Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 38 (Statistical Theory)

The P-star approximation describes the sampling density of a regular maximum-likelihood estimator in sample coordinates consisting of the fitted parameter $\hat{\theta }$ and a suitable ancillary statistic $a$. For a $d$-dimensional parameter, write $\ell (\theta ;\hat{\theta },a)$ for the log-likelihood expressed in these coordinates, and $j(\hat{\theta };\hat{\theta },a)$ for the fitted observed information. The approximation isThe determinant factor supplies the local volume scale, while the exponential retains the full likelihood ratio, rather than replacing it by a quadratic. The normalizing constant is chosen by integrating over the fitted-parameter coordinates; its leading regular large-sample value is $(2\pi {)}^{-d/2}$. These ingredients arise from a statistical saddlepoint approximation. Their use requires an interior, locally unique fit, nonsingular information and suitable smoothness; an ancillary statistic specifies the conditioning surface when the fit alone does not exhaust the relevant sample coordinates. In a regular model the unnormalized leading expression has relative error of order $n^{-1}$ locally; normalization and conditioning matter for refined inference. It is not a universal exact density. Under a smooth one-to-one parameter transformation, the Hessian at the fit transforms without a score term, so its determinant supplies precisely the density Jacobian determinant. This proves the invariance of the construction under one-to-one changes of statistical parameters.

Here put $S={\sum }_i{X}_i$ and $T={\sum }_i{Y}_i$. The log-likelihood, apart from a data-only constant, isThe two score equations giveThe sample sums form a sufficient statistic, and there is no need to retain an additional ancillary in calculating their joint density. Inverting the fitted coordinates gives $S=n/\hat{\lambda }$ and $T=n/(\hat{\psi }\hat{\lambda })$. With parameter order $(\psi ,\lambda )$, the fitted observed information isAlso,Thus the required joint P-star approximation isTo integrate out $\hat{\lambda }$, substitute $u=b/\hat{\lambda }$ inTaking $b=n\lambda (1+\psi /\hat{\psi })$ and rearranging the powers yieldsThe beta function identity ${\int }_0^{\infty }{z}^{n-1}(1+z{)}^{-2n}dz=\Gamma (n{)}^2/\Gamma (2n)$ therefore givesBoth normalized approximations are actually exact. To check the joint assertion independently, $S$ and $T$ have independent gamma distributions with shape $n$ and rates $\lambda$ and $\psi \lambda$. Their joint density is ${\psi }^n{\lambda }^{2n}{S}^{n-1}{T}^{n-1}{e}^{-\lambda S-\psi \lambda T}/\Gamma (n{)}^2$. The inverse-coordinate Jacobian determinant iswhich reproduces the normalized joint expression. This is the P-star density for two exponential rates; its marginal gives the stated F-distribution. Stirling's approximation gives $c_n=(2\pi {)}^{-1}(1-1/(6n)+O(n^{-2}))$, so using only the leading normalizer retains the exact shape but not its exact total mass.

The original PDF's displayed density constant is missing a square on $\Gamma (n)$. Its printed expression integrates to $\Gamma (n)$, rather than one in general. The normalized density above is the one consistent with the stated F-distribution.

For an interest parameter $\psi$ and nuisance parameter $\lambda$, the profile likelihood is $L_p(\psi )=L(\psi ,{\tilde{\lambda }}_{\psi })$, where ${\tilde{\lambda }}_{\psi }$ maximizes the likelihood at fixed $\psi$. Its maximum occurs at the joint maximum-likelihood estimate of $\psi$. It preserves the optimized likelihood ratios for different interest values and is unchanged by one-to-one reparametrization of the nuisance at each fixed $\psi$. It generally is neither a marginal sampling density nor a conditional likelihood, and eliminating nuisance parameters by maximization can introduce appreciable small-sample distortion. Differentiating the constrained score equation gives the curvature formulaat a joint stationary point. The corresponding expected-information expression is the efficient information for the interest parameter. Under the regular conditions of Wilks theorem, twice the profile log-likelihood drop has an asymptotic chi-squared distribution with the dimension of the interest parameter as its degrees of freedom.

For scalar nuisance, the Barndorff-Nielsen modified profile likelihood may be writtenwhere the sample-coordinate derivative holds $\hat{\psi }$ and the ancillary statistic fixed. For vector nuisance the last two quantities are determinants. An equivalent form is ${\ell }_m={\ell }_p+\frac{1}{2}\log |j_{\lambda \lambda }|-\log |{\ell }_{\lambda ;\hat{\lambda }}|$, since differentiating the constrained nuisance score gives ${\ell }_{\lambda ;\hat{\lambda }}=j_{\lambda \lambda }\partial {\tilde{\lambda }}_{\psi }/\partial \hat{\lambda }$. The semicolon denotes differentiation in sample coordinates, not another ordinary parameter derivative. This Jacobian determinant adjustment is essential to the definition.

