Evaluation and selection of models for out-of-sample prediction when the sample size is small relative to the complexity of the data-generating process

In regression with random design, we study the problem of selecting a model that performs well for out-of-sample prediction. We do not assume that any of the candidate models under consideration are correct. Our analysis is based on explicit finite-sample results. Our main findings differ from those of other analyses that are based on traditional large-sample limit approximations because we consider a situation where the sample size is small relative to the complexity of the data-generating process, in the sense that the number of parameters in a `good' model is of the same order as sample size. Also, we allow for the case where the number of candidate models is (much) larger than sample size.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ127 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Conditional predictive inference post model selection

We give a finite-sample analysis of predictive inference procedures after model selection in regression with random design. The analysis is focused on a statistically challenging scenario where the number of potentially important explanatory variables can be infinite, where no regularity conditions are imposed on unknown parameters, where the number of explanatory variables in a "good" model can be of the same order as sample size and where the number of candidate models can be of larger order than sample size. The performance of inference procedures is evaluated conditional on the training sample. Under weak conditions on only the number of candidate models and on their complexity, and uniformly over all data-generating processes under consideration, we show that a certain prediction interval is approximately valid and short with high probability in finite samples, in the sense that its actual coverage probability is close to the nominal one and in the sense that its length is close to the length of an infeasible interval that is constructed by actually knowing the "best" candidate model. Similar results are shown to hold for predictive inference procedures other than prediction intervals like, for example, tests of whether a future response will lie above or below a given threshold.Comment: Published in at http://dx.doi.org/10.1214/08-AOS660 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The distribution of a linear predictor after model selection: Unconditional finite-sample distributions and asymptotic approximations

We analyze the (unconditional) distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. First, we derive the exact finite-sample cumulative distribution function (cdf) of the linear predictor, and a simple approximation to this (complicated) cdf. We then analyze the large-sample limit behavior of these cdfs, in the fixed-parameter case and under local alternatives.Comment: Published at http://dx.doi.org/10.1214/074921706000000518 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Distribution of the Adaptive LASSO Estimator

We study the distribution of the adaptive LASSO estimator (Zou (2006)) in finite samples as well as in the large-sample limit. The large-sample distributions are derived both for the case where the adaptive LASSO estimator is tuned to perform conservative model selection as well as for the case where the tuning results in consistent model selection. We show that the finite-sample as well as the large-sample distributions are typically highly non-normal, regardless of the choice of the tuning parameter. The uniform convergence rate is also obtained, and is shown to be slower than $n^{-1/2}$ in case the estimator is tuned to perform consistent model selection. In particular, these results question the statistical relevance of the `oracle' property of the adaptive LASSO estimator established in Zou (2006). Moreover, we also provide an impossibility result regarding the estimation of the distribution function of the adaptive LASSO estimator.The theoretical results, which are obtained for a regression model with orthogonal design, are complemented by a Monte Carlo study using non-orthogonal regressors.Comment: revised version; minor changes and some material adde

Mystified Consciousness: Rethinking the Rise of the Far Right with Marx and Lacan

Why did the white working classes in the United States and elsewhere turn to the far right instead of uniting with the raced and gendered working class to overthrow capitalism? In this paper, I bring core concepts coined by Karl Marx in conversation with Jacques Lacan to show how the far-right exploited desires and fears around subjects' fundamental non-wholeness, which the insecurities of neo-liberal capitalism have heightened, for its political gain. I explain how the far-right offered its followers several unconscious fantasy objects petit a to cover over subjects' non-wholeness: first, the money fetish, which is also at the center of the American Dream, serves to secure the illusion of wholeness on earth; second, religion offers the illusion of wholeness in the sky, producing subjects who endure rather than rebel against their suffering on earth. Finally, it brands the sexed and raced working classes as inferior to uphold the illusion of the white working-class subjects as superior and whole, which further undermines the creation of a revolutionary proletariat

Method and apparatus for splitting a beam of energy

A wedge shaped beam splitting device is described which has a first surface for splitting an incident beam energy into an externally reflected beam and an internally transmitted beam, a second surface spaced from the first surface splits the internally transmitted beam into an externally transmitted beam and into an internally reflected beam, and intersects the first surface at an angle that impinges the internally transmitted beam on the second surface at an angle of incidence that is less than the minimum angle necessary for substantially total internal reflection and impinges the internally reflected beam on the first surface at an angle of incidence that exceeds the minimum angle necessary for substantially total internal reflection. The device may also be used as a beam combiner