SDR Variance Estimates in Small Domains

Eric Slud; Tim Trudell

Home/Archive/arXiv

arXiv · 2608.17353

SDR Variance Estimates in Small Domains

Eric Slud·Tim Trudell

View original source ↗Read source document ↗Export metadata ↓

Abstract

Successive Difference Replication (SDR) is a replication based method of variance estimation introduced by Fay and Train (1995) for estimators based on complex multistage surveys, especially those including a final systematic sampling stage. The method has been used for many years as the primary variance-estimation methodology in large national household surveys administered by the Census Bureau, including the American Community Survey and also the Current Population Survey's monthly estimates based on self-representing strata. In settings where it is applied, generally no second method of variance estimation has been available, so the performance of SDR has been studied via simulation by various authors, for variances of survey totals and of nonlinear survey estimators. This paper begins with a thorough exposition of the SDR method and review of previously published results on the small-domain biases of SDR variance estimation. It is shown that the number D of cycles used in implementing SDR should be 3 or larger, in order to control the variability of SDR estimates, but need not be larger than 5. Beyond that, the value of D is virtually irrelevant to the occurrence of small-domain bias in SDR. The SDR method is shown via theoretical formulas and simulation to inflate average estimated variances in small domains by amounts that vary systematically with the patterns of attribute means and variances and survey weights in consecutively enumerated strata. The degree of average variance inflation is generally moderate, no more than 15 percent in domains with sample size 20, but can be larger in special settings. Moreover, SDR estimates are extremely variable in small domains, with standard deviations often far larger than any biases.

Explore related subjects

stat.ME

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Eric Slud, Tim Trudell. 2026-08-18. SDR Variance Estimates in Small Domains. https://arxiv.org/abs/2608.17353

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bias-Correction for Privacy-Protected Spatial Autoregressive Models with Application to Restaurant Network Analysis

Spatial autoregressive (SAR) models and their extensions are important tools for studying network effects. However, with an increasing emphasis on data privacy, data providers often implement protection measures that render standard SAR models inapplicable. In this study, we introduce a privacy-protected SAR model that incorporates noise into both the response and covariates to meet privacy requirements. With noise present in both components, the traditional quasi-maximum likelihood estimator becomes difficult to compute because the likelihood function cannot be directly formulated. To bypass this hurdle, we begin with a pseudo-likelihood approach, initially omitting the noise in the covariates. A Newton-Raphson algorithm is then applied to compute the estimator; however, the estimator is biased. To address this, we propose a bias-corrected Newton-Raphson-type algorithm that simultaneously accounts for noise in both the response and covariates. We further show, under appropriate regularity conditions, that the resulting estimator is consistent and asymptotically normal. To further enhance computational efficiency, we also develop a bias-corrected least squares estimator. Several extensions are discussed, and the finite-sample performance of the proposed methods is evaluated through extensive simulations. We apply the proposed methodology to restaurant transaction data from a third-party payment platform. Our method identifies a statistically significant competitive network effect among restaurants and further reveals meaningful restaurant-customer interaction patterns.

stat.ME↗

arXiv2026

A variational framework for modal estimation

Multivariate mode estimation arises in many statistical problems such as inverse problems, multimodal sampling, and density-based clustering, but becomes challenging in moderate to high dimensions, especially when the underlying density is not directly evaluable. We introduce GERVE (Gibbs-measure Entropy-Regularized Variational Estimation), a sample-based method for estimating multivariate modes by approximating Gibbs distributions directly from samples, without estimating or evaluating the density. GERVE uses Gaussian-mixture variational annealing and natural-gradient optimization, producing a mixture concentrated in high-density regions whose component responsibilities also provide a clustering of the observations. We prove theoretical guarantees in two regimes: as the Gibbs temperature goes to zero, the optimal variational mixture concentrates around the global modes of the population density; at fixed positive temperature, we prove existence, consistency, and asymptotic normality of empirical maximizers and propose a bootstrap procedure for uncertainty quantification. Simulations and a real-data experiment show that GERVE accurately recovers modes and produces meaningful clusters.

stat.ME↗

arXiv2026

Objective Model Prior Probabilities in Variable Selection

For many years it was routine to use equal model prior probabilities in Bayesian model uncertainty analysis. At least twenty years ago it became clear that this was problematic, leading to support of much too large models in the increasingly huge model spaces being considered in genomics and other fields. A popular replacement was to adopt a suggestion of Harold Jeffreys for the variable selection problem in which a total of $k$ possible variables are being considered for inclusion in the model: give the collection of all models containing $d$ variables ($d = 0, . . . , k$) prior probability $1/(k + 1)$ and then divide this prior probability equally among the models in the collection. Many other choices of model prior probabilities that impose severe parsimony have also been introduced. We begin by reviewing the problems with using equal model prior probabilities and then discuss some serious problems with the Jeffreys choice. Finally, we introduce and study a number of objective alternative choices of model prior probabilities, from both numerical and theoretical perspectives.

stat.ME↗