Sparse Multivariate Factor Regression

We consider the problem of multivariate regression in a setting where the relevant predictors could be shared among different responses. We propose an algorithm which decomposes the coefficient matrix into the product of a long matrix and a wide matrix, with an elastic net penalty on the former and an $\ell_1$ penalty on the latter. The first matrix linearly transforms the predictors to a set of latent factors, and the second one regresses the responses on these factors. Our algorithm simultaneously performs dimension reduction and coefficient estimation and automatically estimates the number of latent factors from the data. Our formulation results in a non-convex optimization problem, which despite its flexibility to impose effective low-dimensional structure, is difficult, or even impossible, to solve exactly in a reasonable time. We specify an optimization algorithm based on alternating minimization with three different sets of updates to solve this non-convex problem and provide theoretical results on its convergence and optimality. Finally, we demonstrate the effectiveness of our algorithm via experiments on simulated and real data

The role of statistical methodology in simulation

statistical methods;simulation;operations research

The Dirichet-Multinomial model for multivariate randomized response data and small samples

In survey sampling the randomized response (RR) technique can be used to obtain truthful answers to sensitive questions. Although the individual answers are masked due to the RR technique, individual (sensitive) response rates can be estimated when observing multivariate response data. The beta-binomial model for binary RR data will be generalized to handle multivariate categorical RR data. The Dirichlet-multinomial model for categorical RR data is extended with a linear transformation of the masked individual categorical-response rates to correct for the RR design and to retrieve the sensitive categorical-response rates even for small data samples. This specification of the Dirichlet-multinomial model enables a straightforward empirical Bayes estimation of the model parameters. A constrained-Dirichlet prior will be introduced to identify homogeneity restrictions in response rates across persons and/or categories. The performance of the full Bayes parameter estimation method is verified using simulated data. The proposed model will be applied to the college alcohol problem scale study, where students were interviewed directly or interviewed via the randomized response technique about negative consequences from drinking. (Contains 5 tables.