Statistica Sinica 23 (2013), 99-118

doi:http://dx.doi.org/10.5705/ss.2010.254

THE LASSO UNDER POISSON-LIKE

HETEROSCEDASTICITY

Jinzhu Jia, Karl Rohe and Bin Yu

Peking University, University of Wisconsin-Madison and

University of California, Berkeley

Abstract: The performance of the Lasso is well understood under the assumptions of the standard sparse linear model with homoscedastic noise. However, in several applications, the standard model does not describe the important features of the data. This paper examines how the Lasso performs on a non-standard model that is motivated by medical imaging applications. In these applications, the variance of the noise scales linearly with the expectation of the observation. Like all heteroscedastic models, the noise terms in this Poisson-like model are not independent of the design matrix. Under a sparse Poisson-like model for the high-dimension regime that allows the number of predictors ($p$) $\gg$ sample size ($n$), we give necessary and sufficient conditions for the sign consistency of the Lasso estimate. Simulations reveal that the Lasso performs equally well in terms of model selection performance on both Poisson-like data and homoscedastic data (with properly scaled noise variance), across a range of parameterizations.

Key words and phrases: Heteroscedasticity, Lasso, Poisson-like model, sign consistency.