Confidence Intervals for a Binomial Parameter after Observing No Successes

The easily computed, one-sided confidence interval for the binomial parameter provides the basis for an interesting classroom example of scientific thinking and its relationship to confidence intervals. The upper limit can be represented as the sample proportion from a number of “successes” in a future experiment of the same sample size. The upper limit reported by most people corresponds closely to that producing a 95 percent classical confidence interval and has a Bayesian interpretation.     

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.     

On the Multivariate Two-Sample Problem Using Strong Approximations of Empirical Copula Processes

In this article, we establish optimal rates for the strong approximation of empirical copula processes in ?² by sequences of Gaussian processes. These results are applied to investigate Cramér–von Mises-type statistics.     

Adjusted Hazard Rate Estimator Based on a Known Censoring Probability

In most reliability studies involving censoring, one assumes that censoring probabilities are unknown. We derive a nonparametric estimator for the survival function when information regarding censoring frequency is available. The estimator is constructed by adjusting the Nelson–Aalen estimator to incorporate censoring information. Our results indicate significant improvements can be achieved if available information regarding censoring is used. We compare this model to the Koziol–Green model, which is also based on a form of proportional hazards for the lifetime and censoring distributions. Two examples of survival data help to illustrate the differences in the estimation techniques.     

On Nonparametric Estimation of a Reliability Function

This article considers the properties of a nonparametric estimator developed for a reliability function which is used in many reliability problems. Properties such as asymptotic unbiasedness and consistency are proven for the estimator and using U-statistics, weak convergence of the estimator to a normal distribution is shown. Finally, numerical examples based on an extensive simulation study are presented to illustrate the theory and compare the estimator developed in this article with another based directly on the ratio of two empirical distributions studied in Zardasht and Asadi (2010 Zardasht , V. , Asadi , M. ( 2010 ). Evaluation of P(X _t  > Y _t ) when both X _t and Y _t are residual lifetimes of two systems . Statistica Neerlandica 64 : 460 – 481 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Coherent Frequentism: A Decision Theory Based on Confidence Sets

By representing fair betting odds according to one or more pairs of confidence set estimators, dual parameter distributions called confidence posteriors secure the coherence of actions without any prior distribution. This theory reduces to the maximization of expected utility when the pair of posteriors is induced by an exact or approximate confidence set estimator or when a reduction rule is applied to the pair. Unlike the p-value, the confidence posterior probability of an interval hypothesis is suitable as an estimator of the indicator of hypothesis truth since it converges to 1 if the hypothesis is true or to 0 otherwise.     

Nonnested Testing for Competing Autoregressive Dynamic Models Estimated by Instrumental Variables

In this work we investigate nonnested tests for two competing univariate dynamic linear models with autoregressive disturbances, where the motivation for instrumental variable estimation is mainly due to the recognized presence of current endogenous variables in the regression function, either in one or both models. As the previous transformation of both models yields regression functions which are nonlinear in the parameters, the attractive Gauss-Newton regression (GNR) approach, firstly advocated by Davidson and Mackinnon (1981 Davidson , R. , Mackinnon , J. G. ( 1981 ). Several tests for model specification in the presence of alternative hypotheses . Econometrica 49 : 781 – 93 .[Crossref], [Web of Science ®] , [Google Scholar]), will be used to obtain the results.     

On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution

The Lomax (Pareto II) distribution has found wide application in a variety of fields. We analyze the second-order bias of the maximum likelihood estimators of its parameters for finite sample sizes, and show that this bias is positive. We derive an analytic bias correction which reduces the percentage bias of these estimators by one or two orders of magnitude, while simultaneously reducing relative mean squared error. Our simulations show that this performance is very similar to that of a parametric bootstrap correction based on a linear bias function. Three examples with actual data illustrate the application of our bias correction.     

A Bivariate Generalization of Gamma Distribution

In this article, a bivariate generalisation of the gamma distribution is proposed by using an unsymmetrical bivariate characteristic function; an extension to the non central case also receives attention. The probability density functions of the product and ratio of the correlated components of this distribution are also derived. The benefits of introducing this generalized bivariate gamma distribution and the distributions of the product and the ratio of its components will be demonstrated by graphical representations of their density functions. An example of this generalized bivariate gamma distribution to rainfall data for two specific districts in the North West province is also given to illustrate the greater versatility of the new distribution.     

Additive Transformation Models for Recurrent Events

In this article, we propose a class of additive transformation models for recurrent event data, which includes the additive rates model as a special case. The new models offer great flexibility in formulating the effects of covariates on the mean function of recurrent events. Estimating equation approaches are developed for the model parameters, and asymptotic properties of the resulting estimators are established. In addition, a model checking procedure is presented to assess the adequacy of the model. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a bladder cancer study is presented.