This dissertation consists of several tensor-related statistical problems. Chapter 2 studied non-convex constraint approach in solving generalized tensor-regression problem. A general framework was developed to analyze theoretical guarantee for applying projected gradient decent to the generalized tensor regression setting with general tensor low rankness structure. In specific cases of tensor low-rankness, implementable (approximate) projections were devised and the resulting non-convex projected gradient descent algorithms were proven to enjoy equivalent or superior statistical error rate than existing convex regularized approach. Chapter 3 designed and implemented algorithms to solve several low rank tensor regression with both non-convex constraint approach and convex regularization approach. The consistency of the empirical errors with the theoretical ones was studied, and convex and non-convex approaches were compared in simulation setting. An R package named TensorRegression was built to present the algorithms. Chapter 4 constructed novel scan statistics in use of third and fourth order moment tensor. And the extreme value type of limiting laws of such scan statistics were derived under high-dimensional setting with certain sparsity condition. Based on the resulting limiting theory, the asymptotic test for multivariate normality was first made possible in the high dimensional setting.