A Family of Estimators of Finite-Population Distribution Function Using Auxiliary Information

Abstract

This paper considers the problem of estimating the finite-population distribution function and quantiles with the use of auxiliary information at the estimation stage of a survey. We propose the families of estimators of the distribution function of the study variate y using the knowledge of the distribution function of the auxiliary variate x. In addition to ratio, product and difference type estimators, many other estimators are identified as members of the proposed families. For these families the approximate variances are derived, and in addition, the optimum estimator is identified along with its approximate variance. Estimators based on the estimated optimum values of the unknown parameters used to minimize the variance are also given with their properties. Further, the family of estimators of a finitepopulation distribution function using two-phase sampling is given, and its properties are investigated.

Optimum design-based ratio estimators of the distribution function

“…This method consists on taking a large preliminary sample in which the auxiliary characteristic alone is measured, and the control parameter for the auxiliary information is thus estimated with this information. The use of this method can make the estimates more precise than those without the use of such information (see, for example, [18]). …”

A quantile estimator under two-phase sampling for stratification

“…Many quantile estimators can be found in the literature (see, for example, [1][2][3][4][5][6][7][8][9][10]15]), and most of them assume that the values of an auxiliary variable are known for the entire population. However, this situation is not very common in practice.…”

“…However, this situation is not very common in practice. For this reason, Singh et al [15] defined quantile estimators under two-phase sampling. However, such estimators discuss the estimation of the population median, and the extension to the estimation of a general population quantile has not been studied.…”

“…Moreover, these estimators cannot be applied under a general sampling design, since they were defined under simple random sampling without replacement (SRSWOR). An alternative to the estimators defined by Singh et al [15] is to use two-phase sampling for stratification.…”

“…The asymptotic variance and the quantile estimator are also derived in this section. In Section 4, the proposed estimators are compared numerically via simulation studies with the stratification estimator and its variance estimator [15], which are introduced in Section 3. Results derived from simulation studies show that the proposed estimators are more accurate than the stratification estimators, especially in the problem of estimating variances.…”