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After pointing out a drawback in Bartlett's chi-square approximation, we suggest a simple modification and a Gamma approximation to improve Bartlett's M test for homogeneity of variances.

In this article, we propose an empirical likelihood-based test to check the existence of serial correlation in partial linear errors-in-variables models. A nonparametric version of Wilk' theorem is derived, which says that our proposed test has an asymptotic chi-square distribution. Simulation results reveal that the finite sample performance of our proposed test is satisfactory in both size and power.

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.

We prove that the standard EWMA mean chart with asymptotic control limits and the EWMA mean chart with time-varying control limits for monitoring mean changes in a normal process with known mean and known variance are ARL-unbiased. Using the results derived we discuss the effects of estimation of the process mean on ARL.