In this article, we propose a generalized class of exponential factor type estimators for estimation of the finite population distribution function (PDF) using an auxiliary variable in the form of the mean and rank of the auxiliary variable exist. The expressions of the bias and mean square error of the estimators are computed up to the first order approximation. The proposed estimators provide minimum mean square error as compared to all other considered estimators. Three real data sets are used to check the performance of the proposed estimators. Moreover, simulation studies are also carried out to observe the performances of the proposed estimators. The proposed estimators confirmed their superiority numerically as well as theoretically by producing efficient results as compared to all other competing estimators.

In some situations, the population of interest differs significantly in size, for example, in a medical study, the number of patients having a specific disease and the size of health units may vary. Similarly, in a survey related to the income of a household, the household may have a different number of siblings, and then in such situations, we use probability proportional to size sampling. In this article, we have proposed an improved class of estimators for the estimation of population mean on the basis of probability proportional to size (PPS) sampling, using two auxiliary variables. The mathematical expressions of the bias and mean square error (MSE) are derived up to the first order of approximation. Four real datasets and a simulation study are conducted to assess the efficiency of the improved class of estimators. It is found from the real datasets and a simulation study, that the proposed generalized class of estimators produced better results in terms of minimum MSE and higher PRE, as related to other considered estimators. An empirical study is given to support the theoretical results. The theoretical study also demonstrates that the proposed generalized class of estimators outperforms the existing estimators.

In this article, we proposed an improved finite population variance estimator based on simple random sampling using dual auxiliary information. Mathematical expressions of the proposed and existing estimators are obtained up to the first order of approximation. Two real data sets are used to examine the performances of a new improved proposed estimator. A simulation study is also recognized to assess the robustness and generalizability of the proposed estimator. From the result of real data sets and simulation study, it is examining that the proposed estimator give minimum mean square error and percentage relative efficiency are higher than all existing counterparts, which shown the importance of new improved estimator. The theoretical and numerical result illustrated that the proposed variance estimator based on simple random sampling using dual auxiliary information has the best among all existing estimators.

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