Bibliographical Note

In England in the Middle Ages, inheritance data were recorded of tenants who owned land from the Crown. Male adult mortality is estimated from these data. A tenant was allowed to sell his land. Only if he still owned land at death, his age at death was observed; so death was right censored by sell of all the land. The censoring times are not observed because sell of land was never recorded. This makes the estimation problem nonstandard. The age at death is left truncated, because a future tenant had to survive his testator to inherit the title “tenant” and the land and to appear in the dataset. Life span distribution and life expectancy are estimated before and during the outbreak of the Black Death, which started in 1348.     

Asymptotically refined score and GOF tests for inverse Gaussian models

ABSTRACT

The score test and the GOF test for the inverse Gaussian distribution, in particular the latter, are known to have large size distortion and hence unreliable power when referring to the asymptotic critical values. We show in this paper that with the appropriately bootstrapped critical values, these tests become second-order accurate, with size distortion being essentially eliminated and power more reliable. Two major generalizations of the score test are made: one is to allow the data to be right-censored, and the other is to allow the existence of covariate effects. A data mapping method is introduced for the bootstrap to be able to produce censored data that are conformable with the null model. Monte Carlo results clearly favour the proposed bootstrap tests. Real data illustrations are given.     

Partially linear transformation model for length-biased and right-censored data

In this paper, we consider a partially linear transformation model for data subject to length-biasedness and right-censoring which frequently arise simultaneously in biometrics and other fields. The partially linear transformation model can account for nonlinear covariate effects in addition to linear effects on survival time, and thus reconciles a major disadvantage of the popular semiparamnetric linear transformation model. We adopt local linear fitting technique and develop an unbiased global and local estimating equations approach for the estimation of unknown covariate effects. We provide an asymptotic justification for the proposed procedure, and develop an iterative computational algorithm for its practical implementation, and a bootstrap resampling procedure for estimating the standard errors of the estimator. A simulation study shows that the proposed method performs well in finite samples, and the proposed estimator is applied to analyse the Oscar data.     

Semiparametric inference for the recurrent events process by means of a single-index model

In this paper, we introduce new parametric and semiparametric regression techniques for a recurrent event process subject to random right censoring. We develop models for the cumulative mean function and provide asymptotically normal estimators. Our semiparametric model which relies on a single-index assumption can be seen as a dimension reduction technique that, contrary to a fully nonparametric approach, is not stroke by the curse of dimensionality when the number of covariates is high. We discuss data-driven techniques to choose the parameters involved in the estimation procedures and provide a simulation study to support our theoretical results.     

Nonparametric estimation of the bivariate survival function for one modified form of current-status data

In this article, the estimation of the bivariate survival function for one modified form of current-status data is considered. Two types of estimators, which are generalizations of the estimators by Campbell and Földes [G. Campbell and A. Földes, Large sample properties of nonparametric statistical inference, in Nonparametric Statistical Inference, B.V. Gnredenko, M.L. Puri, and I. Vineze, eds., North-Holland, Amsterdam, 1982, pp. 103–122] and Dabrowska [D.M. Dabrowska, Kaplan-Meier estimate on the plane, Ann. Stat. 18 (1988), pp. 1475–1489; D.M. Dabrowska, Kaplan-Meier estimate on the plane: weak convergence, LIL, and the bootstrap, J. Multivariate Anal. 29 (1989), pp. 308–325], are proposed. The consistency of the proposed estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators.     

Estimation in multiple linear regression model with doubly censored data

In this article, we propose three M-estimators for multiple regression model when response variable is subject to double censoring. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to construct interval estimators.     

Logspline Density Estimation under Censoring and Truncation

ABSTRACT. In this paper we consider logspline density estimation for data that may be left-truncated or right-censored. For randomly left-truncated and right-censored data the product-limit estimator is known to be a consistent estimator of the survivor function, having a faster rate of convergence than many density estimators. The product-limit estimator and B-splines are used to construct the logspline density estimate for possibly censored or truncated data. Rates of convergence are established when the log-density function is assumed to be in a Besov space. An algorithm involving a procedure similar to maximum likelihood, stepwise knot addition, and stepwise knot deletion is proposed for the estimation of the density function based upon sample data. Numerical examples are used to show the finite-sample performance of inference based on the logspline density estimation.     

Non-parametric Estimation for the Location of a Change-point in an Otherwise Smooth Hazard Function under Random Censoring

A non-parametric wavelet based estimator is proposed for the location of a change-point in an otherwise smooth hazard function under non-informative random right censoring. The proposed estimator is based on wavelet coefficients differences via an appropriate parametrization of the time-frequency plane. The study of the estimator is facilitated by the strong representation theorem for the Kaplan–Meier estimator established by Lo and Singh (1986). The performance of the estimator is checked via simulations and two real examples conclude the paper.     

NONPARAMETRIC ESTIMATION OF CONDITIONAL CUMULATIVE HAZARDS FOR MISSING POPULATION MARKS

A new function for the competing risks model, the conditional cumulative hazard function, is introduced, from which the conditional distribution of failure times of individuals failing due to cause  j  can be studied. The standard Nelson–Aalen estimator is not appropriate in this setting, as population membership (mark) information may be missing for some individuals owing to random right-censoring. We propose the use of imputed population marks for the censored individuals through fractional risk sets. Some asymptotic properties, including uniform strong consistency, are established. We study the practical performance of this estimator through simulation studies and apply it to a real data set for illustration.     

Smooth nonparametric quantile estimation under censoring: simulations and bootstrap methods

Based on right-censored data from a lifetime distribution F , a smooth nonparametric estimator of the quantile function Q (p) is given by Qn(p)=h 1jjQn(t)K((t-p)/h)dt, where QR(p) denotes the product-limit quantile function. Extensive Monte Carlo simulations indicate that at fixed p this kernel-type quantile estimator has smaller mean squared error than (L(p) for a range of values of the bandwidth h. A method of selecting an "optimal" bandwidth (in the sense of small estimated mean squared error) based on the bootstrap is investigated yielding results consistent with the simulation study. The bootstrap is also used to obtain interval estimates for Q (p) after determining the optimal value of h.