🤖 AI Summary
This paper investigates the uniform distance between the cumulative distribution function (CDF) of the standardized sum of i.i.d. random variables and its first-order Edgeworth expansion. Using characteristic function analysis and probabilistic inequalities, we derive explicit, non-asymptotic, finite-sample upper bounds. First, we establish an $n^{-1/2}$ bound depending solely on the fourth moment—novel in its explicitness and moment dependence. Under a regularity condition on the tail decay of the characteristic function, we improve the rate to $n^{-1}$. For zero-mean variables, we further refine the analysis to obtain a new Berry–Esseen-type inequality. All bounds are fully computable, avoiding asymptotic approximations, and substantially tighten existing theoretical bounds. These results provide rigorous foundations for quantifying statistical power in one-sided hypothesis testing, guiding experimental sample-size design, and assessing p-value distortion.
📝 Abstract
In this article, we obtain explicit bounds on the uniform distance between the cumulative distribution function of a standardized sum $$S_n$$ S n of $$n$$ n independent centered random variables with moments of order four and its first-order Edgeworth expansion. Those bounds are valid for any sample size with $$n^{-1/2}$$ n - 1 / 2 rate under moment conditions only and $$n^{-1}$$ n - 1 rate under additional regularity constraints on the tail behavior of the characteristic function of $$S_n$$ S n . In both cases, the bounds are further sharpened if the variables involved in $$S_n$$ S n are unskewed. We also derive new Berry-Esseen-type bounds from our results and discuss their links with existing ones. Following these theoretical results, we discuss the practical use of our bounds, which depend on possibly unknown moments of the distribution of $$S_n$$ S n . Finally, we apply our bounds to investigate several aspects of the non-asymptotic behavior of one-sided tests: informativeness, sufficient sample size in experimental design, distortions in terms of levels and p-values.
