General affine invariances related to Mahler volume are introduced. We establish their affine isoperimetric inequalities by using a symmetrization scheme that involves a total of 2n elaborately chosen Steiner symmetrizations at a time. The necessity of this scheme, as opposed to the usual Steiner symmetrization, is demonstrated with an example (see the Appendix).

Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges.

No abstract uploaded!

Herein, we use hybrid resampling to address (a) the long-standing problem of inference on change times and changed parameters in change-point ARX-GARCH models, and (b) the challenging problem of valid confidence intervals, after variable selection under sparsity assumptions, for the parameters in linear regression models with high-dimensional stochastic regressors and asymptotically stationary noise. For the latter problem, we introduce consistent estimators of the selected parameters and a resampling approach to overcome the inherent difficulties of post-selection confidence intervals. For the former problem, we use a sequential Monte Carlo for the latent states (representing the change times and changed parameters) of a hidden Markov model. Asymptotic efficiency theory and simulation and empirical studies demonstrate the advantages of the proposed methods.

We prove that given a three manifold with an arbitrary metric $(M^3, g)$ of positive Ricci curvature, there exists a sweepout of $M$ by surfaces of genus $\leq 3$ and areas bounded by $C vol(M^3, g)^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $C vol(M^3, g)^{1/3}$ and prove a systolic inequality for all $M \neq S^3$. The sweepout of surfaces is generated from a min-max minimal surface. If further assuming a positive scalar curvature lower bound, we can get a diameter upper bound for the min-max surface.