Order statistics from the linear-exponential distribution,part I: increasing hazard rate case

For the linear-exponential distribution with increasing hazard rate, exact and explicit expressions for means, product moments and percentage points of order statistics are obtained. Some recurrence relations for both single and product moments of order statistics are also derived. These recurrence relations would enable one to obtain all the higher order moments of order statistics for all sample sizes from those of the lower order     

HAZARD RATE ESTIMATION FOR CENSORED DATA BY WAVELET METHODS

ABSTRACT

We study the estimation of a hazard rate function based on censored data by non-linear wavelet method. We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based hazard rate estimators under randomly censored data. We show this MISE formula, when the underlying hazard rate function and censoring distribution function are only piecewise smooth, has the same expansion as analogous kernel estimators, a feature not available for the kernel estimators. In addition, we establish an asymptotic normality of the nonlinear wavelet estimator.     

LIMIT THEOREMS FOR ASYMPTOTICALLY MINIMAX ESTIMATION OF A DISTRIBUTION WITH INCREASING FAILURE RATE UNDER A RANDOM MIXED CENSORSHIP/TRUNCATION MODEL

ABSTRACT

The search for optimal non-parametric estimates of the cumulative distribution and hazard functions under order constraints inspired at least two earlier classic papers in mathematical statistics: those of Kiefer and Wolfowitz[1] Kiefer, J. and Wolfowitz, J. 1976. Asymptotically Minimax Estimation of Concave and Convex Distribution Functions. Z. Wahrsch. Verw. Gebiete, 34: 73–85. [Crossref], [Web of Science ®] , [Google Scholar] and Grenander[2] Grenander, U. 1956. On the Theory of Mortality Measurement. Part II. Scand. Aktuarietidskrift J., 39: 125–153.  [Google Scholar] respectively. In both cases, either the greatest convex minorant or the least concave majorant played a fundamental role. Based on Kiefer and Wolfowitz's work, Wang3-4 Wang, J.L. 1986. Asymptotically Minimax Estimators for Distributions with Increasing Failure Rate. Ann. Statist., 14: 1113–1131. Wang, J.L. 1987. Estimators of a Distribution Function with Increasing Failure Rate Average. J. Statist. Plann. Inference, 16: 415–427.   found asymptotically minimax estimates of the distribution function F and its cumulative hazard function Λ in the class of all increasing failure rate (IFR) and all increasing failure rate average (IFRA) distributions. In this paper, we will prove limit theorems which extend Wang's asymptotic results to the mixed censorship/truncation model as well as provide some other relevant results. The methods are illustrated on the Channing House data, originally analysed by Hyde.5-6 Hyde, J. 1977. Testing Survival Under Right Censoring and Left Truncation. Biometrika, 64: 225–230. Hyde, J. 1980. “Survival Analysis with Incomplete Observations”. In Biostatistics Casebook 31–46. New York: Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics.       

BAYES EMPIRICAL BAYES APPROACH TO ESTIMATION OF THE FAILURE RATE IN EXPONENTIAL DISTRIBUTION

ABSTRACT

In the empirical Bayes (EB) decision problem consisting of squared error estimation of the failure rate in exponential distribution, a prior Λ is placed on the gamma family of prior distributions to produce Bayes EB estimators which are admissible. A subclass of such estimators is shown to be asymptotically optimal (a.o.). The results of a Monte Carlo study are presented to demonstrate the a.o. property of the Bayes EB estimators.     

SOJOURN TIMES IN NON-HOMOGENEOUS QBD PROCESSES WITH PROCESSOR SHARING

We study sojourn times of customers in a processor sharing queue with a service rate that varies over time, depending on the number of customers and on the state of a random environment. An explicit expression is derived for the Laplace–Stieltjes transform of the sojourn time conditional on the state upon arrival and the amount of work brought into the system. Particular attention is paid to the conditional mean sojourn time of a customer as a function of his required amount of work, and we establish the existence of an asymptote as the amount of work tends to infinity. The method of random time change is then extended to include the possibility of a varying service rate. By means of this method, we explain the well-established proportionality between the conditional mean sojourn time and required amount of work in processor sharing queues without random environment. Based on numerical experiments, we propose an approximation for the conditional mean sojourn time. Although first presented for exponentially distributed service requirements, the analysis is shown to extend to phase-type services. The service discipline of discriminatory processor sharing is also shown to fall within the framework.     

