semiparametric linear hazard regression model one usually starts with a given set of covariates. For each covariate one has at least the following three choices: allow it to have time-varying effect; allow it to have constant effect over time; or exclude it from the model. In this paper we discuss focused information criteria (FIC) to help with this choice. In the spirit of Claeskens and Hjort (2003, 2008), `focused' means that one is interested in one specific quantity, e.g. the probability of survival of a patient with a certain set of covariates up to a given time. The FIC involves estimating the mean squared error of the estimator of the quantity one is interested in, and the chosen model is the one minimising this estimated mean squared error. The focused model selection machinery is extended to allow for weighted versions, leading to a suitable wFIC method that aims at finding models that lead to good estimates of a given list of parameters, such as survival probabilities for a subset of patients or for a specified region of covariate vectors. In addition to developing model selection criteria, methods associated with averaging across the best models are also discussed. We illustrate these methods of model selection in a real data situation.

stat.ME↗

arXiv2026

Model Selection via Focused Information Criteria for Complex Data in Ecology and Evolution

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.

stat.AP↗

arXiv2026

Rejoinder to the discussants of the two JASA articles `Frequentist Model Averaging' and `The Focused Information Criterion', by Nils Lid Hjort and Gerda Claeskens

We are honoured to have our work read and discussed at such a thorough level by several experts. Words of appreciation and encouragement are gratefully received, while the many supplementary comments, thoughtful reminders, new perspectives and additional themes raised are warmly welcomed and deeply appreciated. Our thanks go also to JASA Editor Francisco Samaniego and his editorial helpers for organising this discussion. Space does not allow us answering all of the many worthwhile points raised by our discussants, but in the following we make an attempt to respond to what we perceive of as being the major issues. Our responses are organised by themes rather than by discussants. We shall refer to our two articles as `the FMA paper' (Hjort and Claeskens) and `the FIC paper' (Claeskens and Hjort).

stat.ME↗

arXiv2026

On the time a diffusion process spends along a line

For an arbitrary diffusion process $X$ with time-homogeneous drift and variance parameters $μ(x)$ and $σ^2(x)$, let $V_\varepsilon$ be $1/\varepsilon$ times the total time $X(t)$ spends in the strip $[a+bt-(1/2)\varepsilon,a+bt+(1/2)\varepsilon]$.The limit $V$ as $\varepsilon\rightarrow0$ is the full halfline version of the local time of $X(t)-a-bt$ at zero, and can be thought of as the time $X$ spends along the straight line $x=a+bt$. We prove that $V$ is either infinite with probability 1 or distributed as a mixture of an exponential and a unit point mass at zero, and we give formulae for the parameters of this distribution in terms of $μ(\cdot)$, $σ(\cdot)$, $a$, $b$, and the starting point $X(0)$. The special case ofa Brownian motion is studied in more detail, leading in particular to a full process $V(b)$ with continuous sample paths and exponentially distributed marginals. This construction leads to new families of bivariate and multivariate exponential distributions. Truncated versions of such `total relative time' variables are also studied. A relation is pointed out to a second order asymptotics problem in statistical estimation theory, recently investigated in Hjort and Fenstad (1992a, 1992b).

math.PR↗

arXiv2026

A New Look at the Visual Performance of Nonparametric Hazard Rate Estimators

Nonparametric curve estimation by kernel methods has attracted widespread interest in theoretical and applied statistics. One area of conflict between theory and application relates to the evaluation of the performance of the estimators. Recently, Marron and Tsybakov (1995) proposed {\it visual error criteria} for addressing this issue of controversy in density estimation. Their core idea consists in using integrated alternatives to the Hausdorff distance for measuring the closeness of two sets based onthe Euclidean distance. In this paper, we transfer these ideas to hazard rate estimation from censored data. We are able to derive similar results that help to understand when the application of the new criteria will lead to answers that differ from those given by the conventional approaches.

math.ST↗

arXiv2026

The Best Metal-Grabbing Games Ever: How a Tiny Nation Won the Most Medals (By Far)

For three Winter Olympics in a row, tiny nation Norway has out-medalled everyone else, in 2026 winning 18 golds, 12 silvers, 11 bronzes, i.e.~41 medals, compared to e.g.~12 + 12 + 9 = 33 for the USA, 10 + 6 + 14 = 30 for home team Italy, 8 + 10 + 7 = 26 for powerhouse Germany, etc. Never before have we [pluralis proudiensis] or anyone else won as many as 41 medals at a Winter Olympics. But how impressive is this, really, when we factor in that the number of events has increased so drastically?

stat.AP↗

arXiv2026

On the errors committed by sequences of estimator functionals

Consider a sequence of estimators $\hat θ_n$ which converges almost surely to $θ_0$ as the sample size $n$ tends to infinity. Under weak smoothness conditions, we identify the asymptotic limit of the last time $\hat θ_n$ is further than $\eps$ away from $θ_0$ when $\eps \rightarrow 0^+$. These limits lead to the construction of sequentially fixed width confidence regions for which we find analytic approximations. The smoothness conditions we impose is that $\hat θ_n$ is to be close to a Hadamard-differentiable functional of the empirical distribution, an assumption valid for a large class of widely used statistical estimators. Similar results were derived in Hjort and Fenstad (1992, Annals of Statistics) for the case of Euclidean parameter spaces; part of the present contribution is to lift these results to situations involving parameter functionals. The apparatus we develop is also used to derive appropriate limit distributions of other quantities related to the far tail of an almost surely convergent sequence of estimators, like the number of times the estimator is more than $\eps$ away from its target. We illustrate our results by giving a new sequential simultaneous confidence set for the cumulative hazard function based on the Nelson--Aalen estimator and investigate a problem in stochastic programming related to computational complexity.

