Brownian distance covariance

Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but generalize and extend these classical bivariate measures of dependence. Distance correlation characterizes independence: it is zero if and only if the random vectors are independent. The notion of covariance with respect to a stochastic process is introduced, and it is shown that population distance covariance coincides with the covariance with respect to Brownian motion; thus, both can be called Brownian distance covariance. In the bivariate case, Brownian covariance is the natural extension of product-moment covariance, as we obtain Pearson product-moment covariance by replacing the Brownian motion in the definition with identity. The corresponding statistic has an elegantly simple computing formula. Advantages of applying Brownian covariance and correlation vs the classical Pearson covariance and correlation are discussed and illustrated.Comment: This paper discussed in: [arXiv:0912.3295], [arXiv:1010.0822], [arXiv:1010.0825], [arXiv:1010.0828], [arXiv:1010.0836], [arXiv:1010.0838], [arXiv:1010.0839]. Rejoinder at [arXiv:1010.0844]. Published in at http://dx.doi.org/10.1214/09-AOAS312 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

The bootstrap -A review

The bootstrap, extensively studied during the last decade, has become a powerful tool in different areas of Statistical Inference. In this work, we present the main ideas of bootstrap methodology in several contexts, citing the most relevant contributions and illustrating with examples and simulation studies some interesting aspects

Jackknife empirical likelihood: small bandwidth, sparse network and high-dimension asymptotic

This paper sheds light on inference problems for statistical models under alternative or nonstandard asymptotic frameworks from the perspective of jackknife empirical likelihood. Examples include small bandwidth asymptotics for semiparametric inference and goodness-of- fit testing, sparse network asymptotics, many covariates asymptotics for regression models, and many-weak instruments asymptotics for instrumental variable regression. We first establish Wilks’ theorem for the jackknife empirical likelihood statistic on a general semiparametric in- ference problem under the conventional asymptotics. We then show that the jackknife empirical likelihood statistic may lose asymptotic pivotalness under the above nonstandard asymptotic frameworks, and argue that these phenomena are understood as emergence of Efron and Stein’s (1981) bias of the jackknife variance estimator in the first order. Finally we propose a modi- fication of the jackknife empirical likelihood to recover asymptotic pivotalness under both the conventional and nonstandard asymptotics. Our modification works for all above examples and provides a unified framework to investigate nonstandard asymptotic problems

Jackknife Emperical Likelihood Method and its Applications

In this dissertation, we investigate jackknife empirical likelihood methods motivated by recent statistics research and other related fields. Computational intensity of empirical likelihood can be significantly reduced by using jackknife empirical likelihood methods without losing computational accuracy and stability. We demonstrate that proposed jackknife empirical likelihood methods are able to handle several challenging and open problems in terms of elegant asymptotic properties and accurate simulation result in finite samples. These interesting problems include ROC curves with missing data, the difference of two ROC curves in two dimensional correlated data, a novel inference for the partial AUC and the difference of two quantiles with one or two samples. In addition, empirical likelihood methodology can be successfully applied to the linear transformation model using adjusted estimation equations. The comprehensive simulation studies on coverage probabilities and average lengths for those topics demonstrate the proposed jackknife empirical likelihood methods have a good performance in finite samples under various settings. Moreover, some related and attractive real problems are studied to support our conclusions. In the end, we provide an extensive discussion about some interesting and feasible ideas based on our jackknife EL procedures for future studies

One and One-Half Bound Dichotomous Choice Contingent Valuation

To reduce the potential for response bias on the follow-up bid in multiple-bound discrete choice CVM questions while maintaining much of the efficiency gains of the multiple-bound approach, we introduce the one-and-one-half-bound (OOHB) approach. Despite the fact that the OOHB model uses less information than the double-bound (DB) approach, efficiency gains in moving from single-bound to OOHB capture a large portion of the gain associated with moving from single-bound to DB. In an analysis of survey data, our OOHB estimates demonstrated higher consistency with respect to the follow-up data than the DB estimates and were more efficient as well.contingent valuation, double-bound, one-and-one-half bound

JAWS: Predictive Inference Under Covariate Shift

We propose \textbf{JAWS}, a series of wrapper methods for distribution-free uncertainty quantification tasks under covariate shift, centered on our core method \textbf{JAW}, the \textbf{JA}ckknife+ \textbf{W}eighted with likelihood-ratio weights. JAWS also includes computationally efficient \textbf{A}pproximations of JAW using higher-order influence functions: \textbf{JAWA}. Theoretically, we show that JAW relaxes the jackknife+'s assumption of data exchangeability to achieve the same finite-sample coverage guarantee even under covariate shift. JAWA further approaches the JAW guarantee in the limit of either the sample size or the influence function order under mild assumptions. Moreover, we propose a general approach to repurposing any distribution-free uncertainty quantification method and its guarantees to the task of risk assessment: a task that generates the estimated probability that the true label lies within a user-specified interval. We then propose \textbf{JAW-R} and \textbf{JAWA-R} as the repurposed versions of proposed methods for \textbf{R}isk assessment. Practically, JAWS outperform the state-of-the-art predictive inference baselines in a variety of biased real world data sets for both interval-generation and risk-assessment auditing tasks