Bias in parametric estimation: reduction and useful side-effects

The bias of an estimator is defined as the difference of its expected value from the parameter to be estimated, where the expectation is with respect to the model. Loosely speaking, small bias reflects the desire that if an experiment is repeated indefinitely then the average of all the resultant estimates will be close to the parameter value that is estimated. The current paper is a review of the still-expanding repository of methods that have been developed to reduce bias in the estimation of parametric models. The review provides a unifying framework where all those methods are seen as attempts to approximate the solution of a simple estimating equation. Of particular focus is the maximum likelihood estimator, which despite being asymptotically unbiased under the usual regularity conditions, has finite-sample bias that can result in significant loss of performance of standard inferential procedures. An informal comparison of the methods is made revealing some useful practical side-effects in the estimation of popular models in practice including: i) shrinkage of the estimators in binomial and multinomial regression models that guarantees finiteness even in cases of data separation where the maximum likelihood estimator is infinite, and ii) inferential benefits for models that require the estimation of dispersion or precision parameters

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

New important developments in small area estimation

The purpose of this paper is to review and discuss some of the new important developments in small area estimation (SAE) methods. Rao (2003) wrote a very comprehensive book, which covers all the main developments in this topic until that time and so the focus of this review is on new developments in the last 7 years. However, to make the review more self contained, I also repeat shortly some of the older developments. The review covers both design based and model-dependent methods with emphasis on the prediction of the area target quantities and the assessment of the prediction error. The style of the paper is similar to the style of my previous review on SAE published in 2002, explaining the new problems investigated and describing the proposed solutions, but without dwelling on theoretical details, which can be found in the original articles. I am hoping that this paper will be useful both to researchers who like to learn more on the research carried out in SAE and to practitioners who might be interested in the application of the new methods

Selecting time-series hyperparameters with the artificial jackknife

This article proposes a generalisation of the delete-$d$ jackknife to solve hyperparameter selection problems for time series. This novel technique is compatible with dependent data since it substitutes the jackknife removal step with a fictitious deletion, wherein observed datapoints are replaced with artificial missing values. In order to emphasise this point, I called this methodology artificial delete-$d$ jackknife. As an illustration, it is used to regulate vector autoregressions with an elastic-net penalty on the coefficients. A software implementation, ElasticNetVAR.jl, is available on GitHub

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

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