Uniformly strong consistency and the rates of asymptotic normality for the edge frequency polygons

Abstract

AbstractIn this paper, we primarily focus on the edge frequency polygon estimator of f(x), which represents the probability density function of a sequence of φ-mixing random variables {Xi,i≥1}. We establish the uniformly strong consistency and the convergence rate of asymptotic normality for the edge frequency polygon estimator under suitable conditions. Notably, the convergence rate achieves O(n−1/6), which is more precise compared to the corresponding rate mentioned in the existing literature. Additionally, we present simulation studies to validate the theoretical results.Keywords: Berry–Esseen boundsuniformly strong consistencydensity functionedge frequency polygon estimatorMathematical Subject Classifications: 60E0562G20 AcknowledgementsThe authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and valuable suggestions which helped in improving an earlier version of this paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingSupported by the National Social Science Foundation of China (22BTJ059), the National Natural Science Foundation of China (12201600), and the Natural Science Foundation of Anhui Province (2108085MA06), and the Postdoctoral Science Foundation of China (2022M713056).