arXiv subjects

Emil Aas Stoltenberg

Publications and source records attributed to Emil Aas Stoltenberg.

6 recordsLinked to original sources

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↗

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.

math.ST↗

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 $π$.

math.PR↗

arXiv2022

Regression discontinuity design with right-censored survival data

In this paper the regression discontinuity design is adapted to the survival analysis setting with right-censored data, studied in an intensity based counting process framework. In particular, a local polynomial regression version of the Aalen additive hazards estimator is introduced as an estimator of the difference between two covariate dependent cumulative hazard rate functions. Large-sample theory for this estimator is developed, including confidence intervals that take into account the uncertainty associated with bias correction. As is standard in the causality literature, the models and the theory are embedded in the potential outcomes framework. Two general results concerning potential outcomes and the multiplicative hazards model for survival data are presented.

stat.ME↗

arXiv2020

The standard cure model with a linear hazard

In this paper we introduce a mixture cure model with a linear hazard rate regression model for the event times. Cure models are statistical models for event times that take into account that a fraction of the population might never experience the event of interest, this fraction is said to be {`}cured{'}. The population survival function in a mixture cure model takes the form $S(t) = 1 - π+ π\exp(-\int_0^tα(s)\,d s)$, where $π$ is the probability of being susceptible to the event under study, and $α(s)$ is the hazard rate of the susceptible fraction. We let both $π$ and $α(s)$ depend on possibly different covariate vectors $X$ and $Z$. The probability $π$ is taken to be the logistic function $π(X^{\prime}γ) = 1/\{1+\exp(-X^{\prime}γ)\}$, while we model $α(s)$ by Aalen's linear hazard rate regression model. This model postulates that a susceptible individual has hazard rate function $α(t;Z) = β_0(t) + β_1(t)Z_1 + \cdots + Z_{q-1}β_{q-1}(t)$ in terms of her covariate values $Z_1,\ldots,Z_{q-1}$. The large-sample properties of our estimators are studied by way of parametric models that tend to a semiparametric model as a parameter $K \to \infty$. For each model in the sequence of parametric models, we assume that the data generating mechanism is parametric, thus simplifying the derivation of the estimators, as well as the proofs of consistency and limiting normality. Finally, we use contiguity techniques to switch back to assuming that the data stem from the semiparametric model. This technique for deriving and studying estimators in non- and semiparametric settings has previously been studied and employed in the high-frequency data literature, but seems to be novel in survival analysis.

math.ST↗

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.

math.ST↗