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

In this paper, we primarily focus on the edge frequency polygon estimator of $ f(x) $ f(x), which represents the probability density function of a sequence of phi-mixing random variables $ \{X_i, i\geq 1\} $ {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}) $ 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.