## 1 Introduction

Survival analysis is a branch of statistics which deals with the analysis of time-to-event data (or more generally of event history). Applications of event history analysis are numerous in the medical field, but are also found in economics, engineering and sociology. Usually the investigator collects data on the occurrence times of the outcome variable along with a set of predictors (covariates) such as gender, age, social status, biomarkers of diseases and other variables of a similar nature. The investigator then attempts to determine if and how such covariates influence the occurrence rates (intensity) of the event of interest. Cox's proportional hazards (PH) model is one of the earliest and perhaps the most widely used statistical model which attempts to address such questions. Cox's original model was re-formulated in terms of counting process theory by Andersen & Gill (1982). The re-formulation led to the multiplicative intensity (MI) model which extends the PH model in the sense that it allows multiple events and time varying covariate processes. The PH model and its variants assume that the intensity function of the counting process defined by the events of interest is made up of the product of a baseline nonparametric intensity function and a parametric part consisting of a function of a linear combination of the independent variables. The effects of the covariates are measured through the unknown coefficients of the linear combination (regression parameters) which do not depend on time. This implies that the hazard ratio of two individuals differing just by the level of a given covariate is constant over time. This property which is known as the proportional hazards assumption yields an attractive interpretation of the model in terms of risk ratio and is mathematically tractable. However such an assumption is sometimes not appropriate for the data at hand. In such situations, time varying regression coefficients are required in order to quantify the effect of the covariates on the intensity function. An alternative to the MI model is provided by the additive regression model proposed by Aalen (1980). In this model the intensity function is governed by the covariates as well as the past events through a linear regression with time-varying coefficients. Estimation of Aalen's nonparametric time-varying regression coefficients is performed via weighted least squares and the asymptotic properties of these coefficients are studied using martingale theory for counting processes (Martinussen & Scheike 2006).

Often there is a large number of potential covariates that could be included in such regression models. However it may turn out that only a handful of such covariates is relevant in explaining the outcome of interest. Investigators usually employ either intuitive judgement or a model selection mechanism in order to filter out most of the unimportant covariates and obtain a parsimonious final model. Despite their appeal model selection methods have the disadvantage of introducing bias due to the fact that the descarded covariates may not be completely irrelevant. Therefore, the question of whether to settle for a reduced (uncertain) model or a full (possibly inefficient) model remains open. A way out of this dilemma is to construct James-Stein-type shrinkage estimators which incorporate both models into the estimation process (Saleh 2006). In the classical linear and partially linear regression models and in censored data models, the shrinkage estimators are known to dominate the unrestricted estimators (based on the full model) over the whole parameter space and dominate the restricted estimators (based on the linear hypothesis) except in a small neighborhood of the linear restriction (Ahmed, Doksum, Hossain and You, 2007; Raheem, Ahmed and Doksum, 2012).

In this manuscript we propose James-Stein-Type shrinkage estimators for the nonparametric regression coefficients in Aalen's additive model under a general linear hypothesis about the coefficients.

The manuscript is organized as follows. In Section 'The proposed methodology', we introduce Aalen's additive model, define a general linear hypothesis to be satisfied by the regression coefficients and provide restricted estimators of the coefficients. We study the joint asymptotic normality of the restricted and unrestricted estimators. We then define James-Stein-type shrinkage estimators of the coefficients under the prior uncertain information given in the form of the linear hypothesis. We define and study the integrated distributional quadratic risks of the proposed shrinkage estimators and compare them asymptotically to those of the restricted and unrestricted estimators. In Section 'Empirical Studies', we conduct Monte Carlo simulations examining the small sample performance of the estimators. Furthermore, we compare the performance of the proposed estimators to a recently devised least absolute shrinkage and selection operator (LASSO) estimator as well as to a ridge-type estimator both via simulations and via analysis of data on the survival of primary billiary cirhosis patients.