Optimal design theory for nonlinear regression studies local optimality on a given design space. We identify designs for the Bradley-Terry paired comparison model with small undirected graphs and prove that every saturated, locally D-optimal design is represented by a path. We discuss the case of four alternatives in detail and derive explicit polynomial inequality descriptions for optimality regions in parameter space. Using these regions, for each point in parameter space we can prescribe a locally D-optimal design.
Positive dependence is present in many real world data sets and has appealing stochastic properties. In particular, the notion of multivariate total positivity of order 2 (MTP 2 ) is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce EMTP 2 , an extremal version of MTP 2 . This notion turns out to appear prominently in extremes and, in fact, it is satisfied by many classical models. For a Hüsler-Reiss distribution, the analogue of a Gaussian distribution in extremes, we show that it is EMTP 2 if and only if its precision matrix is a Laplacian of a connected graph. We propose an estimator for the parameters of the Hüsler-Reiss distribution under EMTP 2 as the solution of a convex optimization problem with Laplacian constraint. We prove that this estimator is consistent and typically yields a sparse model with possibly non-decomposable extremal graphical structure. At the example of two data sets, we illustrate this regularization and the superior performance compared to existing methods.
This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was listed as an interesting direction by Chatterjee and Diaconis (2017). For that purpose, we generalize their result for the asymptotic normality of the number of descents in a random permutation and its inverse to other finite reflection groups. This is achieved by applying their proof scheme to signed permutations, i.e. elements of Coxeter groups of type $ \mathtt{B}_n $, which are also known as the hyperoctahedral groups. Furthermore, a similar central limit theorem for elements of Coxeter groups of type $\mathtt{D}_n$ is derived via Slutsky's Theorem and a bound on the Wasserstein distance of certain normalized statistics with local dependency structures and bounded local components is proven for both types of Coxeter groups. In addition, we show a two-dimensional central limit theorem via the Cramér-Wold device.
We study the asymptotic behaviour of the statistic $(\operatorname{des}+\operatorname{ides})_W$ which assigns to an element $w$ of a finite Coxeter group $W$ the number of descents of $w$ plus the number of descents of $w^{-1}$. Our main result is a central limit theorem for the probability distributions associated to this statistic. This answers a question of Kahle-Stump and builds upon work of Chatterjee-Diaconis, Özdemir and Röttger.
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