A Smooth Bootstrap Procedure towards Deriving Confidence Intervals for the Relative Risk

Given a pair of sample estimators of two independent proportions, bootstrap methods are a common strategy towards deriving the associated confidence interval for the relative risk. We develop a new smooth bootstrap procedure, which generates pseudo-samples from a continuous quantile function. Under a variety of settings, our simulation studies show that our method possesses a better or equal performance in comparison with asymptotic theory based and existing bootstrap methods, particularly for heavily unbalanced data in terms of coverage probability and power. We illustrate our procedure as applied to several published data sets.     

Edge Estimation in the Population of a Binary Tree Using Node-Sampling

Suppose a finite population of several vertices, each connected to single or multiple edges. This constitutes a structure of graphical population of vertices and edges. As a special case, the graphical population like a binary tree having only two child vertices associated to parent vertex is taken into consideration. The entire binary tree is divided into two sub-graphs such as a group of left-nodes and a group of right-nodes. This paper takes into account a mixture of graph structured and population sampling theory together and presents a methodology for mean-edge-length estimation of left sub-graph using right edge sub-graph as an auxiliary source of information. A node-sampling procedure is developed for this purpose and a class of estimators is proposed containing several good estimators. Mathematical conditions for minimum bias and optimum mean squared error of the class are derived and theoretical results are numerically supported with a test of 99% confidence intervals. It is shown that suggested class has a sub-class of optimum estimators, and sample-based estimates are closer to the true value of the population parameter.     

Maximum likelihood estimation in a Weibull regression model with type-1 censoring: a Monte Carlo study

Results of the Monte Carlo study of the performance of a maximum likelihood estimation in a Weibull parametric regression model with two explanatory variables are presented. One simulation run contained 1000 samples censored on the average by the amount of 0-30%. Each simulatedsample was generated in a form of two-factor two-level balanced experiment. The confidence intervals were computed using the large-sample normal approximation via the matrix of observed information. For small sample sizes the estimates of the scale parameter b of the loglifetime were significantly negatively biased, which resulted in a poor quality of confidence intervals for b and the low-level quantiles. All estimators improved their quality when the nominal value of b decreased. A moderate amount of censoring improved the quality of point and confidence estimation. The reparametrization b 7 produced rather accurate confidence intervals. Exact confidence intervals for b in case of non-censoring were obtained using the pivotal quantity b/b.     

Conditional statistical inference and quantification of relevance

Summary. We argue that it can be fruitful to take a predictive view on notions such as the precision of a point estimator and the confidence of an interval estimator in frequentist inference. This predictive approach has implications for conditional inference, because it immediately allows a quantification of the concept of relevance for conditional inference. Conditioning on an ancillary statistic makes inference more relevant in this sense, provided that the ancillary is a precision index. Not all ancillary statistics satisfy this demand. We discuss the problem of choice between alternative ancillary statistics. The approach also has implications for the best choice of variance estimator, taking account of correlations with the squared error of estimation itself. The theory is illustrated by numerous examples, many of which are classical.     

Easy-to-construct confidence bands for comparing two simple linear regression lines

This paper shows how to construct confidence bands for the difference between two simple linear regression lines. These confidence bands provide directly the information on the magnitude of the difference between the regression lines over an interval of interest and, as a by-product, can be used as a formal test of the difference between the two regression lines. Various different shapes of confidence bands are illustrated, and particular attention is paid towards confidence bands whose construction only involves critical points from standard distributions so that they are consequently easy to construct.     

A note on new confidence intervals for the difference between two proportions based on an Edgeworth expansion

Zhou and Qin [2004. New intervals for the difference between two independent binomial proportions. J. Statist. Plann. Inference 123, 97–115; 2005. A new confidence interval for the difference between two binomial proportions of paired data. J. Statist. Plann. Inference 128, 527–542] “new confidence intervals” for the difference between two treatment proportions exhibit a severe lack of invariance property that is a compelling reason not to use them.     

DISTRIBUTION-FREE CONFIDENCE PROCEDURE FOR UMBRELLA ORDERINGS

Responses in a one-factor experiment with A; ordered treatments follow an umbrella, ordering if they consist of two piecewise monotone segments, i.e. increasing and then decreasing, or the converse. This paper proposes a non-parametric distribution-free confidence procedure for umbrella orderings, the aim being to identify the treatments that correspond to the optimal effects. It uses a method that joins the seemingly unrelated theories of U-statistics and isotonic regression. A random confidence subset of the ordered treatments is constructed, such that it contains all the unknown peaks (optimal treatments) of an umbrella ordering with any prespecified confidence level. The paper demonstrates that the proposed confidence procedure is nonparametric distribution-free and, further, that the proposed procedure naturally implies a test for umbrella alternatives. Since the proposed confidence procedure is always more informative than tests for umbrella alternatives, it should be used in their place in practice. An example illustrates the proposed procedure.     

Interval estimation for log-linear models with one variable subject to non-ignorable non-response

Summary.  Log-linear models for multiway contingency tables where one variable is subject to non-ignorable non-response will often yield boundary solutions, with the probability of non-respondents being classified in some cells of the table estimated as 0. The paper considers the effect of this non-standard behaviour on two methods of interval estimation based on the distribution of the maximum likelihood estimator. The first method relies on the estimator being approximately normally distributed with variance equal to the inverse of the information matrix. It is shown that the information matrix is singular for boundary solutions, but intervals can be calculated after a simple transformation. For the second method, based on the bootstrap, asymptotic results suggest that the coverage properties may be poor for boundary solutions. Both methods are compared with profile likelihood intervals in a simulation study based on data from the British General Election Panel Study. The results of this study indicate that all three methods perform poorly for a parameter of the non-response model, whereas they all perform well for a parameter of the margin model, irrespective of whether or not there is a boundary solution.     

含零总体中位数的经验似然

当所研究的总体中含有一定比率的零值,而其余的值大于零时,则称为含零总体。文章利用Owen的经验似然方法来构造这类总体的中位数的置信区间,所得结果不需要假设总体服从某种参数模型,并能利用样本中零值的信息,而且构造的置信区间受非零总体偏斜的影响比一些其他非参数方法小一些。同时,随机模拟的结果也显示了这一点。     

Sequential procedures based on M-estimators with discontinuous score functions

Bounded-width sequential confidence intervals and sequential tests for regression parameter based on M-estimators are extended to the case where the score-functions generating the M-estimators have jump-discontinuities. In the context of the asymptotic normality of the stopping variable, for the confidence interval problem, it is observed that the jump-discontinuities induce a slower rate of convergence. The proofs of the main theorems rest on the weak convergence of some related processes and this is also studied.