Mathematics

We investigate Bayesian predictions for Wishart distributions by using the Kullback-Leibler divergence. We compare between the Bayesian predictive distributions based on a recently introduced class of prior distributions, called the family of enriched standard conjugate priors, which includes the Jeffreys prior, the reference prior, and the right invariant prior. We explicitly calculate the risks of Bayesian predictive distributions without using asymptotic expansions and clarify the dependency on the sizes of current and future observations. We also construct a minimax predictive distribution with a constant risk and prove this predictive distribution is not admissible.

Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e.g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry. On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms. In this paper, we propose to fill the void at the intersection between these two schools of thought in OT by proving that the entropy-regularized optimal transport problem between two Gaussian measures admits a closed form. Contrary to the unregularized case, for which the explicit form is given by the Wasserstein-Bures distance, the closed form we obtain is differentiable everywhere, even for Gaussians with degenerate covariance matrices. We obtain this closed form solution by solving the fixed-point equation behind Sinkhorn's algorithm, the default method for computing entropic regularized OT. Remarkably, this approach extends to the generalized unbalanced case -- where Gaussian measures are scaled by positive constants. This extension leads to a closed form expression for unbalanced Gaussians as well, and highlights the mass transportation / destruction trade-off seen in unbalanced optimal transport. Moreover, in both settings, we show that the optimal transportation plans are (scaled) Gaussians and provide analytical formulas of their parameters. These formulas constitute the first non-trivial closed forms for entropy-regularized optimal transport, thus providing a ground truth for the analysis of entropic OT and Sinkhorn's algorithm.

For multivariant Wright-Fisher models in population genetics, we introduce equilibrium states, expressed by fluctuations of probability ratio, in contrast to the traditionally used fluctuations, expressed by the difference between the current value of the random process and its equilibrium value. Then the drift component of the dynamic process of gene frequencies, primarily expressed as a ratio of two quadratic forms, is transformed into a cubic parabola with a certain normalization factor.

In the present paper we consider design criteria which depend on several designs simultaneously. We formulate equivalence theorems based on moment matrices (if criteria depend on designs via moment matrices) or with respect to the designs themselves (for finite design regions). We apply the obtained optimality conditions to the multiple-group random coefficient regression models and illustrate the results by simple examples.

In many applications, we are given access to noisy modulo samples of a smooth function with the goal being to robustly unwrap the samples, i.e., to estimate the original samples of the function. In a recent work, Cucuringu and Tyagi proposed denoising the modulo samples by first representing them on the unit complex circle and then solving a smoothness regularized least squares problem -- the smoothness measured w.r.t the Laplacian of a suitable proximity graph G -- on the product manifold of unit circles. This problem is a quadratically constrained quadratic program (QCQP) which is nonconvex, hence they proposed solving its sphere-relaxation leading to a trust region subproblem (TRS). In terms of theoretical guarantees, ℓ 2 error bounds were derived for (TRS). These bounds are however weak in general and do not really demonstrate the denoising performed by (TRS). In this work, we analyse the (TRS) as well as an unconstrained relaxation of (QCQP). For both these estimators we provide a refined analysis in the setting of Gaussian noise and derive noise regimes where they provably denoise the modulo observations w.r.t the ℓ 2 norm. The analysis is performed in a general setting where G is any connected graph.

Let X 1 ,…, X n be an i.i.d. sample from symmetric stable distribution with stability parameter α and scale parameter γ . Let φ n be the empirical characteristic function. We prove an uniform large deviation inequality: given preciseness ϵ>0 and probability p∈(0,1) , there exists universal (depending on ϵ and p but not depending on α and γ ) constant r ¯ >0 so that P( sup u>0:r(u)≤ r ¯ |r(u)− r ^ (u)|≥ϵ)≤p, where r(u)=(uγ ) α and r ^ (u)=−ln| φ n (u)| . As an applications of the result, we show how it can be used in estimation unknown stability parameter α .

We address the problem of estimating and comparing measures of concordance that arise as Pearson's linear correlation coefficient between two random variables transformed so that they follow the so-called concordance-inducing distribution. The class of such transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases. To answer which transformed rank correlation is best to use, we propose to compare them in terms of their best and worst asymptotic variances on a given set of copulas. A criterion derived from this approach is that concordance-inducing distributions with smaller variances of squared random variables are more preferable. In particular, we show that Blomqvist's beta is the optimal transformed rank correlation, and Spearman's rho outperforms van der Waerden's coefficient. Moreover, we find that Kendall's tau also attains the optimal asymptotic variances that Blomqvist's beta does, although Kendall's tau is not a transformed rank correlation.

We analyze the finite sample mean squared error (MSE) performance of regression trees and forests in the high dimensional regime with binary features, under a sparsity constraint. We prove that if only r of the d features are relevant for the mean outcome function, then shallow trees built greedily via the CART empirical MSE criterion achieve MSE rates that depend only logarithmically on the ambient dimension d . We prove upper bounds, whose exact dependence on the number relevant variables r depends on the correlation among the features and on the degree of relevance. For strongly relevant features, we also show that fully grown honest forests achieve fast MSE rates and their predictions are also asymptotically normal, enabling asymptotically valid inference that adapts to the sparsity of the regression function.

In real life we often deal with independent but not identically distributed observations (i.n.i.d.o), for which the most well-known statistical model is the multiple linear regression model (MLRM) without random covariates. While the classical methods are based on the maximum likelihood estimator (MLE), it is well known its lack of robustness to small deviations from the assumed conditions. In this paper, and based on the Rényi's pseudodistance (RP), we introduce a new family of estimators in case our information about the unknown parameter is given for i.n.i.d.o.. This family of estimators, let say minimum RP estimators (as they are obtained by minimizing the RP between the assumed distribution and the empirical distribution of the data), contains the MLE as a particular case and can be applied, among others, to the MLRM without random covariates. Based on these estimators, we introduce Wald-type tests for testing simple and composite null hypotheses, as an extension of the classical MLE-based Wald test. Influence functions for the estimators and Wald-type tests are also obtained and analysed. Finally, a simulation study is developed in order to asses the performance of the proposed methods and some real-life data are analysed for illustrative purpose.

We treat the change point problem in ergodic diffusion processes from discrete observations. Tonaki et al. (2020) proposed adaptive tests for detecting changes in the diffusion and drift parameters in ergodic diffusion models. When any changes are detected by this method, the next question to be considered is where the change point is. Therefore, we propose the method to estimate the change point of the parameter for two cases: the case where there is a change in the diffusion parameter, and the case where there is no change in the diffusion parameter but a change in the drift parameter. Furthermore, we present rates of convergence and distributional results of the change point estimators. Some examples and simulation results are also given.