First-order asymptotics identify a limit law while discarding the approximation error. Higher-order theory expands that error in powers of \(n^{-1/2}\), revealing skewness, bias, studentization effects, and corrections that can improve coverage. Such expansions require stronger moments, smoothness, and usually a nonlattice condition.
Cumulant scaling
Let \(X_1,\ldots,X_n\) be i.i.d. with mean zero, variance one, and cumulants \(\kappa_r\). For \[ Z_n=\frac1{\sqrt n}\sum_{i=1}^nX_i, \] the \(r\)-th cumulant is \[ \kappa_r(Z_n)=n^{1-r/2}\kappa_r. \] Thus skewness enters at order \(n^{-1/2}\), excess kurtosis at order \(n^{-1}\), and the Gaussian law is the zeroth-order term.
Verification. Cumulants add over independent sums and satisfy \(\kappa_r(aX)=a^r\kappa_r(X)\). Hence \[ \kappa_r(Z_n) =n\,n^{-r/2}\kappa_r =n^{1-r/2}\kappa_r. \]
First Edgeworth correction
One-term Edgeworth expansion. Suppose \(\mathbb E|X|^{3+\delta}\lt\infty\) for some \(\delta>0\) and a Cramer nonlattice condition holds: \[ \limsup_{|t|\to\infty}|\mathbb Ee^{itX}|\lt1. \] Then, uniformly in \(x\), \[ P(Z_n\le x) =\Phi(x) +\frac{\kappa_3}{6\sqrt n}(1-x^2)\phi(x) +o(n^{-1/2}). \]
Proof outline. Expand the log characteristic function near zero: \[ \log\mathbb Ee^{itX/\sqrt n} =-\frac{t^2}{2n} +\frac{\kappa_3(it)^3}{6n^{3/2}} +O(n^{-2}|t|^4). \] Multiplication by \(n\) and exponentiation give \[ \varphi_{Z_n}(t) =e^{-t^2/2} \left\{1+\frac{\kappa_3(it)^3}{6\sqrt n}\right\} +O(n^{-1}) \] on an expanding central frequency range. Fourier inversion identifies the correction because derivatives of the Gaussian density produce Hermite polynomials. The Cramer condition controls the outer-frequency integral, and smoothing yields uniformity in \(x\).
For lattice variables, an additional oscillating continuity-correction term appears. A smooth polynomial correction alone is therefore not uniformly valid for a Bernoulli sum.
Cornish-Fisher quantiles
First corrected quantile. If \(z_\alpha=\Phi^{-1}(\alpha)\), then the \(\alpha\)-quantile of \(Z_n\) is \[ q_{\alpha,n} =z_\alpha+ \frac{\kappa_3}{6\sqrt n}(z_\alpha^2-1) +o(n^{-1/2}). \]
Proof. Write \(q_{\alpha,n}=z_\alpha+a/\sqrt n+o(n^{-1/2})\). Taylor-expand \[ \Phi(q_{\alpha,n}) =\alpha+\frac{a}{\sqrt n}\phi(z_\alpha)+o(n^{-1/2}) \] and evaluate the Edgeworth correction at \(z_\alpha\). Setting the sum equal to \(\alpha\) gives \[ a+\frac{\kappa_3}{6}(1-z_\alpha^2)=0. \]
Studentization
For \[ T_n=\frac{\sqrt n\,\bar X}{S_n}, \] the randomness of \(S_n\) changes the correction polynomial. Under standardized mean zero and variance one, \[ P(T_n\le x) =\Phi(x) +\frac{\kappa_3}{6\sqrt n}(2x^2+1)\phi(x) +o(n^{-1/2}) \] under suitable conditions and convention for \(S_n\). Studentization does not merely replace an unknown scale: it changes higher-order skewness and often makes bootstrap calibration second-order accurate.
Coverage error
Definition. A confidence procedure has coverage error \(O(n^{-a})\) if \[ \sup_{\theta\in\Theta_0} \left| P_\theta\{\theta\in C_n\}-(1-\alpha) \right|=O(n^{-a}) \] over the stated parameter class.
A first-order normal interval commonly has one-sided error \(O(n^{-1/2})\). Symmetric two-sided errors can cancel their first skewness term and achieve \(O(n^{-1})\) in regular problems. A properly studentized bootstrap can often reduce error by another order, but the claim depends on smoothness, moments, and uniformity.
Bartlett correction
Correction principle. Suppose a likelihood-ratio statistic \(W_n\) for \(q\) restrictions satisfies \[ \mathbb E_0W_n=q+\frac b n+O(n^{-2}), \] and its Edgeworth expansion has the corresponding Bartlett form. Then \[ W_n^{\mathrm{B}} =\frac{W_n}{1+b/(qn)} \] has expectation \(q+O(n^{-2})\) and can improve the chi-square distributional error from order \(n^{-1}\) to \(n^{-2}\).
Expectation calculation. Since \[ \{1+b/(qn)\}^{-1} =1-\frac b{qn}+O(n^{-2}), \] multiplication gives \[ \mathbb EW_n^{\mathrm B} =\left(q+\frac bn+O(n^{-2})\right) \left(1-\frac b{qn}+O(n^{-2})\right) =q+O(n^{-2}). \] Improvement of the full distribution requires the stronger Bartlett expansion, not the mean calculation alone.
Signed likelihood roots
For a scalar target \(\psi\), the signed likelihood root is \[ r=\operatorname{sign}(\hat\psi-\psi_0) \sqrt{2\{\ell(\hat\theta)-\ell(\tilde\theta)\}}. \] It is asymptotically standard normal. Modified roots of the form \[ r^\star=r+\frac1r\log\frac{u}{r} \] incorporate nuisance-parameter and sample-space curvature and can attain third-order tail accuracy in regular models. The quantity \(u\) depends on ancillary or tangent-exponential constructions; it is not determined by the ordinary observed information alone.
Saddlepoint approximation
If \(K(t)=\log\mathbb Ee^{tX}\) and \(\hat t\) solves \(K'(\hat t)=x\), the saddlepoint density approximation for the sample mean has the form \[ \hat f_{\bar X}(x) =\left\{\frac{n}{2\pi K''(\hat t)}\right\}^{1/2} \exp[-n\{\hat t x-K(\hat t)\}]. \] The exponent is the Legendre transform governing large deviations, while the prefactor accounts for local curvature. Saddlepoint methods can remain accurate in tails where a finite Edgeworth polynomial deteriorates.
Limits of higher-order claims
Near parameter boundaries, under weak identification, with growing dimension, or for nonsmooth estimators, the expansion parameter may not be \(n^{-1/2}\) and regular corrections can be misleading. Higher-order accuracy is meaningful only after the first-order local experiment and its uniformity class have been established.