Antónia Földes

Strong consistency properties of nonparametric estimators for randomly censored data, II: Estimation of density and failure rate

This article is Part II of a two-part study. Properties of the product-limit estimator established in the previous part [2] are now used to prove the strong consistency of some nonparametric density and failure rate estimators which can be used with randomly censored data.

The local time of iterated Brownian motion

We define and study the local time process {L*(x,t);x∈ℝ1,t≥0} of the iterated Brownian motion (IBM) {H(t):=W1(|W2(t)|); t≥0}, whereW1(·) andW2(·) are independent Wiener processes.

Global Strassen-type theorems for iterated Brownian motions

A class of iterated processes is studied by proving a joint functional limit theorem for a pair of independent Brownian motions. This Strassen method is applied to prove global (t --> [infinity]), as well as local (t --> 0), LIL type results for various iterated processes. Similar results are also proved for iterated random walks via invariance.

Strong approximation of additive functionals

LetX1,X2,...be a sequence of i.i.d. random variables and putS0=0,Sn=X1+...+Xn. A strong approximation type result is given forAN=∑i=1Nf(Si) whereF(x),x∈R is a real valued function. A similar result is given for ∫0tg(B(s))ds. Some weak convergence type implications are also discussed.

Strong consistency properties of nonparametric estimators for randomly censored data, I: The product-limit estimator

AbstractIn reliability and survival-time studies one frequently encounters the followingrandom censorship model:X1,Y1,X2,Y2,… is an independent sequence of nonnegative rvs, theXns having common distributionF and theYns having common distributionG, Zn=min{Xn,Yn},Tn=I[Xn<-Yn]; ifXn represents the (potential) time to death of then-th individual in the sample andYn is his (potential) censoring time thenZn represents the actual observation time andTn represents the type of observation (Tn=O is a censoring,Tn=1 is a death). One way to estimateF from the observationsZ1.T1,Z2,T2, … (and without recourse to theXns) is by means of theproduct limit estimator

STRASSEN THEOREMS FOR A CLASS OF ITERATED PROCESSES

\hat F_n

How big are the increments of the local time of a recurrent random walk

(Kaplan andMeier [6]). It is shown that