Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 38 / 6

Saddlepoint approximation methods retain the nonquadratic shape of a distribution through its cumulant-generating function. Suppose $X_1,\dots ,X_n$ are independent and identically distributed, $K(t)=\log E(e^{t{X}_1})$ is finite on an open interval containing zero, and $K^{\prime \prime }>0$. For an interior target mean $x$, solve the saddlepoint equation $K^{\prime }(t)=x$. Introduce exponential tilting byUnder the tilted distribution the mean and variance are $K^{\prime }(t)=x$ and $K^{\prime \prime }(t)$. For the product sample the Radon-Nikodym derivative is $e^{t{\sum }_i{X}_i-nK(t)}$. Consequently the exact density relation for the sample mean isA local central limit approximation to the tilted density at its own mean gives $f_{\overline{X},t}(x)\simeq \sqrt{n/(2\pi K^{\prime \prime }(t))}$. HenceFor the sum $S_n$, the equivalent expression is $f_{S_n}(s)\simeq e^{nK(t)-ts}/\sqrt{2\pi n{K}^{\prime \prime }(t)}$, with $K^{\prime }(t)=s/n$. Under the usual smooth nonlattice density conditions this approximation has relative error of order $n^{-1}$ for a fixed interior target. In the tilted Edgeworth expansion the order $n^{-1/2}$ odd term vanishes at the mean. The exponent retains the Legendre-transform rate $I(x)=tx-K(t)$, rather than a quadratic Taylor approximation to that rate, explaining the useful tail and skewness behavior. Normalization is not automatic, and accuracy statements need their regularity and range restrictions.

For a normal distribution, $K(t)=\mu t+{\sigma }^2{t}^2/2$ and $t=(x-\mu )/{\sigma }^2$. Substitution recovers exactly the normal density of the sample mean. For an exponential distribution of rate $\lambda$, $K(t)=-\log (1-t/\lambda )$, $t=\lambda -1/x$ and $K^{\prime \prime }(t)=x^2$. The approximation becomesIt has exactly the gamma density shape. Replacing its leading constant by the exact normalizing constant makes it exact, illustrating why exactness of shape and exactness of the unnormalized approximation are separate questions.

For a vector sample mean, tilt by $e^{t^{\mathsf{T}}y-K(t)}$, solve $\nabla K(t)=x$, and use the local multivariate normal density at the tilted mean. This givesJoint and marginal saddlepoint densities can also be divided to approximate conditional densities, for example after conditioning on a nuisance sufficient statistic. The p* approximation uses likelihood and fitted-information coordinates to obtain a closely related approximation to a maximum-likelihood estimator's density, usually conditional on an ancillary statistic.

Tail probabilities need a corresponding integration approximation. The Lugannani-Rice saddlepoint tail approximation usesand givesHere $\Phi ,\varphi$ are the standard normal distribution functions. The upper-tail correction has the opposite sign: $1-\Phi (w)+\varphi (w)(1/u-1/w)$. Inverting a cumulative distribution function introduces a pole in addition to the density's saddlepoint; treating both supplies the correction term, which is why simply integrating a normal approximation at the untilted mean is less accurate. At $x=K^{\prime }(0)$ the apparent singularities cancel. With $v=K^{\prime \prime }(0)$ and $k=K^{\prime \prime \prime }(0)$, expansions give $w=t\sqrt{nv}(1+kt/(3v)+O(t^2))$ and $u=t\sqrt{nv}(1+kt/(2v)+O(t^2))$, so $1/w-1/u\to k/(6\sqrt{n}{v}^{3/2})$. For the normal distribution the correction is zero everywhere.

These methods require care for lattice distributions, where summation and lattice corrections replace the density inversion, and near boundaries or singular covariance matrices. A distribution with no moment-generating function in a neighborhood of zero does not satisfy the starting assumptions. A successful saddlepoint approximation is therefore a structured refinement of regular asymptotic inference, not a formula guaranteed uniformly in every tail or for every model.

Partition a regular parameter as $(\psi ,\lambda )$, with interest $\psi$ and nuisance $\lambda$. Orthogonal statistical parameters have zero cross-block of expected Fisher information,This may hold at one parameter value or throughout a parametrization. It is expected-information orthogonality, and does not generally force the observed information cross-block to vanish in each sample.

First, the asymptotic covariance of the joint maximum-likelihood estimator is the inverse information. Orthogonality makes this matrix block diagonal, giving first-order asymptotic independence of interest and nuisance fits. The efficient information with unknown nuisance isFor orthogonal parameters it equals $I_{\psi \psi }$. Thus estimating the nuisance creates no first-order information loss relative to knowing it, at the orthogonal point. This is a local asymptotic statement, not a general finite-sample independence theorem.

Second, orthogonality stabilizes constrained nuisance estimates. Differentiating ${\ell }_{\lambda }(\psi ,{\tilde{\lambda }}_{\psi })=0$ givesWhen the expected cross-information vanishes, regular sampling fluctuations give $j_{\lambda \psi }=O_p(\sqrt{n})$ and $j_{\lambda \lambda }=O_p(n)$. The derivative is therefore $O_p(n^{-1/2})$ locally. Changing the interest parameter by $O_p(n^{-1/2})$ changes its constrained nuisance fit by only $O_p(n^{-1})$, smaller than the ordinary nuisance estimation error. This reduces sensitivity to how the nuisance is fitted and simplifies higher-order likelihood adjustments.

