We study the generalized trace regression with a near low-rank regression coefficient matrix, which extends notion of sparsity for regression coefficient vectors. Specifically, given a matrix covariate X, the probability density function of the response Y is f (Y| X)= c (Y) exp ( 1 Y + b ()), where = tr ( T X). This model accommodates various types of responses and embraces many important problem setups such as reduced-rank regression, matrix regression that accommodates a panel of regressors, matrix completion, among others. We estimate through minimizing empirical negative log-likelihood plus nuclear norm penalty. We first establish a general theory and then for each specific problem, we derive explicitly the statistical rate of the proposed estimator. They all match the minimax rates in the linear trace regression up to logarithmic factors. Numerical studies confirm the rates we established and

Consider a normal model with unknown mean bounded by a known constant. This paper deals with minimax estimation of the squared mean. We establish an expression for the asymptotic minimax risk. This result is applied in nonparametric estimation of quadratic functionals.

Variable selection is vital to statistical data analyses. Many of procedures in use are ad hoc stepwise selection procedures, which are computationally expensiveandignore stochastic errors in the variable selection process of previous steps. An automatic and simultaneous variable selection procedure can be obtained by using a penalized likelihood method. In traditional linear models, the best subset selection and stepwise deletion methods coincide with a penalized leastsquares method when design matrices are orthonormal. In this paper, we propose a few new approaches to selecting variables for linear models, robust regression models and generalized linear models based on a penalized likelihood approach. A family of thresholding functions are proposed. The LASSO proposed by Tibshirani (1996) is a member of the penalized leastsquares with the # #-penalty. A smoothly clipped absolute deviation (SCAD) penalty function is introduced to ameliorate the properties of # #-penalty. A unied algorithm is introduced, which is backed up by statistical theory. The new approaches are compared with the ordinary leastsquares methods, the garrote method by Breiman (1995) and the LASSO method by Tibshirani (1996). Our simulation results show that the newly proposed methods compare favorably with other approaches as an automatic variable selection technique. Because of simultaneous selection of variables and estimation of parameters, we are able to give a simple estimated standard error formula, which is tested to be accurate enough for practical applications. Two real data examples illustrate the versatility and eectiveness of the

When people in a society want to make inference about some parameter, each person may want to use data collected by other people. Information (data) exchange in social networks is usually costly, so to make reliable statistical decisions, people need to weigh the benefits and costs of information acquisition. Conflicts of interests and coordination problems will arise in the process. Classical statistics does not consider peoples incentives and interactions in the data-collection process. To address this imperfection, this work explores multi-agent Bayesian inference problems with a game theoretic social network model. Motivated by our interest in aggregate inference at the societal level, we propose a new concept, <i>finite population learning</i>, to address whether with high probability, a large fraction of people in a given finite population network can make good inference. Serving as a foundation, this concept enables

Statistical and machine learning theory has developed several conditions ensuring that popular estimators such as the Lasso or the Dantzig selector perform well in high-dimensional sparse regression, including the restricted eigenvalue, compatibility, and q sensitivity properties. However, some of the central aspects of these conditions are not well understood. For instance, it is unknown if these conditions can be checked efficiently on any given dataset. This is problematic, because they are at the core of the theory of sparse regression. Here we provide a rigorous proof that these conditions are NP-hard to check. This shows that the conditions are computationally infeasible to verify, and raises some questions about their practical applications. However, by taking an average-case perspective instead of the worst-case view of NP-hardness, we show that a particular condition, q sensitivity, has certain