# 2016-Statistical Theory

Statistical Theory
Chair: Jianfeng Yao (University of Hong Kong)
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HAOSUI DUANMU, University of Toronto
Nonstandard Complete Class Theorem  [PDF]

For finite parameter spaces under finite loss, there is a close link between optimal frequentist decision procedures and Bayesian procedures: every admissible procedure is Bayes. Using nonstandard analysis, we introduce the notion of a hyperfinite statistical decision problem and study the class of nonstandard Bayesian decision procedure-namely, those whose average risk with respect to some prior is within an infinitesimal of the optimal Bayes risk. We give some sufficient regularity conditions on standard statistical decision problems that imply every admissible procedure is nonstandard Bayes, and conditions such that nonstandard Bayes procedures are in fact Bayes ones.

ROMAIN KADJE KENMOGNE, Université de Montréal
Density of the Ratio of Two Normal Random Variables  [PDF]

The random vector $(X, Y)$ has a multinormal distribution and we are looking for the distribution of the ratio $X / Y$. This issue was addressed in a long paper (T. PHAM GIA et al.) often cited on Google Scholar. The purpose of this work is to significantly reduce the length of the proofs and to explain how to numerically solve estimation problems in a Bayesian setting. We show that the density of $X / Y$ is a mixture of new densities belonging to a family. We discuss the properties of this family. Some convergence results are given.

FÉLIX CAMIRAND LEMYRE, Université de Sherbrooke
Nonparametric Measures of Local Causality and Tests of Non-Causality in Time Series  [PDF]

To study the causal relationships in a process $(Y_t,Z_t)_{t\in \mathbb{Z}}$, a widely-used approach is to consider the Granger causality. In the case of Markovian processes, the notion is based on the joint distribution of $(Y_t,Z_{t-1})$ given $Y_{t-1}$. The Granger causality measures proposed so far are global, which means that if the relationship between $Y_t$ and $Z_{t-1}$ changes with the value of $Y_{t-1}$, this will not be captured. To circumvent this limitation, local Granger causality indices based on the conditional copula of $(Y_t,Z_{t-1})$ given $Y_{t-1}=x$ are here proposed, and the asymptotic behavior of nonparametric estimators is obtained for $\alpha$-mixing processes.

VICTOR VEITCH, University of Toronto
Models and Inference for Sparse Random Graphs Using Exchangeable Random Measures  [PDF]

We introduce a class of random graphs that meets many of the desiderata for a foundation for statistical analysis of real-world networks. The class of random graphs is defined by a probabilistic symmetry: invariance of the distribution of each graph to an arbitrary relabelings of its vertices. We interpret a symmetric simple point process on $\mathbb{R}_+^2$ as the edge set of a random graph, and formalize the probabilistic symmetry as joint exchangeability of the point process. We give a representation theorem for the class of random graphs satisfying this symmetry.

JUN YANG, University of Toronto
Meta-Bayesian Analysis  [PDF]

The optimality'' of the Bayesian approach to inference does not hold when the model is misspecified. As essentially every statistical model is misspecified, this raises the question: what is a prior? We formalize the problem of choosing a (surrogate) prior as a Bayesian decision theory task, and develop theory and algorithms for choosing optimal priors. The resulting theory, which we call meta-Bayesian analysis, gives priors a pragmatic interpretation and has some surprising consequences. For example, the optimal prior may depend on the number of data points you plan to observe, and the number of predictions you expect to make.

ZHIYANG ZHOU, Simon Fraser University
A Generalized General Minimum Lower Order Confounding Criterion for Non-regular Designs  [PDF]

We extend the work of Zhang et al. [Statistica Sinica 18, 1689--1705] for non-regular designs and propose two new concepts, i.e., the generalized aliasing effect-number pattern (G$_2$-AENP) and generalized general minimum lower order confounding (G$_2$-GMC). It proves that (i) isomorphic designs have the same G$_2$-AENP and (ii) the generalized minimum aberration (GMA) and minimum moment aberration (MMA) can both be treated as ones that optimize functions over the G$_2$-AENP. That is, the G$_2$-GMC criterion is more sensitive in the recognition of non-isomorphic designs.