A Note on the Large Sample Properties of Linearization, Jackknife and Balanced Repeated Replication Methods for Stratified Samples

Abstract

Krewski and Rao consider inference for a (nonlinear) function of a vector of finite population means $\theta = g(\bar{Y}).$ For a sequence of finite populations with increasing number of strata, they demonstrate that $\hat{\theta} = g(\bar{y})$ is asymptotically normal, where $\bar{y}$ is the usual unbiased stratified estimator of $\bar{Y}.$ Additionally, they demonstrate that $(\hat{\theta} - \theta)/\upsilon^{1/2}(\hat{\theta})$ is asymptotically a standard normal distribution, where $\upsilon(\hat{\theta})$ is a variance estimator obtained using linearization, jackknife or balanced repeated replication (BRR) methods. In this note we extend their results to when the partial first derivatives $(g_1(\mu), g_2(\mu),\ldots, g_p(\mu)) \equiv 0,$ where $\mu$ is the limit of $\bar{Y}$ with increasing number of strata. We explore the asymptotic distribution of $(\hat{\theta} - \theta)/\upsilon^{1/2}(\hat{\theta})$ and show (1) that it is no longer normal and (2) that it depends upon which variance estimator is used. We describe an application of these results to hypothesis testing using complex survey data.