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The causal inference literature has provided a clear formal definition of confounding expressed in terms of counterfactual independence. The literature has not, however, come to any consensus on a formal definition of a confounder, as it has given priority to the concept of confounding over that of a confounder. We consider a number of candidate definitions arising from various more informal statements made in the literature. We consider the properties satisfied by each candidate definition, principally focusing on (i) whether under the candidate definition control for all \"confounders\" suffices to control for \"confounding\" and (ii) whether each confounder in some context helps eliminate or reduce confounding bias. Several of the candidate definitions do not have these two properties. Only one candidate definition of those considered satisfies both properties. We propose that a \"confounder\" be defined as a pre-exposure covariate C for which there exists a set of other covariates X such that effect of the exposure on the outcome is unconfounded conditional on (X, C) but such that for no proper subset of (X, C) is the effect of the exposure on the outcome unconfounded given the subset. We also provide a conditional analogue of the above definition; and we propose a variable that helps reduce bias but not eliminate bias be referred to as a \"surrogate confounder.\" These definitions are closely related to those given by Robins and Morgenstern [Comput. Math. Appl. 14 (1987) 869-916]. The implications that hold among the various candidate definitions are discussed.

Exclusive particles (e.g. just) express exclusivity inferences by negating focus alternatives to the sentence they modify. Grosz (2012) observes that they can sometimes give rise to what he calls minimal sufficiency readings, which seem to affirm, rather than negate, focus alternatives. Grosz proposes to analyze them in terms of the rank-order reading, a kind of scalar reading of exclusive particles that is independently attested. Coppock & Beaver (2014) put forward a similar analysis based on their unified semantics for different uses of exclusive particles. We point out that these previous accounts fail to capture the distribution of minimal sufficiency readings, in particular, the relevance of distributivity, and propose an alternative analysis where the scalar component of the minimal sufficiency reading comes from a covert version of even, rather than from the exclusive particle itself. Empirical support for this comes from the generalization that an overt even can be added to sentences that have minimal sufficiency readings without changing the meaning, but not to sentences that do not allow for minimal sufficiency readings. We argue that our account not only captures the distribution of the minimal sufficiency reading, but also derives the inferences involved in the minimal sufficiency reading compositionally together with the standardly assumed semantics for exclusive particles and even.

Recently, ultra high-throughput sequencing of RNA (RNA-Seq) has been developed as an approach for analysis of gene expression. By obtaining tens or even hundreds of millions of reads of transcribed sequences, an RNA-Seq experiment can offer a comprehensive survey of the population of genes (transcripts) in any sample of interest. This paper introduces a statistical model for estimating isoform abundance from RNA-Seq data and is flexible enough to accommodate both single end and paired end RNA-Seq data and sampling bias along the length of the transcript. Based on the derivation of minimal sufficient statistics for the model, a computationally feasible implementation of the maximum likelihood estimator of the model is provided. Further, it is shown that using paired end RNA-Seq provides more accurate isoform abundance estimates than single end sequencing at fixed sequencing depth. Simulation studies are also given.

The Rao-Blackwell theorem offers a procedure for converting a crude unbiased estimator of a parameter θ into a \"better\" one, in fact unique and optimal if the improvement is based on a minimal sufficient statistic that is complete. In contrast, behind every minimal sufficient statistic that is not complete, there is an improvable Rao-Blackwell improvement. This is illustrated via a simple example based on the uniform distribution, in which a rather natural Rao-Blackwell improvement is uniformly improvable. Furthermore, in this example the maximum likelihood estimator is inefficient, and an unbiased generalized Bayes estimator performs exceptionally well. Counterexamples of this sort can be useful didactic tools for explaining the true nature of a methodology and possible consequences when some of the assumptions are violated. [Received December 2014. Revised September 2015.]

This is the first paper dealing with the study of minimum and maximum principle sufficiency properties for nonsmooth variational inequalities by using gap functions in the setting of Hadamard manifolds. We also provide some characterizations of these two sufficiency properties. We conclude the paper with a discussion of the error bounds for nonsmooth variational inequalities in the setting of Hadamard manifolds.

In this paper, under several hypotheses and using a dual gap functional, weak sharp solutions are studied for a multidimensional variational-type inequality governed by (ρ,b,d)-convex multiple integral functional. Moreover, a relation between the minimum principle sufficiency property and weak sharpness of solutions for the considered multidimensional variational-type inequality is established.

In this paper, we consider a variational-type inequality and study its weak sharp solutions in terms of a dual gap function, under certain assumptions and convex notion of a functional. Moreover, we aim to constitute the relationship between the minimum principle sufficiency property and weak sharp solutions of the considered variational-type inequality. A numerical result is also constructed to give a better insight for the main derived result.

Sufficiency and minimal sufficiency are examined in settings in which the parameter space and the sample space are both finite, where these properties can be presented in terms of linear algebra. Minimal sufficient partitions are identified constructively. This provides direct, hands-on treatment of these topics at a level accessible to a wide audience.

This paper shows that the classes of sufficient and minimal sufficient σfields are closed under products. The results are used to construct several examples that throw some light on the study of the relationship between minimal sufficiency and invariance, a problem posed in Hall, Wijsman and Ghosh (1965).

Weak sharp type solutions are analyzed for a variational integral inequality defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided.