This dissertation examines some prediction and estimations problems that arise in "high dimensions", increasingly prevalent settings characterized by the presence of a large number of observations and a large number of variables.

Chapter 1 provides an overview and briefly discusses some challenges in a large-dimensional framework.

Chapter 2 considers factor modeling, an effective tool for extracting information from large panels of data, and extends the classical linear factor analytic approach to accommodate nonlinearities, which is made possible by employing the kernel method. This chapter also establishes the theoretical guarantees, discusses the generality of the proposed approach and considers a forecasting application.

Chapter 3 explores an estimation problem in the context of group testing. It proposes a methodology that is based on $\ell_1$-norm sparse recovery which explicitly leverages the fact that the high-dimensional vector of interest is likely to be sparse in certain applications. The theoretical properties are investigated and extensive numerical simulations are provided.

Chapter 4 studies estimation of a large-dimensional covariance matrix. This chapter develops an estimator that is suitable for consistent estimation of large matrices with sparse eigenvectors and error components. It also derives the theoretical properties of the proposed method and provides a numerical experiment.