Estimation of Finite Population Mean and Superpopulation Parameters when the Sampling Design is Informative and Nonresponse Mechanism is Nonignorable

In this paper we study the joint treatment of not missing at random response mechanism and informative sampling for survey data. This is the most general situation in surveys and other combinations of sampling informativeness and response mechanisms can be considered as special cases. The proposed method combines two methodologies used in the analysis of sample surveys for the treatment of informative sampling and the nonignorable nonresponse mechanism. One incorporates the dependence of the first order inclusion probabilities on the study variable, while the other incorporates the dependence of the probability of nonresponse on unobserved or missing observations. The main purpose here is the estimation of finite population mean and superpopulation parameters when the sampling design is informative and nonresponse mechanism is nonignorable. Under four scenarios of sampling design and nonresponse mechanism, we obtained the method of moment estimators of finite population mean, with their biases and mean square errors. Furthermore, a four-step estimation method is introduced for the estimation of superpopulation parameters under informative sampling and nonignorable nonresponse mechanism. New relationships between moments of response, nonresponse, sample, sample-complement and population distributions were derived. Most estimators for finite population mean known from sampling surveys can be derived as a special case of the results derived in this paper