Comparison of Risk Ratios of Shrinkage Estimators in High Dimensions

In this paper, we analyze the risk ratios of several shrinkage estimators using a balanced loss function. The James–Stein estimator is one of a group of shrinkage estimators that has been proposed in the existing literature. For these estimators, sufficient criteria for minimaxity have been established, and the James–Stein estimator’s minimaxity has been derived. We demonstrate that the James–Stein estimator’s minimaxity is still valid even when the parameter space has infinite dimension. It is shown that the positive-part version of the James–Stein estimator is substantially superior to the James–Stein estimator, and we address the asymptotic behavior of their risk ratios to the maximum likelihood estimator (MLE) when the dimensions of the parameter space are infinite. Finally, a simulation study is carried out to verify the performance evaluation of the considered estimators.

A Comparison of Model-Assisted Estimators, With and Without Data-Driven Transformations of Auxiliary Variables, With Application to Forest Inventory

Forest information is requested at many levels and for many purposes. Sampling-based national forest inventories (NFIs) can provide reliable estimates on national and regional levels. By combining expensive field plot data with different sources of remotely sensed information, from airplanes and/or satellite platforms, the precision in estimators of forest variables can be improved. This paper focuses on the design-based model-assisted approach to using NFI data together with remotely sensed data to estimate forest variables for small areas, where the variables studied are total growing stock volume, volume of Norway spruce (Picea abies), and volume of broad-leaved trees. Remote sensing variables may be highly correlated with one another and some may have poor predictive ability for target forest variables, and therefore model selection and/or coefficient shrinkage may be appropriate to improve the efficiency of model-assisted estimators of forest variables. For this purpose, one can use modern shrinkage estimators based on lasso, ridge, and elastic net regression methods. In a simulation study using real NFI data, Sentinel 2 remote-sensing data, and a national airborne laser scanning (ALS) campaign, we show that shrinkage estimators offer advantages over the (weighted) ordinary least-squares (OLS) estimator in a model-assisted setting. For example, for a sample size n of about 900 and with 72 auxiliary variables, the RMSE was up to 41% larger when based on OLS. We propose a data-driven method for finding suitable transformations of auxiliary variables, and show that it can improve estimators of forest variables. For example, when estimating volume of Norway spruce, using a smaller expert selection of auxiliary variables, transformations reduced the RMSE by up to 10%. The overall best results in terms of RMSE were obtained using shrinkage estimators and a larger set of 72 auxiliary variables. However, for this larger set of variables, the use of transformations yielded at most small improvements of RMSE, and at worst large increases of RMSE, except in combination with ridge and elastic net regression.

Climate Risks and the Realized Volatility Oil and Gas Prices: Results of an Out-Of-Sample Forecasting Experiment

We extend the widely-studied Heterogeneous Autoregressive Realized Volatility (HAR-RV) model to examine the out-of-sample forecasting value of climate-risk factors for the realized volatility of movements of the prices of crude oil, heating oil, and natural gas. The climate-risk factors have been constructed in recent literature using techniques of computational linguistics, and consist of daily proxies of physical (natural disasters and global warming) and transition (U.S. climate policy and international summits) risks involving the climate. We find that climate-risk factors contribute to out-of-sample forecasting performance mainly at a monthly and, in some cases, also at a weekly forecast horizon. We demonstrate that our main finding is robust to various modifications of our forecasting experiment, and to using three different popular shrinkage estimators to estimate the extended HAR-RV model. We also study longer forecast horizons of up to three months, and we account for the possibility that policymakers and forecasters may have an asymmetric loss function.

A Meta-Analysis for Simultaneously Estimating Individual Means with Shrinkage, Isotonic Regression and Pretests

Meta-analyses combine the estimators of individual means to estimate the common mean of a population. However, the common mean could be undefined or uninformative in some scenarios where individual means are “ordered” or “sparse”. Hence, assessments of individual means become relevant, rather than the common mean. In this article, we propose simultaneous estimation of individual means using the James–Stein shrinkage estimators, which improve upon individual studies’ estimators. We also propose isotonic regression estimators for ordered means, and pretest estimators for sparse means. We provide theoretical explanations and simulation results demonstrating the superiority of the proposed estimators over the individual studies’ estimators. The proposed methods are illustrated by two datasets: one comes from gastric cancer patients and the other from COVID-19 patients.

On shrinkage estimators improving the James-Stein estimator under balanced loss function

In this paper, we are interested in estimating a multivariate normal mean under the balanced loss function using the shrinkage estimators deduced from the Maximum Likelihood Estimator (MLE). First, we consider a class of estimators containing the James-Stein estimator, we then show that any estimator of this class dominates the MLE, consequently it is minimax. Secondly, we deal with shrinkage estimators which are not only minimax but also dominate the James- Stein estimator.

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.