Nonparametric estimation of a convex bathtub-shaped hazard function

In this paper, we study the nonparametric maximum likelihood estimator (MLE) of a convex hazard function. We show that the MLE is consistent and converges at a local rate of $n^{2/5}$ at points $x_0$ where the true hazard function is positive and strictly convex. Moreover, we establish the pointwise asymptotic distribution theory of our estimator under these same assumptions. One notable feature of the nonparametric MLE studied here is that no arbitrary choice of tuning parameter (or complicated data-adaptive selection of the tuning parameter) is required.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ202 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The Likelihood of Mixed Hitting Times

We present a method for computing the likelihood of a mixed hitting-time model that specifies durations as the first time a latent L\'evy process crosses a heterogeneous threshold. This likelihood is not generally known in closed form, but its Laplace transform is. Our approach to its computation relies on numerical methods for inverting Laplace transforms that exploit special properties of the first passage times of L\'evy processes. We use our method to implement a maximum likelihood estimator of the mixed hitting-time model in MATLAB. We illustrate the application of this estimator with an analysis of Kennan's (1985) strike data.Comment: 35 page

A mixed model approach for structured hazard regression

The classical Cox proportional hazards model is a benchmark approach to analyze continuous survival times in the presence of covariate information. In a number of applications, there is a need to relax one or more of its inherent assumptions, such as linearity of the predictor or the proportional hazards property. Also, one is often interested in jointly estimating the baseline hazard together with covariate effects or one may wish to add a spatial component for spatially correlated survival data. We propose an extended Cox model, where the (log-)baseline hazard is weakly parameterized using penalized splines and the usual linear predictor is replaced by a structured additive predictor incorporating nonlinear effects of continuous covariates and further time scales, spatial effects, frailty components, and more complex interactions. Inclusion of time-varying coefficients leads to models that relax the proportional hazards assumption. Nonlinear and time-varying effects are modelled through penalized splines, and spatial components are treated as correlated random effects following either a Markov random field or a stationary Gaussian random field. All model components, including smoothing parameters, are specified within a unified framework and are estimated simultaneously based on mixed model methodology. The estimation procedure for such general mixed hazard regression models is derived using penalized likelihood for regression coefficients and (approximate) marginal likelihood for smoothing parameters. Performance of the proposed method is studied through simulation and an application to leukemia survival data in Northwest England

Variable Selection for Nonparametric Gaussian Process Priors: Models and Computational Strategies

This paper presents a unified treatment of Gaussian process models that extends to data from the exponential dispersion family and to survival data. Our specific interest is in the analysis of data sets with predictors that have an a priori unknown form of possibly nonlinear associations to the response. The modeling approach we describe incorporates Gaussian processes in a generalized linear model framework to obtain a class of nonparametric regression models where the covariance matrix depends on the predictors. We consider, in particular, continuous, categorical and count responses. We also look into models that account for survival outcomes. We explore alternative covariance formulations for the Gaussian process prior and demonstrate the flexibility of the construction. Next, we focus on the important problem of selecting variables from the set of possible predictors and describe a general framework that employs mixture priors. We compare alternative MCMC strategies for posterior inference and achieve a computationally efficient and practical approach. We demonstrate performances on simulated and benchmark data sets.Comment: Published in at http://dx.doi.org/10.1214/11-STS354 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A simple GMM estimator for the semi-parametric mixed proportional hazard model

Ridder and Woutersen (2003) have shown that under a weak condition on the baseline hazard, there exist root-N consistent estimators of the parameters in a semiparametric Mixed Proportional Hazard model with a parametric baseline hazard and unspeci�ed distribution of the unobserved heterogeneity. We extend the Linear Rank Estimator (LRE) of Tsiatis (1990) and Robins and Tsiatis (1991) to this class of models. The optimal LRE is a two-step estimator. We propose a simple one-step estimator that is close to optimal if there is no unobserved heterogeneity. The e¢ ciency gain associated with the optimal LRE increases with the degree of unobserved heterogeneity.

Bayesian Regularisation in Structured Additive Regression Models for Survival Data

During recent years, penalized likelihood approaches have attracted a lot of interest both in the area of semiparametric regression and for the regularization of high-dimensional regression models. In this paper, we introduce a Bayesian formulation that allows to combine both aspects into a joint regression model with a focus on hazard regression for survival times. While Bayesian penalized splines form the basis for estimating nonparametric and flexible time-varying effects, regularization of high-dimensional covariate vectors is based on scale mixture of normals priors. This class of priors allows to keep a (conditional) Gaussian prior for regression coefficients on the predictor stage of the model but introduces suitable mixture distributions for the Gaussian variance to achieve regularization. This scale mixture property allows to device general and adaptive Markov chain Monte Carlo simulation algorithms for fitting a variety of hazard regression models. In particular, unifying algorithms based on iteratively weighted least squares proposals can be employed both for regularization and penalized semiparametric function estimation. Since sampling based estimates do no longer have the variable selection property well-known for the Lasso in frequentist analyses, we additionally consider spike and slab priors that introduce a further mixing stage that allows to separate between influential and redundant parameters. We demonstrate the different shrinkage properties with three simulation settings and apply the methods to the PBC Liver dataset

Limit distribution theory for maximum likelihood estimation of a log-concave density

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $\varphi_0=\log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.Comment: Published in at http://dx.doi.org/10.1214/08-AOS609 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Simple GMM Estimator for the Semi-Parametric Mixed Proportional Hazard Model

Ridder and Woutersen (2003) have shown that under a weak condition on the baseline hazard there exist root-N consistent estimators of the parameters in a semiparametric Mixed Proportional Hazard model with a parametric baseline hazard and unspecified distribution of the unobserved heterogeneity. We extend the Linear Rank Estimator (LRE) of Tsiatis (1990) and Robins and Tsiatis (1991) to this class of models. The optimal LRE is a two-step estimator. We propose a simple first-step estimator that is close to optimal if there is no unobserved heterogeneity. The efficiency gain associated with the optimal LRE increases with the degree of unobserved heterogeneity.mixed proportional hazard, linear rank estimation, counting process