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This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modified likelihood ratio statistic has χ 2 2 null limiting distribution.

Empirical likelihood (EL) ratio statistic on θ = g(x) is constructed based on the inverse probability weighted imputation approach in a nonparametric regression model Y = g(x) + ε (x ∈ [0, 1] p ) with fixed designs and missing responses, which asymptotically has χ 2 1 distribution. This result is used to obtain a EL based confidence interval on θ.

This paper considers two estimators of θ = g(x) in a nonparametric regression model Y = g(x) + ε(x ∈ (0, 1) p ) with missing responses: Imputation and inverse probability weighted estimators. Asymptotic normality of the two estimators is established, which is used to construct normal approximation based confidence intervals on θ.