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A general partial Cramér's condition for Edgeworth expansion of a function of sample means with applications

Yashi Wei, Jiang Hu, Zhidong Bai

TL;DR

The paper introduces a general partial Cramér's condition (GPCC) that unifies classical Cramér and partial Cramér conditions to justify Edgeworth expansions for smooth functions of sample means. Under GPCC, it proves a general Edgeworth expansion for $W_n=n^{1/2}(H(ar{{f Z}})-H({meta}))$, including cases with discrete components, and verifies validity for a broad class of statistics such as Pearson's correlation, ratio-of-means, and Z-score test statistics. It provides explicit expansions up to second order, derives coefficients from cumulants and derivatives, and demonstrates improved finite-sample accuracy via extensive simulations across continuous and mixed-type data. The results extend Edgeworth methodology to statistics that were previously intractable under standard Cramér-type conditions, offering practical guidance for accurate finite-sample inference in multivariate settings.

Abstract

A large class of statistics can be formulated as smooth functions of sample means of random vectors. In this paper, we propose a general partial Cramér's condition (GPCC) and apply it to establish the validity of the Edgeworth expansion for the distribution function of these functions of sample means. Additionally, we apply the proposed theorems to several specific statistics. In particular, by verifying the GPCC, we demonstrate for the first time the validity of the formal Edgeworth expansion of Pearson's correlation coefficient between random variables with absolutely continuous and discrete components. Furthermore, we conduct a series of simulation studies that show the Edgeworth expansion has higher accuracy.

A general partial Cramér's condition for Edgeworth expansion of a function of sample means with applications

TL;DR

The paper introduces a general partial Cramér's condition (GPCC) that unifies classical Cramér and partial Cramér conditions to justify Edgeworth expansions for smooth functions of sample means. Under GPCC, it proves a general Edgeworth expansion for , including cases with discrete components, and verifies validity for a broad class of statistics such as Pearson's correlation, ratio-of-means, and Z-score test statistics. It provides explicit expansions up to second order, derives coefficients from cumulants and derivatives, and demonstrates improved finite-sample accuracy via extensive simulations across continuous and mixed-type data. The results extend Edgeworth methodology to statistics that were previously intractable under standard Cramér-type conditions, offering practical guidance for accurate finite-sample inference in multivariate settings.

Abstract

A large class of statistics can be formulated as smooth functions of sample means of random vectors. In this paper, we propose a general partial Cramér's condition (GPCC) and apply it to establish the validity of the Edgeworth expansion for the distribution function of these functions of sample means. Additionally, we apply the proposed theorems to several specific statistics. In particular, by verifying the GPCC, we demonstrate for the first time the validity of the formal Edgeworth expansion of Pearson's correlation coefficient between random variables with absolutely continuous and discrete components. Furthermore, we conduct a series of simulation studies that show the Edgeworth expansion has higher accuracy.

Paper Structure

This paper contains 23 sections, 26 theorems, 246 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. General partial Cramér's condition
  4. Statement of the general Edgeworth expansion
  5. Edgeworth expansion under GPCC
  6. A special case
  7. Applications and numerical examples
  8. A valid Edgeworth expansion of Pearson's correlation coefficient
  9. A valid Edgeworth expansion for the ratio of sample means
  10. A valid Edgeworth expansion of the Z-score test statistic
  11. Simulation experiments for correlation
  12. Simulation experiments for ratio of sample means
  13. Simulation experiments for Z-score test statistic
  14. Technical proof of Theorem \ref{['distance']}
  15. Proofs of main results
  16. ...and 8 more sections

Key Result

Theorem 1

Assume that the distribution function $G_1$ of $\textbf{Z}_1$ has a finite $s$-th absolute moment for some integer $s \ge 3$. Additionally, assume the conditional distribution $G^*_1$ given $C_n$ satisfies the GPCC Order 1. Let ${\bf U}$ and $\chi_{\bf v}$ be the covariance matrix and ${\bf v}$-th c for some $s'$, $0\le s' \le s$, we have that where $d$ is a suitable positive constant, $c(s,k)$ a

Figures (4)

  • Figure 1: The Edgeworth expansion for continuous-continuous case at $n=50, 100$
  • Figure 2: The Edgeworth expansions for continuous-discrete case at $n=50, 100$
  • Figure 3: The Edgeworth expansions for the ratio of sample means for different sample sizes
  • Figure 4: The Edgeworth expansions for the Z-score statistic for different sample sizes

Theorems & Definitions (53)

  • Definition 1: GPCC
  • Remark 1
  • Example 1
  • Example 2
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Corollary 1
  • ...and 43 more