Nonparametric inference for linear models via an analytic framework for the wild bootstrap

Abstract

Many standard approaches for conducting statistical inference on regression parameters rely heavily on parametric assumptions and asymptotic results. The wild bootstrap (Mammen, 1993) was developed as a nonparametric means to estimate a sampling distribution and is particularly useful when conducting statistical inference for linear models. With wide-reaching applications, the wild bootstrap can be used in a variety of settings where distributional assumptions are violated or are difficult to verify. While the wild bootstrap has attractive properties, as big-data becomes increasingly prevalent in society computationally intensive resampling schemes such as bootstrapping become less appealing and impractical. In this work, an analytic framework for computing confidence regions and intervals in a variety of linear models is developed. The use of the concentration of measure phenomenon paired with the appealing properties of the wild bootstrap leads to a more computationally efficient, nonparametric way to perform statistical inference for regression parameters. The methodology is first introduced for the coefficients in least squares regression, and is then adapted to consider the more complex settings of ridge and LASSO regression. Lastly, this analytic approach is discussed in the context of generalized linear models, focusing on the case of overdispersion in Poisson regression.