Emil Aas Stoltenberg

• 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.

• 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.

• Stoltenberg, Emil Aas; Mykland, Per A. & Zhang, Lan (2022). A CLT FOR SECOND DIFFERENCE ESTIMATORS WITH AN APPLICATION TO VOLATILITY AND INTENSITY. Annals of Statistics. ISSN 0090-5364. 50(4), p. 2072–2095. doi: 10.1214/22-AOS2176. Full text in Research Archive Show summary

  In this paper, we introduce a general method for estimating the quadratic covariation of one or more spot parameter processes associated with continuous time semimartingales, and present a central limit theorem that has this class of estimators as one of its applications. The class of estimators we introduce, that we call Two-Scales Quadratic Covariation ( TSQC ) estimators, is based on sums of increments of second differences of the observed processes, and the intervals over which the differences are computed are rolling and overlapping. This latter feature lets us take full advantage of the data, and, by sufficiency considerations, ought to outperform estimators that are based on only one partition of the observational window. Moreover, a two-scales approach is employed to deal with asymptotic bias terms in a systematic manner, thus automatically giving consistent estimators without having to work out the form of the bias term on a case-to-case basis. We highlight the versatility of our central limit theorem by applying it to a novel leverage effect estimator that does not belong to the class of TSQC estimators. The principal empirical motivation for the present study is that the discrete times at which a continuous time semimartingale is observed might depend on features of the observable process other than its level, such as its spot-volatility process. As an application of the TSQC estimators, we therefore show how it may be used to estimate the quadratic covariation between the spot-volatility process and the intensity process of the observation times, when both of these are taken to be semimartingales. The finite sample properties of this estimator are studied by way of a simulation experiment, and we also apply this estimator in an empirical analysis of the Apple stock. Our analysis of the Apple stock indicates a rather strong correlation between the spot volatility process of the log-prices process and the times at which this stock is traded and hence observed.

• 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.

• Stoltenberg, Emil Aas; Nordeng, Hedvig Marie Egeland; Ystrøm, Eivind & Samuelsen, Sven Ove (2020). The cure model in perinatal epidemiology. Statistical Methods in Medical Research. ISSN 0962-2802. 29(10), p. 2783–2794. doi: 10.1177/0962280220904092. Full text in Research Archive Show summary

  In the statistical literature, the class of survival analysis models known as cure models has received much attention in recent years. Cure models seem not, however, to be part of the statistical toolbox of perinatal epidemiologists. In this paper, we demonstrate that in perinatal epidemiological studies where one investigates the relation between a gestational exposure and a condition that can only be ascertained after several years, cure models may provide the correct statistical framework. The reason for this is that the hypotheses being tested often concern an unobservable outcome that, in view of the hypothesis, should be thought of as occurring at birth, even though it is only detectable much later in life. The outcome of interest can therefore be viewed as a censored binary variable. We illustrate our argument with a simple cure model analysis of the possible relation between gestational exposure to paracetamol and attention-deficit hyperactivity disorder, using data from the Norwegian Mother, Father and Child Cohort Study conducted by the Norwegian Institute of Public Health, and information about the attention-deficit hyperactivity disorder diagnoses obtained from the Norwegian Patient Registry.

• 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.

• 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.

View all works in NVA

• 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 & 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.

• Stoltenberg, Emil Aas & Hjort, Nils Lid (2018). Models and inference for on-off data via clipped Ornstein–Uhlenbeck processes. Full text in Research Archive

View all works in NVA