Lattice Corrections in Edgeworth Expansions

Why a Lattice Needs a Correction

A continuous Edgeworth approximation treats probability as if it were spread smoothly along the real line. A lattice-valued random variable instead places its probability at separated points. When a CDF is formed, the last included point contributes a boundary effect: integrating a smooth density only up to that point misses about half of the probability cell centered there. The correction is of the same order as the first skewness correction, so it cannot be ignored in a first-order approximation.

We derive the effect for \(S_n\sim\operatorname{Binomial}(n,p)\), with fixed \(0<p<1\), and write \(q=1-p\). Its mean and standard deviation are \(np\) and \(\sigma=\sqrt{npq}\). The lattice spacing after standardization is \(h=1/\sigma\). The binomial model gives a convenient setting in which both the local expansion and the CDF correction can be made explicit.

Local Edgeworth Approximation for the Binomial

Set \(x_j=(j-np)/\sigma\), and let \(\phi(x)=(2\pi)^{-1/2}e^{-x^2/2}\) be the standard normal density. The standardized third cumulant of the sum is \(\lambda_3=(q-p)/\sigma\). With \(H_3(x)=x^3-3x\), the first-order local expansion is

The factor \(h\) matters: this is a probability at one lattice point, not a density value. Here is a form of the approximation strong enough to sum into a CDF without relying on an uncontrolled accumulation of errors.

To see why this bound has the stated form, apply Stirling's formula with remainder to \(n!,j!\), and \((n-j)!\) in \(P(S_n=j)=\binom njp^jq^{n-j}\). On the range \(|x_j|\le n^{1/12}\), both \(j\) and \(n-j\) are bounded below by a positive constant times \(n\), for large \(n\). Thus the Stirling remainders are \(O(n^{-1})\). Taylor expansion of the logarithms about \(j=np\) gives the normal log-density term \(-x_j^2/2\), the standardized skewness term \(\lambda_3H_3(x_j)/6\), and a remaining log error bounded by \(C n^{-1}(1+|x_j|^4)\). The skewness term is bounded in absolute value by \(C n^{-1/2}(1+|x_j|^3)\), which tends uniformly to zero on this range. Exponentiating, its quadratic contribution is bounded by \(C n^{-1}(1+|x_j|^6)\), giving the displayed probability bound.

From Lattice Sums to the Continuity-Corrected CDF

For an integer \(k\), put \(z_k=(k+\tfrac12-np)/\sigma=x_k+h/2\). The half-step is the upper edge of the cell centered at \(x_k\). The following result gives the CDF approximation, including an error bound that accounts for the full sum rather than just a bounded standardized region.

Proof. First restrict attention to lattice points satisfying \(|x_j|\le n^{1/12}\). The local expansion's remainders summed over any subset of these points are bounded by

The sum is bounded uniformly in \(n\): a Gaussian times a fixed polynomial is integrable, and its Riemann sums on a grid of mesh \(h\to0\) are uniformly bounded. The points outside this range do not cause a hidden CDF error. Hoeffding's inequality gives

The corresponding normal and polynomial-times-normal tails are also exponentially small in \(n^{1/6}\). Consequently, truncating the probability sum and the approximating sums to the stated range costs \(o(n^{-1})\), uniformly in the cutoff \(k\).

Now sum the two leading local terms up to \(j=k\). The points \(x_j\) are cell centers with spacing \(h\), and the union of cells through \(x_k\) ends at \(z_k=x_k+h/2\). For a twice differentiable function \(g\) with integrable second derivative, Taylor's theorem on each cell shows that the difference between its midpoint value times \(h\) and its integral over that cell is bounded by a constant times \(h^2\int_{\text{cell}}|g''(u)|\,du\). Summing gives an \(O(h^2)\) bound, uniformly over partial sums. Apply this to \(g=\phi\), for which \(\int_{\mathbb R}|\phi''(u)|\,du<\infty\). Hence

Apply the same midpoint argument to \(g=H_3\phi\), whose second derivative is also integrable. Its sum is its integral through \(z_k\), up to \(O(h^2)\). Since it is multiplied by \(\lambda_3/6=O(n^{-1/2})\), this quadrature error is \(O(n^{-3/2})\). Direct differentiation gives

so \(\int_{-\infty}^{z_k}H_3(u)\phi(u)\,du=(1-z_k^2)\phi(z_k)\). Combining the summed local remainders, the tail bounds, and these two integral approximations proves the theorem. \(\square\)

The Periodic Term and Other Cutoffs

The half-step is the simplest expression of a periodic lattice correction. To see the periodicity, consider an integer-valued variable and a real, not necessarily integer, cutoff \(t\). Write \(m=\lfloor t\rfloor\), so the event \(S_n\le t\) is exactly \(S_n\le m\). Its continuity-corrected boundary is \(m+\tfrac12\), not \(t\). Relative to integrating to \(t\), the shift is \(m+\tfrac12-t\), which repeats with period one as \(t\) moves across successive lattice cells. A first-order Taylor expansion of the normal integral makes the boundary contribution approximately \((m+\tfrac12-t)\phi((t-np)/\sigma)/\sigma\), with the standardized location understood to the accuracy of this first-order expansion. This is the periodic, sawtooth-shaped correction; at an integer cutoff it reduces to the familiar half-step.

For a lattice with maximal span \(d\), the same geometry uses cells of width \(d\): a sum through a lattice point is matched to an integral ending halfway between that point and the next point of the supporting lattice. The standardized cell width is \(d/\sigma\) when \(\sigma\) is the standard deviation of the variable being approximated. The precise periodic term depends on where the requested cutoff falls within a lattice cell.

Common Mistake and Practical Use

A common but invalid argument is to derive a local expansion only for bounded \(x_j\), sum it from the bottom of the support through \(k\), and claim an \(O(n^{-1})\) CDF remainder. A pointwise error bound on bounded standardized ranges says nothing about points outside those ranges, and even a small local error can accumulate when many terms are added. The correction is to use a local bound on a growing range, sum its weighted errors, and separately control the omitted tails. The binomial proof above does all three; the stated CDF error therefore follows from more than a formal integration of the local expansion.

The continuity-corrected Edgeworth formula is useful when a binomial CDF or tail probability is needed and a normal approximation alone is not adequate. Its main components have distinct roles: \(\Phi(z_k)\) supplies the continuous bulk approximation, the half-step aligns the boundary with the lattice, and the skewness term adjusts for binomial asymmetry. The \(O(n^{-1})\) statement assumes fixed \(p\) strictly between zero and one; it does not assert reliable relative accuracy for extremely small tail probabilities.