On Mean Estimation with Imputation in Two- Phase Sampling Design

A sample survey remains incomplete in presence of missing data and one of the substitution techniques of missing observations is known as imputation. A number of imputation methods are available in literature using auxiliary information, for example, Mean method of imputation, Ratio method of imputation, Compromised method of imputation and so on. These suggested methods are based on either population parameter of auxiliary variable or available information (both study and auxiliary variable) in the sample. Also, the number of available observations is considered as a constant but practically, it is not possible, the missing values may vary from sample to sample i.e. it may be considered as random variable. If population mean of auxiliary variable is unknown, then all these methods fail to perform. In such situations the idea of two-phase sampling is used for estimating population parameters. This paper presents the estimation of mean in presence of missing data under two-phase sampling scheme while the numbers of available observations are considered as random variable. The bias and m.s.e of suggested estimators are derived in the form of population parameters using the concept of large sample approximation. Numerical study is performed over two populations using the expressions of bias and m.s.e and efficiency compared with existing estimators.