Order-restricted inference for means with missing values

Missing values appear very often in many applications, but the problem of missing values has not received much attention in testing order-restricted alternatives. Under the missing at random (MAR) assumption, we impute the missing values nonparametrically using kernel regression. For data with imputation, the classical likelihood ratio test designed for testing the order-restricted means is no longer applicable since the likelihood does not exist. This article proposes a novel method for constructing test statistics for assessing means with an increasing order or a decreasing order based on jackknife empirical likelihood (JEL) ratio. It is shown that the JEL ratio statistic evaluated under the null hypothesis converges to a chi-bar-square distribution, whose weights depend on missing probabilities and nonparametric imputation. Simulation study shows that the proposed test performs well under various missing scenarios and is robust for normally and nonnormally distributed data. The proposed method is applied to an Alzheimer's disease neuroimaging initiative data set for finding a biomarker for the diagnosis of the Alzheimer's disease.