13 research outputs found
Wishart laws and variance function on homogeneous cones
We present a systematic study of Riesz measures and their natural exponential
families of Wishart laws on a homogeneous cone. We compute explicitly the
inverse of the mean map and the variance function of a Wishart exponential
family.Comment: 24 pages, Probab. Math. Statist (2019
Challenges in Statistical Theory: Complex Data Structures and Algorithmic Optimization
Technological developments have created a constant incoming stream of complex new data structures that need analysis. Modern statistics therefore means mathematically sophisticated new statistical theory that generates or supports innovative data-analytic methodologies for complex data structures. Inherent in many of these methodologies are challenging numerical optimization methods. The proposed workshop intends to bring together experts from mathematical statistics as well as statisticians involved in serious modern applications and computing. The primary goal of this meeting was to advance the mathematical and methodological underpinnings of modern statistics for complex data. Particular focus was given to the advancement of theory and methods under non-stationarity and complex dependence structures including (multivariate) financial time series, scientific data analysis in neurosciences and bio-physics, estimation under shape constraints, and highdimensional discrimination/classification
Modeling and Forecasting of Realized Covariance Matrices of Asset Returns using State-Space Models
This thesis comprises three self-contained essays on the modeling and prediction of realized covariance
matrices of asset returns using state-space models
On Riesz and Wishart distributions associated with decomposable undirected graphs
Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4] and [8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.Decomposable undirected graphs Exponential families Graphical models Riesz distribution Siegel integral Wishart distribution
Estimating Structured High-Dimensional Covariance and Precision Matrices: Optimal Rates and Adaptive Estimation
This is an expository paper that reviews recent developments on optimal estimation of structured high-dimensional covariance and precision matrices. Minimax rates of convergence for estimating several classes of structured covariance and precision matrices, including bandable, Toeplitz, sparse, and sparse spiked covariance matrices as well as sparse precision matrices, are given under the spectral norm loss. Data-driven adaptive procedures for estimating various classes of matrices are presented. Some key technical tools including large deviation results and minimax lower bound arguments that are used in the theoretical analyses are discussed. In addition, estimation under other losses and a few related problems such as Gaussian graphical models, sparse principal component analysis, factor models, and hypothesis testing on the covariance structure are considered. Some open problems on estimating high-dimensional covariance and precision matrices and their functionals are also discussed
On Riesz and Wishart distributions associated with decomposable undirected graphs
Classical Wishart distributions on the open convex cones of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson & WOjnar (2004a,b). This work also extends the Wishart distributions of type I in Letac & Massam (2007) and, more importantly, presents an alternative point of view on the latter paper