Integrating Edgeworth Densities to Approximate CDFs

From a Density Approximation to a CDF Approximation

An Edgeworth density expansion describes how the density of a standardized sum differs from the standard normal density. Probabilities such as \(P(Z_n \leq x)\) and \(P(Z_n > x)\), however, are areas under that density. To use the expansion for these probabilities, each Hermite-polynomial correction must be integrated, with its sign and boundary term handled carefully.

The key feature is that a Gaussian density multiplied by a Hermite polynomial is, up to a sign, a derivative of the same density multiplied by a lower-order Hermite polynomial. Consequently, integration reduces the polynomial's order by one and produces a boundary term at the evaluation point.

The Density Expansion and Its Integrated Form

Let \(X_1,\ldots,X_n\) be independent and identically distributed with mean \(\mu\), variance \(\sigma^2>0\), and finite moments sufficient for the density expansion below. Define the standardized sum \(Z_n=\sum_{i=1}^n(X_i-\mu)/(\sigma\sqrt n)\). Write \(\gamma_1=E[(X_1-\mu)^3]/\sigma^3\) for the standardized skewness and \(\gamma_2=E[(X_1-\mu)^4]/\sigma^4-3\) for the excess kurtosis. The \(-3\) in \(\gamma_2\) makes it the fourth cumulant on the standardized scale.

Use the probabilists' Hermite polynomials \(H_k(x)=(-1)^k\phi(x)^{-1}\phi^{(k)}(x)\), where \(\phi\) and \(\Phi\) are the standard normal density and CDF. The polynomials needed here include \(H_2(x)=x^2-1\), \(H_3(x)=x^3-3x\), \(H_4(x)=x^4-6x^2+3\), and \(H_5(x)=x^5-10x^3+15x\). In this convention, a second-order Edgeworth density expansion has the form below.

The integrated remainder condition is one way to make the transfer from density to CDF precise; it is not a claim that every local density expansion automatically has this property. The calculation below is conditional on the stated expansion. Its central identity follows directly from the definition of the Hermite polynomials.

Why Integration Lowers the Hermite Order

By definition, \(\phi(x)H_k(x)=(-1)^k\phi^{(k)}(x)\). Therefore, for \(k\geq 1\), \((\phi(x)H_{k-1}(x))'=-\phi(x)H_k(x)\). The boundary term at \(-\infty\) vanishes because every derivative of \(\phi\) is a polynomial times \(\phi\), and a polynomial times a Gaussian density tends to zero as \(x\) tends to either infinity.

Here is the full derivation of the CDF formula. Integrate the density expansion from \(-\infty\) to \(x\). The normal term integrates to \(\Phi(x)\). Applying the identity to the \(H_3\), \(H_4\), and \(H_6\) terms gives, respectively, \(-\phi(x)H_2(x)\), \(-\phi(x)H_3(x)\), and \(-\phi(x)H_5(x)\). Finally, the integrated remainder is uniformly bounded: \(\left|\int_{-\infty}^{x}r_n(t)\,dt\right|\leq\int_{\mathbb R}|r_n(t)|\,dt=o(n^{-1})\). Thus the CDF formula follows, including its uniform remainder. Since \(P(Z_n>x)=1-P(Z_n\leq x)\), taking the complement reverses the signs of all the corrections and gives the upper-tail formula. \(\square\)

Signs, Endpoints, and Boundary Terms

For a CDF, the integral runs from \(-\infty\) up to \(x\). The boundary at \(-\infty\) vanishes, leaving \(-\phi(x)H_{k-1}(x)\). For an upper tail, the integral runs from \(x\) to \(+\infty\); the boundary at \(+\infty\) vanishes instead, leaving \(+\phi(x)H_{k-1}(x)\). The two correction signs are opposite because the CDF and upper tail are complements.

A useful check is that each correction integrates to zero over the entire real line: the boundary expression \(\phi(x)H_{k-1}(x)\) vanishes at both ends. Thus the density corrections do not change total mass. But this does not mean the CDF correction has the same sign everywhere: the Hermite polynomials change sign, and the sign also depends on the skewness and kurtosis coefficients.

Common Mistake: Reusing the CDF Sign for the Upper Tail

A common error is to write the upper-tail approximation with the same minus sign as the CDF correction. For example, integrating \(\phi(t)H_k(t)\) from \(x\) to \(+\infty\) and writing \(-\phi(x)H_{k-1}(x)\) overlooks which endpoint remains after integration. The correct calculation is \(\int_x^\infty\phi(t)H_k(t)\,dt=[-\phi(t)H_{k-1}(t)]_x^\infty=+\phi(x)H_{k-1}(x)\). Alternatively, take the complement of the CDF approximation; this also reverses every correction sign.

A second mistake is to integrate only the polynomial part and omit the Gaussian factor's boundary contribution. Hermite corrections are not ordinary polynomial corrections by themselves: their integrals are evaluated through the derivative of \(\phi H_{k-1}\). Retaining that factor is what makes the correction vanish at the tails and preserves the total mass of the expansion.