Ryszard Zieliński

Stability of the Bayesian estimator of the Poisson mean under the inexactly specified gamma prior

The Bayesian estimator of the mean of the Poisson distribution under the gamma prior ([alpha]0, [beta]0) is stable (robust) in the sense that if the prior runs over the set {([alpha], [beta]0): [alpha][epsilon][[alpha]0-[delta], [alpha]0+[delta]]}, then the oscillat estimator with the oscillation O([delta]2) is constructed; it also minimizes the oscillation of the posterior risk when the shape parameter runs over a finite interval.

Best exact nonparametric confidence intervals for quantiles

Well-known nonparametric confidence intervals for quantiles are of the form (X i : n , X j : n ) with suitably chosen order statistics X i : n and X j : n , but typically their coverage levels differ from those prescribed. It appears that the coverage level of the confidence interval of the form (X i : n , X j : n ) with random indices I and J can be rendered equal, exactly to any predetermined level γ ∈ (0, 1). Best in the sense of minimum E(J − I), i.e., ‘the shortest’, two-sided confidence intervals are constructed. If no two-sided confidence interval exists for a given γ, the most accurate one-sided confidence intervals are constructed.

Best equivariant nonparametric estimator of a quantile

Given q[set membership, variant](0,1) and a sample X1,X2,...,Xn from an unknown , an estimator T*=T*(X1,X2,...,Xn) of the qth quantile of the distribution F is constructed such that EFF(T*)-q[less-than-or-equals, slant]EFF(T)-q for all and for all , where is a non-parametric family of distributions and is a class of estimators. It is shown that T*=Xj:n for a suitably chosen jth order statistic. The best median-unbiased estimator is also constructed.

A Distribution-Free Median-Unbiased Quantile Estimator

Given λ∈(0-,l), let xλ(F) denote the unique λ-quantile of the distribution F. A distribution-free median-unbiased estimator of xλ(F) is explicitly constructed

Uniform strong consistency of sample quantiles

It is well known that if xq(F) is the unique qth quantile of a distribution function F, then Xk(n):n with k(n)/n --> q is a strongly consistent estimator of xq(F). However, for every [var epsilon] >0 and for every, even very large n, supF[set membership, variant]F,PF{Xk(n):n--Xq(F)>[var epsilon]}=1. This is a consequence of the fact that in the family of all distribution functions with uniquely defined qth quantile the almost sure convergence Xk(n):n --> xq(F) is not uniform. A simple necessary and sufficient condition for the uniform strong consistency of Xk(n):n is given.

Small-Sample Quantile Estimators in a Large Nonparametric Model

The large nonparametric model in this note is a statistical model with the family ℱ of all continuous and strictly increasing distribution functions. In the abundant literature of the subject, there are many proposals for nonparametric estimators that are applicable in the model. Typically the kth order statistic X k:n is taken as a simplest estimator, with k = [nq], or k = [(n + 1)q], or k = [nq] + 1, etc. Often a linear combination of two consecutive order statistics is considered. In more sophisticated constructions, different L-statistics (e.g., Harrel–Davis, Kaigh–Lachenbruch, Bernstein, kernel estimators) are proposed. Asymptotically the estimators do not differ substantially, but if the sample size n is fixed, which is the case of our concern, differences may be serious. A unified treatment of quantile estimators in the large, nonparametric statistical model is developed.

Robust Testing Serial Correlation in AR(1) Processes in the Presence of a Single Additive Outlier

Abstract The effect of a single additive outlier on the test of serial correlation in first order autoregressive processes is presented. We show that the test based on the least square estimator is highly sensitive to a single additive outlier: if the outlier is large then both the size and the power of the test are close to zero! A test based on the median of ratios of consecutive observations (Hurwicz, L. (1950). Least-Squares Bias in Time Series. In: Koopmas, T. C., ed. Statistical Inference in Dynamic Economic Models. Wiley.) and a test based on the normalized median of products of two consecutive observations (Haddad, John N. (2000). On robust estimation in the first-order autoregressive process. Communication in Statistics, Theory and Methods 29(1.1):45–54.) are discussed. A thorough simulation study as well as some theoretical considerations allow us to conclude that both are reasonably robust.

A SIMPLE IMPROVEMENT OF THE KAPLAN-MEIER ESTIMATOR

ABSTRACT Though widely used, the celebrated Kaplan-Meier estimator suffers from a disadvantage: it may happen, and in small and moderate samples it often does, that even if the difference between two consecutive times t 1 and t 2 ( ) is considerably large, for the values of the Kaplan-Meier estima-tor KM(t 1) and KM(t 2) we may have KM . Although that is a general problem in estimating a smooth and monotone distribution function from small or moderate samples, in the context of estimating survival probabilities the disadvantage is particularly annoying. In the paper we discuss a local smoothing of the Kaplan-Meier estimator based on an approximation by the Weibull distribution function. It appears that Mean Square Error and Mean Absolute Deviation of the smoothed estimator is significantly smaller. It follows also from the Pitman Closeness Criterion that the new version of the estimator can be recommended.

Bias of the lse estimator of the first order autoregressive model under tukey contamination

Results of an exhaustive study of the bias of the least square estimator (LSE) of an first order autoregression coefficient α in a contaminated Gaussian model are presented. The model describes the following situation. The process is defined as Xt = α Xt-1 + Yt . Until a specified time T, Yt are iid normal N(0, 1). At the moment T we start our observations and since then the distribution of Yt, t≥T, is a Tukey mixture T(eσ) = (1 – e)N(0,1) + eN(0, σ2). Bias of LSE as a function of α and e, and σ2 is considered. A rather unexpected fact is revealed: given α and e, the bias does not change montonically with σ (“the magnitude of the contaminant”), and similarly, given α and σ, the bias is not growing with e (“the amount of contaminants”).

An aperiodic pseudorandom number generator

Abstract A pseudorandom number generator with infinite period is constructed. Results of statistical tests are presented.