Nonparametric estimation of survival functions for doubly censored observations when the lifetime hazard rate is proportionally related to the censoring hazard rates

Let T, X and Y be non-negative random variables, where T is the time of occurrence of an event of interest, X and Y being the lefl and right censoring variables respectively.

In this paper we propose a nonparametric estimator of the survival function, ST, when T, X and Y are supposed to be independent and their corresponding hazard rates are proportionally related. In this way, our results extend Ebrahimi's work (1985) to the doubly censored data case.     

Decomposition for multivariate extremal processes

The probability distribution of an extremal process in R^d with independent max-increments is completely determined by its distribution function. The df of an extremal process is similar to the cdf of a random vector. It is a monotone function on (0, ∞) × R^d with values in the interval [0,1]. On the other hand the probability distribution of an extremal process is a probability measure on the space of sample functions. That is the space of all increasing right continuous functions y: (0, ∞) → R^d with the topology of weak convergence. A sequence of extremal processes converges in law if the probability distributions converge weakly. This is shown to be equivalent to weak convergence of the df's.

An extremal process Y: [0, ∞) → R^d is generated by a point process on the space [0, ∞) × [-∞, ∞)^d and has a decomposition Y = X v Z as the maximum of two independent extremal processes with the same lower curve as the original process. The process X is the continuous part and Z contains the fixed discontinuities of the process Y. For a real valued extremal process the decomposition is unique: for a multivariate extremal process uniqueness breaks down due to blotting.     

Empirical power comparisons of two mrpp rank tests

Simulated powers of the MRPP two-sample rank test statistic ?₁- are compared with the powers of the MRPP test statistic ?₂(the two-sided Wilcoxon-Mann-Whitney test) for large samples from several underlying populations. Powers are obtained using two approximate distributions of ?₁ involving three and four moments

respectively, The use of the fourth moment indicates that an approximation to the null distribution of ? based on four moments can perform better     

The Effects of Misconceptions on the Properties of Friedman's Test

Friedman's test is a widely used rank-based alternative to the analysis of variance (ANOVA) F-test for identifying treatment differences in a randomized complete block design. Many texts provide incomplete or misleading information about when Friedman's test may be appropriately applied. We discuss the assumptions needed for the test and common misconceptions. We show via simulation that when the variance or skew of the treatment distributions differ, application of Friedman's test to detect differences in treatment location can result in Type I error probabilities larger than the nominal α, and even when α is unaffected, the power of the test can be less than expected.     

Minicommnications

This section of the periodical is reserved for Conclusions (Results), Comments, Conjectures, and Microcommunications.     

A Class of Markov Chain Models for Average Run Length Computations for Autocorrelated Processes

The average run length (ARL) of conventional control charts is typically computed assuming temporal independence. However, this assumption is frequently violated in practical applications. Alternative ARL computations have often been conducted via time consuming and yet not necessarily very accurate simulations. In this article, we develop a class of Markov chain models for evaluating the run length performance of traditional control charts for autocorrelated processes. We show extensions from the univariate AR(1) model to the general multivariate VARMA(p, q) time series. The results of the proposed method are highly comparable to those of simulations and with significantly less computational overhead.     

On the Computation of Non-Central Chi-Square Distribution Function

The cumulative distribution function of the non-central chi-square is very important in calculating the power function of some statistical tests. On the other hand it involves an integral which is difficult to obtain. In literature some workers discussed the evaluation and the approximation of the c.d.f. of the non-central chi-square [see references (2)]. In the present work two computational formulae for computing the cumulative distribution function of the non-central chi-square distribution are given, the first one deals with the case of any degrees of freedom (odd and even), and the second deals with the case of odd degrees of freedom. Numerical illustrations are discussed.     

Small sample set estimation of a baseline ranking

Ranking objects by a panel of judges is commonly used in situations where objective attributes cannot easily be measured or interpreted. Under the assumption that the judge independently arrive at their rankings by making pair wise comparisons among the objects in an attempt to reproduce a common baseline ranking w₀, we develop and explore confidence regions and Bayesian highest posterior density credible regions for w₀ with emphasis on very small sample sizes.     

Two stage fixed width confidence intervals for the common location parameter of several exponential distributins

Two stage sampling schemes are introduced for use in estimating the common location parameter (guarantee time) of two or more exponential distributions with a confidence interval of prespecified width whose coverage probability is at least a given nominal value. Exact expressions for all moments of order r ≥ 1 of the associated two stage sample sizes and for the actual coverage probabilities are derived. The performance of the procedures in a variety of two population, moderate fixed sample size cases is examined via numerical studies involving both exact calculations and Monte Carlo simulations. No new tables are needed to implement any of the proposed methods. A modified two stage procedure is recommended for practical use     

Oil the Equivalence of pavlov-ushakov and kaplan-meier estimators of survival probability in multiply censored sample

It is proved that the unbiased estimator of survival probability in a multiply censored sample suggested by Pavlov & Ushakov (1980, 1984) is equivalent to the Kaplan-Meier Product-Limit estimator.