math.ST↗

arXiv2026

Estimation, inference and model selection for jump regression models

We consider regression models with data of the type $y_i=m(x_i)+\varepsilon_i$, where the $m(x)$ curve is taken locally constant, with unknown levels and jump points. We investigate the large-sample properties of the minimum least squares estimators, finding in particular that jump point parameters and level parameters are estimated with respectively $n$-rate precision and $\sqrt{n}$-rate precision, where $n$ is sample size. Bayes solutions are investigated as well and found to be superior. We then construct jump information criteria, respectively AJIC and BJIC, for selecting the right number of jump points from data. This is done by following the line of arguments that lead to the Akaike and Bayesian information criteria AIC and BIC, but which here lead to different formulae due to the different type of large-sample approximations involved.

stat.ME↗

arXiv2026

Confidence in confidence distributions!

The recent article `Satellite conjunction analysis and the false confidence theorem' (Balch, Martin, and Ferson, 2019, Proceedings of the Royal Society, Series A) 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 setup where exact analysis is attainable. We also show that a different and frequentist method, involving confidence distributions, is free of the false confidence syndrome.

math.ST↗

arXiv2026

Local Bayesian Regression

This paper develops a class of Bayesian non- and semiparametric methods for estimating regression curves and surfaces. The main idea is to model the regression as locally linear, and then place suitable local priors on the local parameters. The method requires the posterior distribution of the local parameters given local data, and this is found via a suitably defined local likelihood function. When the width of the local data window is large the methods reduce to familiar fully parametric Bayesian methods, and when the width is small the estimators are essentially nonparametric. When noninformative reference priors are used the resulting estimators coincide with recently developed well-performing local weighted least squares methods for nonparametric regression. Each local prior distribution needs in general a centre parameter and a variance parameter. Of particular interest are versions of the scheme that are more or less automatic and objective in the sense that they do not require subjective specifications of prior parameters. We therefore develop empirical Bayes methods to obtain the variance parameter and a hierarchical Bayes method to account for uncertainty in the choice of centre parameter. There are several possible versions of the general programme, and a number of its specialisations are discussed. Some of these are shown to be capable of outperforming standard nonparametric regression methods, particularly in situations with several covariates.

stat.ME↗

arXiv2026

Upper Bounds for the I-MSE and max-MSE of Kernel Density Estimators

The performance of kernel density estimators is usually studied via Taylor expansions and asymptotic approximation arguments, in which the bandwidth parameter tends to zero with increasing sample size. In contrast, this paper focusses directly on the finite-sample situation. Informative upper bounds are derived both for the integrated and the maximal mean squared error function. Results are reached for the traditional case, where the kernel is a probability density function, under various sets of assumptions on the underlying density to be estimated. Results are also derived for the important non-conventional case of the sinc kernel, which is not integrable and also takes negative values. We pin-point ways in which the sinc-based estimator performs better than the conventional kernel estimators. When proving our results we rely on methods related to characteristic and empirical characteristic functions.

math.ST↗

arXiv2026

Confidence Distributions and Related Themes

This is the guest editors' general introduction to a Special Issue of the Journal of Statistical Planning and Inference, dedicated to confidence distributions and related themes. Confidence distributions (CDs) are distributions for parameters of interest, constructed via a statistical model after analysing the data. As such they serve the same purpose for the frequentist statisticians as the posterior distributions for the Bayesians. There have been several attempts in the literature to put up a clear theory for such confidence distributions, from Fisher's fiducial inference and onwards. There are certain obstacles and difficulties involved in these attempts, both conceptually and operationally, which have contributed to the CDs being slow in entering statistical mainstream. Recently there is a renewed surge of interest in CDs and various related themes, however, reflected in both series of new methodological research, advanced applications to substantive sciences, and dissemination and communication via workshops and conferences. The present special issue of the JSPI is a collection of papers emanating from the {\it Inference With Confidence} workshop in Oslo, May 2015. Several of the papers appearing here were first presented at that workshop. The present collection includes however also new research papers from other scholars in the field.

stat.ME↗

arXiv2026

Combining Information Across Diverse Sources: The II-CC-FF Paradigm

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 $ψ_1,\ldots,ψ_k$ from data sources $1,\ldots,k$. 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 $C_j(ψ_j)$ for the relevant focus parameter $ψ_j$ associated with data source $j$. Second, Confidence Conversion (CC) techniques are used to translate the confidence distributions to confidence log-likelihood functions, say $\ell_{{\rm con},j}(ψ_j)$. Finally, the Focused Fusion (FF) step uses relevant and context-driven techniques to construct a confidence distribution for the primary focus parameter $ϕ=ϕ(ψ_1,\ldots,ψ_k)$, 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.

stat.ME↗

arXiv2026

On Expectation Propagation and the Probabilistic Editor in some simple mixture problems

As for other latent-variable problems, exact Bayesian analysis is typically not practicable for mixture problems and approximate methods have been developed. Variational Bayes tends to produce approximate posterior distributions for parameters that are too tightly concentrated in having variances that are too small. The paper identifies a few mixture problems in which Expectation Propagation and variations thereof lead to approximate posterior distributions that asymptotically exhibit `correct' variances and therefore stand to provide reliable interval estimates for the unknown parameter or parameters.

math.ST↗