Inference for mixtures of symmetric distributions

This article discusses the problem of estimation of parameters in finite mixtures when the mixture components are assumed to be symmetric and to come from the same location family. We refer to these mixtures as semi-parametric because no additional assumptions other than symmetry are made regarding the parametric form of the component distributions. Because the class of symmetric distributions is so broad, identifiability of parameters is a major issue in these mixtures. We develop a notion of identifiability of finite mixture models, which we call k-identifiability, where k denotes the number of components in the mixture. We give sufficient conditions for k-identifiability of location mixtures of symmetric components when k=2 or 3. We propose a novel distance-based method for estimating the (location and mixing) parameters from a k-identifiable model and establish the strong consistency and asymptotic normality of the estimator. In the specific case of L_2-distance, we show that our estimator generalizes the Hodges--Lehmann estimator. We discuss the numerical implementation of these procedures, along with an empirical estimate of the component distribution, in the two-component case. In comparisons with maximum likelihood estimation assuming normal components, our method produces somewhat higher standard error estimates in the case where the components are truly normal, but dramatically outperforms the normal method when the components are heavy-tailed.Comment: Published at http://dx.doi.org/10.1214/009053606000001118 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rank-Based Procedures for Mixed Paired and Two-Sample Designs

This paper presents a rank-based procedure for parameter estimation and hypothesis testing when the data are a mixture of paired observations and independent samples. Such a situation may arise when comparing two treatments. When both treatments can be applied to a subject, paired data will be generated. When it is not possible to apply both treatments, the subject will be randomly assigned to one of the treatment groups. Our rank-based procedure allows us to use the data from the paired sample and the independent samples to make inferences about the difference in the mean responses. The rank-based procedure uses both types of data by combining the Wilcoxon signed-rank statistic and the Wilcoxon-Mann-Whitney statistic. The exact and asymptotic distributions of the test statistic under the null hypothesis are determined as well as the limiting distribution of the point estimate. We also consider the Pitman efficacy of our rank-based procedure and its efficiency with respect to mean-based procedures

Inference on multivariate M-estimators based on bivariate censored data

Consider two random variables subject to random right censoring, like the survival times of twins or two recurrence times of a certain disease recorded on the same individual. Of interest is the estimation of a multivariate M -functional of these two random variables. Asymptotic results for the proposed estimator are established under general conditions on the criterion function of the M-estimator. The obtained results include the asymptotic normality of the estimator, its local power and the construction of a confidence region. Our primary example of interest is the bivariate L1 median. We compare the small sample behavior of this estimator with that of the vector of marginal medians by means of a simulation study. Finally, we consider a data set on kidney dialysis patients and estimate the median time to two different infections for these individuals

Two-sample inference based on one-sample Wilcoxon signed rank statistics

A two-sample test is studied which rejects the null hypothesis of equal population medians when two Wilcoxon distribution free confidence intervals are disjoint. A confidence interval for the difference in population medians is constructed by subtracting the endpoints of two one-sample confidence intervals. Two different ways to select the one-sample intervals are presented. A solution that specifies equal confidence coefficients for the one-sample intervals is recommended. All solutions are shown to have the same asymptotic (Pitman) efficiency as the Mann-Whitney two-sample test.Mann-Whitney-Wilcoxon test nonparametric test sign test nonparametric confidence intervals notched box plots

Robust nonparametric statistical methods

One-Sample ProblemsIntroduction Location Model Geometry and Inference in the Location Model Examples Properties of Norm-Based InferenceRobustness Properties of Norm-Based Inference Inference and the Wilcoxon Signed-Rank Norm Inference Based on General Signed-Rank Norms Ranked Set Sampling L1 Interpolated Confidence Intervals Two-Sample AnalysisTwo-Sample ProblemsIntroduction Geometric MotivationExamples Inference Based on the Mann-Whitney-WilcoxonGeneral Rank Scores

Almost fully efficient and robust simultaneous estimation of location and scale parameters: A minimum distance approach

Simultaneous robust estimates of location and scale parameters are derived from minimizing a minimum distance criterion function. The criterion function measures the squared distance between the pth power (p> 0) of the empirical distribution function and the pth power of the imperfectly determined model distribution function over the real line. We show that the estimator is uniquely defined, asymptotically bivariate normal and has positive breakdown. When p = 0.56 the estimator is almost fully efficient at the normal model. Efficiencies are 0.9999 and 0.9998 for the location and scale parameters, respectively. Some other choices of p values produce highly efficient and robust estimates as well. It is shown that the location estimator has maximum breakdown point 0.5 independent of p when the scale is known. If the true and target models are both symmetric, then the location estimator is consistent for the center of the symmetry even if the true model is imperfectly determined.Cramer-von Mises distance Robustness Breakdown point Efficiency