Jackknife Variance Estimation

There are two basic approaches to estimation of the variance for survey data: the Taylor linearization method and the resampling method. The resampling method includes the jackknife, balanced repeated replication (Fay's method as a variant), and bootstrap methods. The resampling method calls for creation of many replicate samples (“replicates” in short) taken from the original sample (called also the full or parent sample). Each resampling method uses a unique way of creating the replicates. Each replicate provides a point estimate of the population parameter of interest and the variability among the replicate estimates forms the basis of estimating the variance of the point estimate.

Let θ be the population parameter to be estimated from the sample and let R replicates be created, from which R replicate point estimates, j, j =1,2,…,R of θ are obtained. Then the jackknife variance estimator is given by , where Cj are scaling constants to correct the bias and is the point estimate based on the full sample.

A special case of the jackknife variance estimator for the sample mean under simple random sampling is a helpful way to understand the idea behind the method. In this case the maximum number of replicates that can be created in a replicate sample is formed by deleting one sample unit at a time—the number of replicates is equal to the sample size n. Then the jackknife variance formula is given by where f = n/N is the finite population correction, N is the size of the finite population under study, is the sample mean of the j-th replicate sample, and the sample mean for the full sample. Note that in this special case, cj = (1 − f) (n − l)/n for all j = 1,2,3,…,n. It is not difficult to show that the variance estimator is equivalent to the usual variance estimation formula . Therefore, the jackknife variance estimator is an unbiased variance estimator for this special case.

However, even when the point estimate is a complex one, such as a ratio estimate, the jackknife variance estimator still gives approximately correct variance estimate if the sample size is large (the technical term for this property is consistency). The same is true for other point estimates that are denned in a smooth functional form of sample (weighted) totals or averages—survey weights should be used especially when unequal probability sampling has been used. Such point estimates for which the jackknife is consistent include the ratio, regression coefficient, and correlation coefficient, but not the median or, more generally, quantiles.

When the sample design is complex with stratification and clustering (in one or multi-stage), estimation of the variance of a survey estimate is not simple. The usual approach is to assume that the primary sampling units (PS Us) have been selected with replacement within strata, although replacement sampling is seldom used in practice. Then the variance estimator can be written in a form of PSU level aggregates of the variables involved in the definition of the point [Page 404][Page 405]estimate. This approach easily incorporates the cluster effect on the variance estimate. So it simplifies the variance estimation substantially. However, the price of this simplification is that the variance estimate is generally overestimated as a result of the assumption of replacement sampling of the PSUs. The overestimation is not serious if the PSU level sampling fraction is small, and mild overestimation is generally accepted—this leads to mildly conservative results when the variance estimate is used in statistical inference. However, if the sampling fraction is not small, the overestimation can be substantial, and incorporation of the finite population correction (fpc) may be helpful to reduce the overestimation. (This issue applies not only to the jackknife method but also to the Taylor method.)

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