Strong Gaussian Approximations of Product-Limit and Quantile Processes for Strong Mixing and Censored Data

In this article, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate O[(log n)^?λ] for some λ > 0. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.     

A Distribution-Free Control Chart Based on Order Statistics

In this article, we introduce a new distribution-free Shewhart-type control chart that takes into account the location of a single order statistic of the test sample (such as the median) as well as the number of observations in that test sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution, and the average run length (ARL) are all derived. A key advantage of the chart is that, due to its nonparametric nature, the false alarm rate and in-control run length distribution are the same for all continuous process distributions, and so will be naturally robust. Tables are provided for the implementation of the chart for some typical ARL values and false alarm rates. The empirical study carried out reveals that the new chart is preferable from a robustness point of view in comparison to a classical Shewhart-type chart and also the nonparametric chart of Chakraborti et al. (2004 Chakraborti , S. , van der Laan , P. , van de Wiel , M. A. ( 2004 ). A class of distribution-free control charts . J. Roy. Statist. Soc. Ser. C-Appl. Statist. 53 ( 3 ): 443 – 462 .[Web of Science ®] , [Google Scholar]).     

Renyi Entropy Properties of Order Statistics

In this article, we discuss some properties of Renyi entropy and Renyi information of order statistics. Some bounds for Renyi entropy of order statistics are obtained. Also, we relate Renyi entropy ordering of order statistics to Renyi entropy ordering and other well known orderings of parent random variables. Then it is proved that the Renyi information between order statistics and parent random variable is distribution free, and it is shown, as expected, the distance is minimum for the median.     

Free Knot Splines with RJMCMC in Survival Data Analysis

The B-spline representation is a common tool to improve the fitting of smooth nonlinear functions, it offers a fitting as a piecewise polynomial. The regions that define the pieces are separated by a sequence of knots. The main difficulty in this type of modeling is the choice of the number and the locations of these knots. The Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm provides a solution to simultaneously select these two parameters by considering the knots as free parameters. This algorithm belongs to the MCMC techniques that allow simulations from target distributions on spaces of varying dimension. The aim of the present investigation is to use this algorithm in the framework of the analysis of survival time, for the Cox model in particular. In fact, the relation between the hazard ratio function and the covariates being assumed to be log-linear, this assumption is too restrictive. Thus, we propose to use the RJMCMC algorithm to model the log hazard ratio function by a B-spline representation with an unknown number of knots at unknown locations. This method is illustrated with two real data sets: the Stanford heart transplant data and lung cancer survival data. Another application of the RJMCMC is selecting the significant covariates, and a simulation study is performed.     

Relations for Moments of Progressively Type-II Censored Order Statistics from Log-Logistic Distribution with Applications to Inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a log-logistic distribution. The use of these relations in a systematic recursive manner would enable the computation of all the means, variances and covariances of progressively Type-II right censored order statistics from the log-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁,…, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Malik (1987 Balakrishnan , N. , Malik , H. J. ( 1987 ). Moments of order statistics from truncated log-logistic distribution . J. Statist. Plann. Infer. 17 : 251 – 267 .[Crossref], [Web of Science ®] , [Google Scholar]) and Balakrishnan et al. (1987 Balakrishnan , N. , Malik , H. J. , Puthenpura , S. ( 1987 ). Best linear unbiased estimation of location and scale parameters of the log-logistic distribution . Commun. Statist. Theor. Meth. 16 : 3477 – 3495 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The moments so determined are then utilized to derive best linear unbiased estimators for the scale- and location-scale log-logistic distributions. A comparison of these estimates with the maximum likelihood estimates is made through Monte Carlo simulation. The best linear unbiased predictors of progressively censored failure times is then discussed briefly. Finally, a numerical example is presented to illustrate all the methods of inference developed here.     

On the Use of Superpopulation Model in Estimation of Population Mean in Two-Occasion Rotation Patterns

The present work is an attempt to study the estimation of the population mean on the current occasion in two-occasion successive (rotation) sampling under a superpopulation model. Six different estimators are proposed for estimating the current population mean in two-occasion successive (rotation) sampling. Optimum replacement policies and performances of the proposed estimators have been discussed. Results are interpreted via empirical studies.     

Use of Quasi-Ranges in the Estimation of Scale Parameter of a Symmetric Distribution

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the scale parameter of a symmetric distribution using quasi-ranges is considered. We also prove a sufficient condition for the non negativity of the scale estimator obtained by the above method. Further, we obtain necessary and sufficient conditions for the derived estimators to be constant multiple of the sample range.     

On Blest’s Measure of Kurtosis Adjusted for Skewness

We reconsider the derivation of Blest’s (2003) skewness adjusted version of the classical moment-based coefficient of kurtosis and propose an adaptation of it which generally eliminates the effects of asymmetry a little more successfully. Lower bounds are provided for the two skewness adjusted kurtosis moment measures as functions of the classical coefficient of skewness. The results from a Monte Carlo experiment designed to investigate the sampling properties of numerous moment-based estimators of the two skewness adjusted kurtosis measures are used to identify those estimators with lowest mean squared error for small to medium sized samples drawn from distributions with varying levels of asymmetry and tailweight.     

An Improved Test for Continuous Local Martingales

We present a new test for the “continuous martingale hypothesis”. That is, a test for the hypothesis that observed data are from a process which is a continuous local martingale. The basis of the test is an embedded random walk at first passage times, obtained from the well-known representation of a continuous local martingale as a continuous time-change of Brownian motion. With a variety of simulated diffusion processes our new test shows higher power than existing tests using either the crossing tree or the quadratic variation, including the situation where non-negligible drift is present. The power of the test in the presence of jumps is also explored with a variety of simulated jump diffusion processes. The test is also applied to two sequences of high-frequency foreign exchange trade-by-trade data. In both cases the continuous martingale hypothesis is rejected at times less than hourly and we identify significant dependence in price movements at these small scales.     

Effective Rotation Patterns Under Non Response in Two-Occasion Successive Sampling

This article intends to develop some effective rotation patterns with the aid of attractive imputation methods when the problems of non response occur in two-occasion successive sampling. Utilizing the information on p (p ??1) auxiliary variables regression methods of imputation have been considered and subsequently multiple linear regression type estimators are proposed to estimate the current population mean in two-occasion successive sampling. Proposed estimators are compared with the estimator for same situations but in the absence of non-response. Optimum replacement strategies of the respective estimators have been discussed and results are interpreted with the help of empirical studies. Conclusions and suitable recommendations are made.