Approximation of Finite Population Totals Using Lagrange Polynomial

Lamin Kabareh,Thomas Mageto,Benjamin Muema

doi:10.4236/ojs.2017.74048

Abstract

Approximation of finite population totals in the presence of auxiliary information is considered. A polynomial based on Lagrange polynomial is proposed. Like the local polynomial regression, Horvitz Thompson and ratio estimators, this approximation technique is based on annual population total in order to fit in the best approximating polynomial within a given period of time (years) in this study. This proposed technique has shown to be unbiased under a linear polynomial. The use of real data indicated that the polynomial is efficient and can approximate properly even when the data is unevenly spaced.

Highlights

This study is using an approximation technique to approximate the finite population total called the Lagrange polynomial that doesn’t require any selection of bandwidth as in the case of local polynomial regression estimator
Like the local polynomial regression, Horvitz Thompson and ratio estimators, this approximation technique is based on annual population total in order to fit in the best approximating polynomial within a given period of time in this study
The chart in (Figure 2) below comprises of two linear polynomials that have uniformly approximated the function in green in order to give a better approximate to the population total in 2019

Summary

Introduction

This study is using an approximation technique to approximate the finite population total called the Lagrange polynomial that doesn’t require any selection of bandwidth as in the case of local polynomial regression estimator. In 1940, Cochran made an important contribution to the modern sampling theory by suggesting methods of using the auxiliary information for the purpose of estimation in order to increase the precision of the estimates [2] He developed the ratio estimator to estimate the population mean or the total of the study variable y. The most common way of defining a more efficient class of estimators than usual ratio (product) and sample mean estimator is to include one or more unknown parameters in the estimators whose optimum choice is made by minimizing the corresponding mean square error or variance Sometimes, such modifications or generalizations are made by mixing two or more estimators with unknown weights whose optimum values are determined which generally depend upon population parameters.

Proposed Lagrange Polynomial

Asymptotic Properties of Polynomial Approximations

Conclusion