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Jackknife Variance Estimation

A replicate-weight variance estimation method for complex survey designs that constructs one replicate per primary sampling unit by deleting that PSU (and reweighting the rest), then estimates variance from how much the estimator changes across all the delete-one-PSU replicates.

Sampling Theory & Survey MethodsAdvancedFree

In Plain English

Jackknife variance estimation adapts the general leave-one-out jackknife idea specifically to complex survey designs. Instead of deleting one observation at a time from an ordinary dataset, this version deletes one primary sampling unit, an entire cluster, like a school or a neighborhood, at a time, reweighting the remaining units within that same stratum to compensate, and recomputes the survey estimate on what's left. Doing this once for every PSU in the sample produces a whole set of replicate estimates, and how much those replicate estimates bounce around relative to the full-sample estimate is exactly what tells you the estimator's sampling variance, with all of the design's stratification and clustering automatically baked into the procedure.

Definition

Jackknife variance estimation for complex surveys (distinct from the general delete-one jackknife used for simple i.i.d. data) constructs one replicate per primary sampling unit: for stratum $h$ with $n_{h}$ PSUs, replicate $(hk)$ deletes the $k$ -th PSU from stratum $h$ and reweights the remaining $n_{h} - 1$ PSUs in that stratum by a factor of $n_{h} / (n_{h} - 1)$ (weights outside stratum $h$ are unchanged), and the estimator $\hat{θ}_{(hk)}$ is recomputed using these adjusted replicate weights. The variance estimate sums the squared deviations of each replicate from the full-sample estimate, appropriately scaled by stratum, $Var_{J K} (\hat{θ}) = \sum_{h = 1}^{H} \frac{n _{h} - 1}{n _{h}} \sum_{k = 1}^{n_{h}} (\hat{θ}_{(hk)} - \hat{θ})^{2}$ , correctly incorporating the design's stratification and clustering structure into the resulting standard error; jackknife variance estimation is one of the three standard replicate-weight approaches (alongside balanced repeated replication and the bootstrap) used to produce publishable replicate weight sets for complex survey microdata, and unlike BRR, it does not require exactly two PSUs per stratum, working with any number of PSUs per stratum instead.

Formula

Var_{J K} (\hat{θ}) = h = 1 \sum H \frac{n _{h} - 1}{n _{h}} k = 1 \sum n_{h} (\hat{θ}_{(hk)} - \hat{θ})^{2}

Notation

\hat{θ}_{(hk)}

—Replicate estimate with the k-th PSU of stratum h deleted

n_{h}

—Number of PSUs in stratum h

H

—Number of strata

Properties

The survey jackknife's key structural difference from the general delete-one jackknife (used for simple i.i.d. data) is that it deletes and reweights at the level of the primary sampling unit within each stratum, not individual observations, correctly reflecting that PSUs, not individual final-stage observations, carry the design's actual randomization.
Unlike balanced repeated replication, which requires exactly two PSUs per stratum, the survey jackknife works with any number of PSUs per stratum (as few as two or many more), making it more broadly applicable across different survey designs, which is one reason it is a common default choice for published replicate weight sets.
The survey jackknife, like BRR, produces a number of replicates roughly proportional to the number of PSUs in the sample, for surveys with a very large number of PSUs this can mean hundreds of replicate weight columns, a computational and data-storage cost that is nonetheless usually far more tractable than deriving an explicit Taylor series linearization for every statistic of interest.

At a Glance

Mechanismdelete one PSU at a time, reweight the rest of its stratum, recompute

Number of replicatesequal to the total number of PSUs across all strata

Works withany number of PSUs per stratum, unlike BRR's exactly-two requirement

Distinct fromthe general delete-one jackknife, which operates on i.i.d. data, not PSUs

Last updated September 8, 2026← Back to the Dictionary