Sándor Csörgő

The rate of strong uniform consistency for the product-limit estimator

SummaryThe maximal deviation of the product-limit estimate from the estimated distribution function is investigated. As a consequence of a functional law of the iterated logarithm, the log log law is proved for this deviation on appropriate half lines, with the precise constants. This result implies that the log log law need not hold in general for the maximal deviation on the whole line. Then a general asymptotic order of magnitude is determined for the latter deviation. This order depends on the joint behaviour of the censoring and censored distributions in a well-defined way. Corresponding to specific joint behaviours, several limsup results are deduced as consequence including all the previously known log log-type laws in improved form. The results are also extended to the variable censoring model.

What portion of the sample makes a partial sum asymptotically stable or normal

SummaryLet a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<α≦2 be given. We show that if at each stage n a number kn depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on kn the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.

Bootstrapping Empirical Functions

In our previous sections we have established weak convergence of mean residual life, total time of test, Lorenz and Goldie processes to appropriate Gaussian processes. Apart from a few special cases (cf. Section 8), these limiting processes are functions of the underlying distributions. Consequently when testing for statistical hypotheses for example, one would have to compute the resulting limiting distribution for each F of interest. The same is true when trying to use our results for constructing confidence bands for the theoretical functionals of the said empirical processes. This type of problems can be solved by adapting the bootstrap method, proposed by Efron (1979, 1982), to the present situation. As we will see, the bootstrap method is simply a Monte Carlo simulation determined by the given observations.

A correction to and improvement of «strong approximations of some biometric estimates under random censorship»

SummaryIn the proof of the main result of the original paper there is an error. Instead of repairing that proof to get just the original result, at the critical spot we improve the proof and obtain a much better result. In particular, we approximate the product-limit and empirical cumulative hazard processes by suitable copies of the corresponding limiting Gaussian processes with rates of approximation that on appropriate fixed half lines reduce to the rates of Komlós, Major and Tusnády for the uncensored empirical process.

Mean residual life processes

We summarise now a convergence theory for the mean residual life process zn of (1.15) as a consequence of the preceding section.

Upper and lower classes for triangular arrays

SummaryWe derive characterizations of upper and lower classes in the law of the iterated logarithm for row sums of triangular arrays of Gaussian random variables. We also derive strong approximations for more general triangular arrays by Gaussian arrays. By combining these results we deduce characterizations of upper and lower classes for row sums from general arrays.

Testing for linearity

A nonparametric large sample test is proposed for testing the linearity of a regression model with independent and identically distributed errors satisfying only a very mild tail condition. The statistic is based on the functional least squares estimator of the slope vector. The test is applied to the stack loss data.

The Basic Setting for the Approximations and Various Preliminaries

Without loss of generality we assume that our basic sequence X1,X2,... is defined on an appropriate probability space (Ω, A, P) such that for the resulting uniform empirical process

Discussion of Results on Empirical Lorenz Processes

{\alpha _n}\left( y \right) = {n^{1/2}}\left( {y - {E_n}\left( y \right)} \right),\quad 0 \leq y \leq 1