STRETCHING THE NET: MULTIDIMENSIONAL REGULARIZATION

This paper derives asymptotic risk (expected loss) results for shrinkage estimators with multidimensional regularization in high-dimensional settings. We introduce a class of multidimensional shrinkage estimators (MuSEs), which includes the elastic net, and show that—as the number of parameters to estimate grows—the empirical loss converges to the oracle-optimal risk. This result holds when the regularization parameters are estimated empirically via cross-validation or Stein’s unbiased risk estimate. To help guide applied researchers in their choice of estimator, we compare the empirical Bayes risk of the lasso, ridge, and elastic net in a spike and normal setting. Of the three estimators, we find that the elastic net performs best when the data are moderately sparse and the lasso performs best when the data are highly sparse. Our analysis suggests that applied researchers who are unsure about the level of sparsity in their data might benefit from using MuSEs such as the elastic net. We exploit these insights to propose a new estimator, the cubic net, and demonstrate through simulations that it outperforms the three other estimators for any sparsity level.

Asymptotic Risk of Bézier Simplex Fitting

The B'ezier simplex fitting is a novel data modeling technique which utilizes geometric structures of data to approximate the Pareto set of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from skeletons of a simplex, whereas the all-at-once fitting uses a non-stratified sampling which treats a simplex as a single object. In this paper, we analyze the asymptotic risks of those B'ezier simplex fitting methods and derive the optimal subsample ratio for the inductive skeleton fitting. It is shown that the inductive skeleton fitting with the optimal ratio has a smaller risk when the degree of a B'ezier simplex is less than three. Those results are verified numerically under small to moderate sample sizes. In addition, we provide two complementary applications of our theory: a generalized location problem and a multi-objective hyper-parameter tuning of the group lasso. The former can be represented by a B'ezier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a B'ezier simplex of degree three where the all-at-once fitting gets an advantage.

On uniform asymptotic risk of averaging GMM estimators

This paper studies the averaging GMM estimator that combines a conservative GMM estimator based on valid moment conditions and an aggressive GMM estimator based on both valid and possibly misspecified moment conditions, where the weight is the sample analog of an infeasible optimal weight. We establish asymptotic theory on uniform approximation of the upper and lower bounds of the finite‐sample truncated risk difference between any two estimators, which is used to compare the averaging GMM estimator and the conservative GMM estimator. Under some sufficient conditions, we show that the asymptotic lower bound of the truncated risk difference between the averaging estimator and the conservative estimator is strictly less than zero, while the asymptotic upper bound is zero uniformly over any degree of misspecification. The results apply to quadratic loss functions. This uniform asymptotic dominance is established in non‐Gaussian semiparametric nonlinear models.

A Critical Review Of The Basel Margin Of Conservatism Requirement In A Retail Credit Context

The Basel II accord (2006) includes guidelines to financial institutions for the estimation of regulatory capital (RC) for retail credit risk. Under the advanced Internal Ratings Based (IRB) approach, the formula suggested for calculating RC is based on the Asymptotic Risk Factor (ASRF) model, which assumes that a borrower will default if the value of its assets were to fall below the value of its debts. The primary inputs needed in this formula are estimates of probability of default (PD), loss given default (LGD) and exposure at default (EAD). Banks for whom usage of the advanced IRB approach have been approved usually obtain these estimates from complex models developed in-house. Basel II recognises that estimates of PDs, LGDs, and EADs are likely to involve unpredictable errors, and then states that, in order to avoid over-optimism, a bank must add to its estimates a margin of conservatism (MoC) that is related to the likely range of errors. Basel II also requires several other measures of conservatism that have to be incorporated. These conservatism requirements lead to confusion among banks and regulators as to what exactly is required as far as a margin of conservatism is concerned. In this paper, we discuss the ASRF model and its shortcomings, as well as Basel II conservatism requirements. We study the MoC concept and review possible approaches for its implementation. Our overall objective is to highlight certain issues regarding shortcomings inherent to a pervasively used model to bank practitioners and regulators and to potentially offer a less confusing interpretation of the MoC concept.