Kernel Estimation of Rate Function for Recurrent Event Data

Abstract.  Recurrent event data are largely characterized by the rate function but smoothing techniques for estimating the rate function have never been rigorously developed or studied in statistical literature. This paper considers the moment and least squares methods for estimating the rate function from recurrent event data. With an independent censoring assumption on the recurrent event process, we study statistical properties of the proposed estimators and propose bootstrap procedures for the bandwidth selection and for the approximation of confidence intervals in the estimation of the occurrence rate function. It is identified that the moment method without resmoothing via a smaller bandwidth will produce a curve with nicks occurring at the censoring times, whereas there is no such problem with the least squares method. Furthermore, the asymptotic variance of the least squares estimator is shown to be smaller under regularity conditions. However, in the implementation of the bootstrap procedures, the moment method is computationally more efficient than the least squares method because the former approach uses condensed bootstrap data. The performance of the proposed procedures is studied through Monte Carlo simulations and an epidemiological example on intravenous drug users.     

Deconvolution with supersmooth distributions

Nonparametric deconvolution problems require one to recover an unknown density when the data are contaminated with errors. Optimal global rates of convergence are found under the weighted L_p-loss (1 ≤ p ≤ ∞). It appears that the optimal rates of convergence are extremely low for supersmooth error distributions. To resolve this difficulty, we examine how high the noise level can be for deconvolution to be feasible, and for the deconvolution estimate to be as good as the ordinary density estimate. It is shown that if the noise level is not too high, nonparametric Gaussian deconvolution can still be practical. Several simulation studies are also presented.     

On the asymptotic normality of the L1- and L2-errors in histogram density estimation

The L₁ and L₂-errors of the histogram estimate of a density f from a sample X₁,X₂,…,Xn using a cubic partition are shown to be asymptotically normal without any unnecessary conditions imposed on the density f. The asymptotic variances are shown to depend on f only through the corresponding norm of f. From this follows the asymptotic null distribution of a goodness-of-fit test based on the total variation distance, introduced by Györfi and van der Meulen (1991). This note uses the idea of partial inversion for obtaining characteristic functions of conditional distributions, which goes back at least to Bartlett (1938).     

Model diagnostics for smoothing spline ANOVA models

The author proposes some simple diagnostics for assessing the necessity of selected terms in smoothing spline ANOVA models. The elimination of practically insignificant terms generally enhances the interpretability of the estimates and sometimes may also have inferential implications. The diagnostics are derived from Kullback‐Leibler geometry and are illustrated in the settings of regression, probability density estimation, and hazard rate estimation.     

Loss functions for estimation of extrema with an application to disease mapping

It is often of interest to find the maximum or near maxima among a set of vector‐valued parameters in a statistical model; in the case of disease mapping, for example, these correspond to relative‐risk “hotspots” where public‐health intervention may be needed. The general problem is one of estimating nonlinear functions of the ensemble of relative risks, but biased estimates result if posterior means are simply substituted into these nonlinear functions. The authors obtain better estimates of extrema from a new, weighted ranks squared error loss function. The derivation of these Bayes estimators assumes a hidden‐Markov random‐field model for relative risks, and their behaviour is illustrated with real and simulated data.     

Confidence Intervals for Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

Analysis of Competing Risks by Using Bayesian Smoothing

We consider the competing risks set-up. In many practical situations, the conditional probability of the cause of failure given the failure time is of direct interest. We propose to model the competing risks by the overall hazard rate and the conditional probabilities rather than the cause-specific hazards. We adopt a Bayesian smoothing approach for both quantities of interest. Illustrations are given at the end.     

PATTERN RECOGNITION VIA ROBUST SMOOTHING WITH APPLICATION TO LASER DATA

Nowadays airborne laser scanning is used in many territorial studies, providing point data which may contain strong discontinuities. Motivated by the need to interpolate such data and preserve their edges, this paper considers robust nonparametric smoothers. These estimators, when implemented with bounded loss functions, have suitable jump‐preserving properties. Iterative algorithms are developed here, and are equivalent to nonlinear M‐smoothers, but have the advantage of resembling the linear Kernel regression. The selection of their coefficients is carried out by combining cross‐validation and robust‐tuning techniques. Two real case studies and a simulation experiment confirm the validity of the method; in particular, the performance in building recognition is excellent.     

Assessing Effects on Long-term Survival after Early Termination of Randomized Trials

In this paper we present an approach to using historical control data to augment information from a randomized controlled clinical trial, when it is not possible to continue the control regimen to obtain the most reliable and valid assessment of long term treatment effects. Using an adjustment procedure to the historical control data, we investigate a method of estimating the long term survival function for the clinical trial control group and for evaluating the long term treatment effect. The suggested method is simple to interpret, and particularly motivated in clinical trial settings when ethical considerations preclude the long term follow-up of placebo controls. A simulation study reveals that the bias in parameter estimates that arises in the setting of group sequential monitoring will be attenuated when long term historical control information is used in the proposed manner. Data from the first and second National Wilms' Tumor studies are used to illustrate the method.