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standard PFS definition. When δ p  = 1 and δ d  = 0 or 1, we refer to those patients with observed progression times as having progression-PFS events ( X s  =  T p ). When δ p  = 0 and δ d  = 1, we refer to those patients as having a death–PFS event ( X s  =  T d ). The true PFS time T is actually between U p and T d in the presence of death–PFS events. Thus, there exists a clear systematic bias, where X s will generally overestimate T . This systematic bias is what our proposed PFS estimators will attempt to correct. 2.2 Kaplan–Meier estimator We assume the

how to adapt the Owen–Jager–Wellner method to calculate exact, nonparametric, likelihood-based confidence bands for , starting from the Kaplan–Meier estimator and inverting a modified Berk–Jones statistic. Of course, since these bands are likelihood-based, they inherit all the familiar, attractive properties of the empirical likelihood method – they are range preserving, transformation invariant, and their shape is determined by the observed data. To compute quantiles of the exact null distribution for finite samples of size n , we use the recursions of Noé [ 3

likelihood based simultaneous confidence bands for differences and ratios of two distribution functions from independent samples of right-censored survival data. The proposed confidence bands provide a flexible way of comparing treatments in biomedical settings, and bring empirical likelihood methods to bear on important target functions for which only Wald-type confidence bands have been available in the literature. The approach is illustrated with a real data example. KEYWORDS: Kaplan--Meier estimator, Nelson--Aalen estimator, plug-in, right censoring 1 Introduction The

given X, i.e., finding a function m∗(X) which achieves the minimum of the mean squared error: E ∣∣m∗(X) − Y ∣∣2 = min m E |m(X) − Y |2 . It is well known that the solution of this problem is the regression function r(x) defined by r(x) = E (Y | X = x) . AMS 1991 subject classification: Primary: 62G20; Secondary: 62E20 Key words and phrases: Asymptotic normality, censored data, Kaplan–Meier estimator, kernel, nonparametric regression, rate of convergence, strong consistency 160 Guessoum -- Ould-Saïd In statistics literature, many papers deal with estimating the

asymptotically normally distributed. A simulation study shows that the proposed estimator performs well when compared with competing alternatives. The various methods are illustrated with a real data set. KEYWORDS: Kendall's tau, dependence, Horvitz-Thompson estimator, Kaplan-Meier estimator, martingales Author Notes: Partial funding for this work was provided by the Natural Sciences and Engineering Research Council of Canada and the Fonds québécois de la recherche sur la nature et les technologies. The authors would like to thank Professor Phyllis K. Mansfield for granting

values in an infinite-dimension space. An estimator of the conditional quantile is given and, under some regularity conditions, among which the small-ball probability for the covariate, its uniform strong convergence with rates is established. Keywords. Censored data, conditional distribution function, infinite dimension, KaplanMeier estimator, kernel estimator, small-ball probability. 2010 Mathematics Subject Classification. 62G05, 62G20. 1 Introduction It has been well known from a robustness approach that the mean is sensible to outliers (see Hampel et al. [31

dichotomized into a binary variable with an empiric cut-off at the 75th percentile of its distribution. The distribution of baseline variables between patients with and without MPV above the empiric cut-off was compared with Wilcoxon’s rank-sum tests, χ 2 -tests, and Fisher’s exact tests, respectively. Median follow-up was estimated with a reverse Kaplan-Meier estimator according to Schemper and Smith. Analysis of time-to-death was performed with Kaplan-Meier estimators, log-rank tests, and uni- and multivariable Cox proportional hazards models, respectively. The

## Abstract

Research background: Enterprises are an important element of the economy, which explains that the analysis of their duration on the market is an important and willingly undertaken research topic. In the case of complex problems like this, considering only one type of event, which ends the duration, is often insufficient for full understanding.

Purpose: In this paper there is an analysis of the duration of enterprises on the market, taking into account various reasons for the termination of their business activity as well as their characteristics.

Research methodology: A survival analysis can be used to study duration on the market. However, the possibility of considering the waiting time for only one type of event is its important limitation. One solution is to use competing risks. Various competing risks models (naive Kaplan-Meier estimator, subdistribution model, subhazard and cause-specific hazard) are presented and compared with an indication of their advantages and weakness.

Results: The competing risks models are estimated to investigate the impact of the causes of an enterprises liquidation on duration distribution. The greatest risk concerns enterprises with a natural person as the owner (regardless of the reason of failure). For each of the competing risks, it is also indicated that there is a section of activity which adversely affects the ability of firms to survive on the market.

Novelty: A valuable result is considering the reasons for activity termination in the duration analysis for enterprises from the Mazowieckie Voivodeship.

E-maiil; etaufer@cs.unitn.it ReceixvecÖ for ROSE December 4, 2001 Abstiracet — VVe consider a Standard random censorship model where either the failure times and/or the censoirinjg times exhibit long-range dependence whereas one of the two maybe short ränge dependent. VVe ar.'e iinterested in investigating the asymptotic properties of the Kaplan-Meier estimator in such a situatüöfii. In Order to conaider an oa comprclicnaivc oa poaaiblc modcl wo analyzc scquoncoa of random variables of the form G(£j)> where £j represent a stationary Gamma process with long

normal and obtain a convolution theorem for estimating functionals of the parameter T. We illustrate the approach with several examples. In particular, we obtain short proofs for the efficiency of the Nelson-Aalen estimator and the Kaplan-Meier estimator. 1. Introduction Counting process models with multiplicative intensity process were intro- duced by Aalen (1975, 1978) to describe statistical models for life history data. For expositions and many examples we refer to Andersen et al. (1982), Jacobsen (1982) and Andersen and Borgan (1985). Here we follow