Effective Degrees of Freedom for Balanced Repeated Replication and Paired Jackknife Variance Estimates: A Unified Approach via Stratum Contrasts

TL;DR

This paper unifies the understanding of BRR and jackknife variance estimators in stratified sampling, revealing their independence properties and deriving a practical degrees of freedom formula for confidence intervals.

Contribution

It provides a unified theoretical framework for BRR and jackknife methods, clarifying their independence structures and linking variance estimation to degrees of freedom approximation.

Findings

BRR variance estimator components are independent due to Hadamard matrix properties.

Jackknife components are inherently independent from its construction.

Derived a practical degrees of freedom formula for confidence interval estimation.

Abstract

Balanced repeated replication (BRR) and the jackknife are two widely used methods for estimating variances in stratified samples with two primary sampling units per stratum. While both methods produce variance estimators that can be expressed as sums of squared stratum-level contrasts, they differ fundamentally in their construction and in the dependence structure of their replicate estimates. This article examines the independence properties of the components contributing to these variance estimators. For BRR, we show that although the replicate estimates themselves are correlated, the balancing property of Hadamard matrices collapses the variance estimator into a sum of independent stratum-specific components. For the jackknife, the independence of components follows directly from the construction. Using these independence results, we derive the variance of each variance estimator and establish a direct connection to the Welch-Satterthwaite degrees of freedom approximation. This yields a practical formula for estimating degrees of freedom when constructing confidence intervals for population totals. The derivation highlights the unified treatment of both replication methods and provides insights into their relative efficiency and applicability.