For scalar interest, local orthogonal nuisance reparametrization can often be constructed explicitly. Keep $\psi$ and write the original nuisance as $\lambda (\psi ,\eta )$. The transformed interest score is $U_{\psi }+{\lambda }_{\psi }^{\mathsf{T}}{U}_{\lambda }$. Its covariance with the new nuisance score vanishes whenWith smooth nonsingular nuisance information, this ordinary differential equation has a local solution for suitable initial nuisance coordinates. For several interest coordinates the corresponding equations need compatibility conditions; global orthogonality is not automatic.

The normal model gives a transparent example. For independent observations with mean $\mu$ and variance $\nu$, direct score expectations giveThe cross-information is zero because the centered normal third moment is zero. When fitting $\nu$ at fixed $\mu$,so the change is indeed quadratic in a local change of the mean.

Orthogonality is also useful for removing nuisance effects beyond first order. Take variance $\nu$ as interest and mean $\mu$ as nuisance. Put $Q={\sum }_i(X_i-\overline{X}{)}^2$. The profile log-likelihood is ${\ell }_p(\nu )=-\frac{n}{2}\log \nu -Q/(2\nu )$ up to a constant. The constrained mean is ${\tilde{\mu }}_{\nu }=\overline{X}$ for all $\nu$, so its sample-coordinate Jacobian determinant is one, and $j_{\mu \mu }=n/\nu$. The information adjustment givesThis agrees with the exact likelihood based on $Q/\nu \sim {\chi }_{n-1}^2$: estimating the mean has used one degree of freedom. In this setting the Cox-Reid adjusted profile likelihood has the same form. The exact distribution follows by projecting the centered normal vector onto the $(n-1)$-dimensional subspace orthogonal to the constant vector; an orthonormal basis there supplies $n-1$ independent standard normal coordinates whose squares sum to $Q/\nu$.

The example shows both the benefit and the limits. Orthogonality organizes first-order precision and nuisance stability; it does not eliminate all higher-order nuisance effects. Nor does it identify an ancillary statistic or justify dropping the sample-coordinate factor in every modified profile likelihood. In the exponential regression calculation that factor cancels the information adjustment, so an information-only correction would give a different answer.

An ancillary statistic has a sampling distribution independent of the unknown parameter. It describes a realized aspect of the sample configuration without supplying a marginal likelihood for the parameter. Its observed value can nevertheless be crucial to the precision of inference from the rest of the data. In a transitive transformation model, invariance gives a common distribution under every parameter value, so maximal invariants provide natural ancillaries, such as the standardized configuration in a location-scale family.

Conditional inference uses the sampling distribution of an estimator or sufficient statistic conditional on the observed ancillary. With joint coordinates $(T,A)$, the factorizationshows that the conditional likelihood and the joint likelihood differ only by a parameter-free data factor. Conditional reference distributions, confidence intervals and p-values can still differ markedly from unconditional ones, because they adapt to the realized configuration. For a continuous ancillary one uses a regular conditional distribution or conditional density; dividing by the zero probability of a single ancillary value is not a valid definition.

An exact example makes this adaptation visible. Let independent observations be uniform on $(\theta -1/2,\theta +1/2)$, with $n\ge 2$. Put $U=X_{(1)}$, $V=X_{(n)}$, $R=V-U$ and $C=(U+V)/2$. Specifying the minimum and maximum contributes $n(n-1)$ choices of observations, while each remaining observation must lie between them. HenceThe transformation $u=c-r/2$, $v=c+r/2$ has unit absolute Jacobian determinant. Its support is $0<r<1$, $|c-\theta |<(1-r)/2$. Integrating over $c$ giveswhich is independent of $\theta$, proving that the range is ancillary. Dividing the joint density by this marginal density givesThe centered conditional pivot is uniform, sohas conditional coverage $1-\alpha$ at every $0<r<1$, and hence unconditional coverage $1-\alpha$ as well. A large realized range leaves little room for translating the interval and gives more precise inference. This is range-conditioned uniform location inference. The range and center are not independent: the conditional spread depends on the range, even though the range is ancillary.

Completeness can give independence in other models. Suppose $S$ is a complete sufficient statistic and $A$ is ancillary. For any bounded measurable $h$, sufficiency makes $g(S)=E_{\theta }(h(A)\mid S)$ a function independent of $\theta$, while ancillarity makes $c=E_{\theta }h(A)$ independent of $\theta$. Since $E_{\theta }(g(S)-c)=0$ for every $\theta$, completeness implies $g(S)=c$ almost surely. Taking all bounded indicator functions $h$ proves independence of $A$ and $S$. This is the content of Basu's theorem, with the proof displaying precisely where sufficiency and completeness are used. Without completeness the conclusion fails: in the uniform location example the minimum and maximum are sufficient, and contain the nondegenerate ancillary range.

There are practical qualifications. Ancillaries need not be unique, and some models supply only approximate ancillary coordinates; one must specify the conditioning surface and the accuracy intended. Conditioning on a statistic sufficient for a nuisance parameter is a different device: it may remove that nuisance even though the statistic is not ancillary for the full parameter. Conditioning indiscriminately can discard useful information. In higher-order likelihood inference, the ancillary coordinates specify how the fitted data are varied, entering both the p* density and the sample derivatives in the modified profile likelihood. Their role is therefore substantive, rather than merely a label for an extra statistic.