Estimation of Finite Population Totals in High Dimensional Spaces

Abstract

In this paper, the problem of Nonparametric Estimation of Finite Population Totals in high dimensional datasets is considered. A robust estimator of the Finite Population Total based on Feedforward Backpropagation Neural Network is derived with the aid of a Super-Population Model. This current study is motivated by the fact that Local Polynomials and Kernel methods have in preceding related studies, been shown to provide good estimators for Finite Population Totals but in low dimensions. Even in these situations however, bias at boundary points presents a big challenge when using these estimators in estimating Finite Population parameters. The challenge worsens as the dimension of regressors increase. This is because as the dimension of the Regressor Vectors grows, the Sparseness of the Regressors’ values in the design space becomes unfeasible, resulting in a decrease in the fastest achievable rates of convergence of the Regression Function Estimators towards the target curve, rendering Kernel Methods and Local Polynomials ineffective to address these challenges. This study considers the technique of Artificial Neural Networks which yields robust estimators in high dimensions and reduces the estimation bias with marginal increase in variance. This is due to its Multi-Layer Structure, which can approximate a wide range of functions to any required level of precision. The estimator’s properties are developed, and a comparison with existing estimators was conducted to evaluate the estimator’s performance using real data sets acquired from the United Nations Development Programme 2020. The estimation approach performs well in an example using data from a United Nations Development Programme 2020 on the study of Human Development Index against other factors. The theoretical and practical results imply that the Neural Network estimator is highly recommended for survey sampling estimation of the finite population total.