A Retrial Queue with a Constant Retrial Rate,Server Downs and Impatient Customers

ABSTRACT

In this paper, we consider a retrial queueing system consisting of a waiting line of infinite capacity in front of a single server subject to breakdowns. A customer upon arrival may join the queue (waiting line) or go to the retrial orbit (another queue) to retry for service after a random time. Only the customer at the head of the retrial orbit is allowed to retry for service. Upon retrial, the customer enters the service if the server is idle; otherwise, it may go back to the retrial orbit or leave the system (become impatient). All the interarrival times, service times, server up times, server down times and retrial times are exponential, and all the necessary independence conditions in these variables are assumed. For this system, we provide sufficient conditions under which, for any given number of customers in the orbit, the stationary probability of the number of customers in the waiting line decays geometrically. We also provide explicitly an expression for the decay parameter.     

P-value adjustment for multiple binary endpoints

The p-value-based adjustment of individual endpoints and the global test for an overall inference are the two general approaches for the analysis of multiple endpoints. Statistical procedures developed for testing multivariate outcomes often assume that the multivariate endpoints are either independent or normally distributed. This paper presents a general approach for the analysis of multivariate binary data under the framework of generalized linear models. The generalized estimating equations (GEE) approach is applied to estimate the correlation matrix of the test statistics using the identity and exchangeable working correlation matrices with the model-based as well as robust estimators. The objectives of the approaches are the adjustment of p-values of individual endpoints to identify the affected endpoints as well as the global test of an overall effect. A Monte Carlo simulation was conducted to evaluate the overall family wise error (FWE) rates of the single-step down p-value adjustment approach from two adjustment methods to three global test statistics. The p-value adjustment approach seems to control the FWE better than the global approach Applications of the proposed methods are illustrated by analyzing a carcinogenicity experiment designed to study the dose response trend for 10 tumor sites, and a developmental toxicity experiment with three malformation types: external, visceral, and skeletal.     

Statistical Analysis of Coins Lost in Circulation

The loss rate of coins in the Israeli pound (IL) series issued in Israel in 1960–1979 is estimated, and the connection between it and the characteristics of the coins is examined. A lifetime model with time-dependent hazard rate and an estimation procedure for the model parameters are presented. The estimates are then used to forecast the number of coins surviving in circulation and the median lifetime of different denominations. The data are random samples drawn from the coins in circulation at a point in time, the sampled coins being sorted by date of issue.     

Measuring the Influence of Unemployment Insurance on Unemployment Experiences

This article investigates the comprehensive effects of unemployment insurance (UI) policies on the amount of time and unemployment that individuals report between jobs. The econometric model jointly determines the effects of UI on the lengths of nonemployment spells, the classification of these spells as unemployment, and the likelihood of collecting program benefits. The model carefully attempts to isolate variation in UI benefits attributable to differences in generosity across programs to avoid biases in estimating policy effects induced by other contaminating sources of benefit variation. Using data on men from the National Longitudinal Survey of Youth, the empirical results find (a) UI recipients typically experience longer spells between jobs, at least up to the exhaustion of UI benefits, and report substantially larger fractions of these spells as unemployment; (b) weekly benefit amounts exert no significant influence on the likelihood of UI recipiency, on the length of spells between jobs, or on the fraction of these spells classified as unemployment; and (c) increases in weeks of UI eligibility raise the likelihood of UI collection and lengthen the number of weeks of unemployment between jobs by inducing long spells to become longer and not by altering short-duration behavior.     

Bayes estimators of the average failure rates

In the present paper we have proposed a Bayesian approach for making inferences from accelerated life tests which do not require distributional assumptions