Study On Ratio Estimators And Inclusion Probabilities Under AP Design

In socioeconomic investigations, people need to know the statistical characteristic of study population, such as, population total or mean etc. Simple estimation is simple, but estimation precision is not high. This is especially true in the cases of missing data. Therefore,how to improve estimation accuracy of population parameter is a constant research topic.It is generally known that rational use of auxiliary information can improve survey precision. When auxiliary information can be used, unequal probability sampling design could have higher e?ciency. πPS sampling design whose inclusion probability is proportional to its size is representative of unequal probability sampling design without replacement. How to implement πPS sampling design and compute or calculate approximately inclusion probability under this sampling design is a research hotspot as well. In this paper, methods to solve the above problems are introduced mainly:Firstly, known auxiliary variables which have a positive correlation with the study variables can be always obtained in practice. This paper presents a kind of new ratio estimators of population mean on the base of exploring fully information of auxiliary variables, such as, population mean, quantile, coe?cient of kurtosis, coe?cient of skewness, coe?cient of correlation etc. These estimators have improved existing estimators by using the two aspects of information of the auxiliary variable. Their mean square errors are calculated by Taylor formula and are theoretically compared with the mean square errors(or variances) of simple estimation, the traditional ratio estimation and the previous estimations. We can obtain conditions that accuracy of these new estimations is better than the other estimators. Finally the data is simulated according to Monte Carlo method and verify the validity of these new ratio estimators further.Secondly, sampling survey often appears inevitably missing data and the causes might be context-sensitive of survey or carelessness of investigators. Problem for missing data usually adopts overlooked method or imputation methods. This paper puts forward a series of estimators of population mean in missing sample data. Proposed estimators estimate population mean by using the coe?cient of variation of auxiliary variables and their bias and mean square errors are computed by the Taylor formula. The precision is measured by the mean square error. Proposed estimators are compared with existing and classical estimations and conditions that e?ciency of the proposed estimators is higher than previous estimators are obtained. A practical example shows that proposed estimators are e?ective.Thirdly, Jens Olofsson(2011) introduced the 2PπPS sampling design, which is a kind of approximate πPS sampling design with fixed sample size. At the same time, an algorithm to calculate first-order and second-order inclusion probabilities under 2PπPS sampling design was given. Zaizai(2013) proposed another non-rejective approximate πPS sampling design with fixed sample size, which is called AP sampling design. In this paper, we propose an algorithm to calculate inclusion probabilities under AP sampling design. The proposed algorithm uses a recursive way to get exact expressions of first-order and second-order inclusion probabilities. Population mean is estimated by Horvitz-Thompson estimator and variance of this estimator is obtained under the AP sampling design. Finally, we give three real examples to compare numerically the precision of AP sampling design, other classical unequal probability sampling designs and simple random sampling. The examples show that the e?ciency of AP sampling design is better than the other sampling designs.