An official website of the United States government
Abstract When the dimension of data is comparable to or larger than the number of data samples, principal components analysis (PCA) may exhibit problematic high-dimensional noise. In this work, we propose an empirical Bayes PCA method that reduces this noise by estimating a joint prior distribution for the principal components. EB-PCA is based on the classical Kiefer–Wolfowitz non-parametric maximum likelihood estimator for empirical Bayes estimation, distributional results derived from random matrix theory for the sample PCs and iterative refinement using an approximate message passing (AMP) algorithm. In theoretical ‘spiked’ models, EB-PCA achieves Bayes-optimal estimation accuracy in the same settings as an oracle Bayes AMP procedure that knows the true priors. Empirically, EB-PCA significantly improves over PCA when there is strong prior structure, both in simulation and on quantitative benchmarks constructed from the 1000 Genomes Project and the International HapMap Project. An illustration is presented for analysis of gene expression data obtained by single-cell RNA-seq.
more »
« less
Robust Empirical Bayes Confidence Intervals
We construct robust empirical Bayes confidence intervals (EBCIs) in a normal means problem. The intervals are centered at the usual linear empirical Bayes estimator, but use a critical value accounting for shrinkage. Parametric EBCIs that assume a normal distribution for the means (Morris (1983b)) may substantially undercover when this assumption is violated. In contrast, our EBCIs control coverage regardless of the means distribution, while remaining close in length to the parametric EBCIs when the means are indeed Gaussian. If the means are treated as fixed, our EBCIs have an average coverage guarantee: the coverage probability is at least 1 − α on average across the n EBCIs for each of the means. Our empirical application considers the effects of U.S. neighborhoods on intergenerational mobility.
more »
« less
- PAR ID:
- 10389999
- Date Published:
- Journal Name:
- Econometrica
- Volume:
- 90
- Issue:
- 6
- ISSN:
- 0012-9682
- Page Range / eLocation ID:
- 2567 to 2602
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract Many large‐scale surveys collect both discrete and continuous variables. Small‐area estimates may be desired for means of continuous variables, proportions in each level of a categorical variable, or for domain means defined as the mean of the continuous variable for each level of the categorical variable. In this paper, we introduce a conditionally specified bivariate mixed‐effects model for small‐area estimation, and provide a necessary and sufficient condition under which the conditional distributions render a valid joint distribution. The conditional specification allows better model interpretation. We use the valid joint distribution to calculate empirical Bayes predictors and use the parametric bootstrap to estimate the mean squared error. Simulation studies demonstrate the superior performance of the bivariate mixed‐effects model relative to univariate model estimators. We apply the bivariate mixed‐effects model to construct estimates for small watersheds using data from the Conservation Effects Assessment Project, a survey developed to quantify the environmental impacts of conservation efforts. We construct predictors of mean sediment loss, the proportion of land where the soil loss tolerance is exceeded, and the average sediment loss on land where the soil loss tolerance is exceeded. In the data analysis, the bivariate mixed‐effects model leads to more scientifically interpretable estimates of domain means than those based on two independent univariate models.more » « less
-
Bayes estimators are well known to provide a means to incorporate prior knowledge that can be expressed in terms of a single prior distribution. However, when this knowledge is too vague to express with a single prior, an alternative approach is needed. Gamma-minimax estimators provide such an approach. These estimators minimize the worst-case Bayes risk over a set Γ of prior distributions that are compatible with the available knowledge. Traditionally, Gamma-minimaxity is defined for parametric models. In this work, we define Gamma-minimax estimators for general models and propose adversarial meta-learning algorithms to compute them when the set of prior distributions is constrained by generalized moments. Accompanying convergence guarantees are also provided. We also introduce a neural network class that provides a rich, but finite-dimensional, class of estimators from which a Gamma-minimax estimator can be selected. We illustrate our method in two settings, namely entropy estimation and a prediction problem that arises in biodiversity studies.more » « less
-
Summary Since the introduction of fiducial inference by Fisher in the 1930s, its application has been largely confined to relatively simple, parametric problems. In this paper, we present what might be the first time fiducial inference is systematically applied to estimation of a nonparametric survival function under right censoring. We find that the resulting fiducial distribution gives rise to surprisingly good statistical procedures applicable to both one-sample and two-sample problems. In particular, we use the fiducial distribution of a survival function to construct pointwise and curvewise confidence intervals for the survival function, and propose tests based on the curvewise confidence interval. We establish a functional Bernstein–von Mises theorem, and perform thorough simulation studies in scenarios with different levels of censoring. The proposed fiducial-based confidence intervals maintain coverage in situations where asymptotic methods often have substantial coverage problems. Furthermore, the average length of the proposed confidence intervals is often shorter than the length of confidence intervals for competing methods that maintain coverage. Finally, the proposed fiducial test is more powerful than various types of log-rank tests and sup log-rank tests in some scenarios. We illustrate the proposed fiducial test by comparing chemotherapy against chemotherapy combined with radiotherapy, using data from the treatment of locally unresectable gastric cancer.more » « less
-
Summary Modern empirical work in regression discontinuity (RD) designs often employs local polynomial estimation and inference with a mean square error (MSE) optimal bandwidth choice. This bandwidth yields an MSE-optimal RD treatment effect estimator, but is by construction invalid for inference. Robust bias-corrected (RBC) inference methods are valid when using the MSE-optimal bandwidth, but we show that they yield suboptimal confidence intervals in terms of coverage error. We establish valid coverage error expansions for RBC confidence interval estimators and use these results to propose new inference-optimal bandwidth choices for forming these intervals. We find that the standard MSE-optimal bandwidth for the RD point estimator is too large when the goal is to construct RBC confidence intervals with the smaller coverage error rate. We further optimize the constant terms behind the coverage error to derive new optimal choices for the auxiliary bandwidth required for RBC inference. Our expansions also establish that RBC inference yields higher-order refinements (relative to traditional undersmoothing) in the context of RD designs. Our main results cover sharp and sharp kink RD designs under conditional heteroskedasticity, and we discuss extensions to fuzzy and other RD designs, clustered sampling, and pre-intervention covariates adjustments. The theoretical findings are illustrated with a Monte Carlo experiment and an empirical application, and the main methodological results are available in R and Stata packages.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3982/ECTA18597
