The relative case fatality rates (CFRs) between groups and countries are key measures of relative risk that guide policy decisions regarding scarce medical resource allocation during the ongoing COVID-19 pandemic. In the middle of an active outbreak when surveillance data is the primary source of information, estimating these quantities involves compensating for competing biases in time series of deaths, cases, and recoveries. These include time- and severity- dependent reporting of cases as well as time lags in observed patient outcomes. In the context of COVID-19 CFR estimation, we survey such biases and their potential significance. Further, we analyze theoretically the effect of certain biases, like preferential reporting of fatal cases, on naive estimators of CFR. We provide a partially corrected estimator of these naive estimates that accounts for time lag and imperfect reporting of deaths and recoveries. We show that collection of randomized data by testing the contacts of infectious individuals regardless of the presence of symptoms would mitigate bias by limiting the covariance between diagnosis and death. Our analysis is supplemented by theoretical and numerical results and a simple and fast open-source codebase at https://github.com/aangelopoulos/cfr-covid-19 .

No abstract

No abstract

We study the covariate shift problem in the context of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We focus on two natural families of covariate shift problems defined using the likelihood ratios between the source and target distributions. When the likelihood ratios are uniformly bounded, we prove that the kernel ridge regression (KRR) estimator with a carefully chosen regularization parameter is minimax rate-optimal (up to a log factor) for a large family of RKHSs with regular kernel eigenvalues. Interestingly, KRR does not require full knowledge of the likelihood ratio apart from an upper bound on it. In striking contrast to the standard statistical setting without covariate shift, we also demonstrate that a naïve estimator, which minimizes the empirical risk over the function class, is strictly suboptimal under covariate shift as compared to KRR. We then address the larger class of covariate shift problems where likelihood ratio is possibly unbounded yet has a finite second moment. Here, we show via careful simulations that KRR fails to attain the optimal rate. Instead, we propose a reweighted KRR estimator that weights samples based on a careful truncation of the likelihood ratios. Again, we are able to show that this estimator is minimax optimal, up to logarithmic factors.