Statistics provides essential tools for transforming data into meaningful evidence for economic and scientific decision-making, while polynomial functions serve as a fundamental mathematical bridge that supports both economic and theoretical statistics. This paper aims to explore the applications to show how polynomial ideas support statistical learning from elementary computation to advanced estimation. First, in economic statistics, we reformulate the cumulative method for calculating average growth speed as a polynomial root-finding problem. We prove sufficient conditions for the existence and uniqueness of the economically meaningful positive root, thereby providing a rigorous foundation for a textbook procedure that is usually introduced without formal justification. We also extend the result to more general polynomial structures in which some coefficients may be negative. Second, in theoretical statistics, we develop a profile maximum likelihood estimation strategy for the spatial autoregressive nonparametric stochastic frontier model, where the unknown frontier function is approximated through a p-order local polynomial procedure. We derive the concentrated likelihood, construct the corresponding estimators, and provide an implementable bandwidth-selection algorithm. Numerical experiments verify the theoretical conclusions for the cumulative method and examine the finite-sample performance of the proposed estimator. Overall, this paper links polynomial root-finding in economic statistics with local polynomial estimation in nonparametric statistics.
Deng et al. (Thu,) studied this question.