Inference under kernel regression imputation for missing response data is considered. An adjusted empirical likelihood approach to inference for the mean of the response variable is developed. A nonparametric version of Wilks' theorem is proved for the adjusted empirical log-likelihood ratio by showing that it has an asymptotic standard chi-squared distribution, and the corresponding empirical likelihood confidence interval for the mean is constructed. With auxiliary information, an empirical likelihood-based estimator is defined and an adjusted empirical log-likelihood ratio is derived. Asymptotic normality of the estimator is proved. Also, it is shown that the adjusted empirical log-likelihood ratio obeys Wilks' theorem. A simulation study is conducted to compare the adjusted empirical likelihood and the normal approximation methods in terms of coverage accuracies and average lengths of confidence intervals. Based on biases and standard errors, a comparision is also made by simulation between the empirical likelihoodbased estimator and related estimators. Our simulation indicates that the adjusted empirical likelihood method performs competitively and that the use of auxiliary information provides improved inferences.

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Summary A simple procedure of unequal probability sampling without replacement is proposed. It leads to an estimator of the population total having a smaller variance than is obtained by sampling with replacement. Other advantages of the present method are simplicity of calculation and the possibility of estimating exactly the variance of the estimator.

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The missing response problem in linear regression is studied. An adjusted empirical likelihood approach to inference on the mean of the response variable is developed. A nonparametric version of Wilks's theorem for the adjusted empirical likelihood is proved, and the corresponding empirical likelihood con®dence interval for the mean is constructed. With auxiliary information, an empirical likelihood-based estimator with asymptotic normality is de®ned and an adjusted empirical log-likelihood function with asymptotic v 2 is derived. A simulation study is conducted to compare the adjusted empirical likelihood methods and the normal approximation methods in terms of coverage accuracies and average lengths of the con®dence intervals. Based on biases and standard errors, a comparison is also made between the empirical likelihood-based estimator and related estimators by simulation. Our simulation indicates that the adjusted empirical likelihood methods perform competitively and the use of auxiliary information provides improved inferences.