Delete-group Jackknife Estimate in Partially Linear Regression Models with Heteroscedasticity

Consider a partially linear regression model with an unknown vector parameter β, an unknown function g(·), and unknown heteroscedastic error variances. Chen, You[23] proposed a semiparametric generalized least squares estimator (SGLSE) for β, which takes the heteroscedasticity into account to increase efficiency. For inference based on this SGLSE, it is necessary to construct a consistent estimator for its asymptotic covariance matrix. However, when there exists within-group correlation, the traditional delta method and the delete-1 jackknife estimation fail to offer such a consistent estimator. In this paper, by deleting grouped partial residuals a delete-group jackknife method is examined. It is shown that the delete-group jackknife method indeed can provide a consistent estimator for the asymptotic covariance matrix in the presence of within-group correlations. This result is an extension of that in [21].