In the present model setThe log-likelihood is $\ell =-n\log \lambda -\psi {\sum }_i{x}_i-A(\psi )/\lambda$. ConsequentlyThe profile score is $nB/A-{\sum }_i{x}_i$, and its derivative is $-n(C/A-(B/A{)}^2)$. The expression in parentheses is the variance of the fixed design values under positive weights proportional to $Y_i{e}^{-\psi x_i}$. Thus the profile log-likelihood is strictly concave for a nonconstant design. Its score changes from $n{\max }_i{x}_i-{\sum }_i{x}_i>0$ to $n{\min }_i{x}_i-{\sum }_i{x}_i<0$ as $\psi$ goes from $-\infty$ to $+\infty$, giving a unique finite fit. If all $x_i$ coincide, only the mean $\lambda e^{\psi x_i}$ is identifiable and inference on $\psi$ alone is not defined; that exceptional design must be excluded.

The specified ancillary coordinates give $Y_i=\hat{\lambda }\exp (\hat{\psi }{x}_i+a_i)$. At fixed $\hat{\psi },a$, therefore,Meanwhile the constrained observed information isSubstitution provesEquivalently the sample derivative of the nuisance score is ${\ell }_{\lambda ;\hat{\lambda }}=A/({\lambda }^2\hat{\lambda })$, which equals $n/({\tilde{\lambda }}_{\psi }\hat{\lambda })$ at the constrained fit and gives the same cancellation. Thus the modified profile likelihood for exponential regression is proportional to the ordinary profile likelihood. This conclusion uses the given ancillary; no proof of its ancillarity is required.

A transformation model consists of a group $G$ acting on the sample space and on the set of statistical parameters, with the compatibility propertyIn words, transforming a sample from $P_{\theta }$ produces a sample from $P_{g\theta }$. A transitive transformation model is generated from one reference distribution by the group action. For densities, the compatibility property includes the Jacobian determinant of the sample transformation; it is not merely equality of density values at transformed points.

A maximal invariant $A$ is constant on orbits of a group action and separates them: $A(gx)=A(x)$, and $A(x)=A(y)$ implies $y=gx$ for some $g$. An equivariant estimator $T$ of the parameter satisfies $T(gx)=gT(x)$. Equivariance means the estimate follows the transformed parameter, whereas invariance means the reported value stays unchanged.

First suppose the set of statistical parameters can be identified with the group, as happens for a simply transitive action. An equivariant estimator then provides a group-valued fit $\hat{g}(x)$ with $\hat{g}(gx)=g\hat{g}(x)$. Normalize the sample by this fit:Indeed $A(gx)=\hat{g}(x{)}^{-1}{g}^{-1}gx=A(x)$. Conversely, if $A(x)=A(y)$, then $y=\hat{g}(y)\hat{g}(x{)}^{-1}x$, placing $x,y$ in the same orbit of a group action. This establishes maximality, rather than only invariance.

For a transitive action with a nontrivial stabilizer $H$ of a reference parameter ${\theta }_0$, a parameter-valued equivariant estimator does not specify a unique normalizing transformation. Choose a section $h_{\theta }$ with $h_{\theta }{\theta }_0=\theta$ and form $b(x)=h_{T(x)}^{-1}x$. Under $g$,Thus the $H$-orbit of $b(x)$ is invariant. It is maximal: if $b(y)=hb(x)$ with $h\in H$, then $y=h_{T(y)}h{h}_{T(x)}^{-1}x$. Appropriate measurable sections and orbit of a group action coordinates are assumed when treating this as a statistic. This qualification explains why an arbitrary equivariant statistic alone is not automatically a complete set of invariant coordinates.

For the location-scale family, write $X_i=\mu +\sigma Z_i$ with $\sigma >0$ and a fixed joint distribution of $Z$. The group consists of maps $x_i\mapsto a+b{x}_i$ with $b>0$, acting on parameters by $(\mu ,\sigma )\mapsto (a+b\mu ,b\sigma )$. For a nonconstant sample of size $n\ge 2$, take the equivariant estimatorIt gives the maximal invariantIts coordinates satisfy ${\sum }_i{A}_i=0$ and ${\sum }_i{A}_i^2=n$. They stay unchanged under location and positive-scale transformations. If two samples have the same standardized vector, thenso a single group transformation maps the first sample onto the second. The vector retains the observation labels: sorting it would discard information not removed by this location-scale group. The construction is algebraic and does not require the underlying distribution to have finite moments. Constant samples can be assigned a separate invariant value; they form a single exceptional orbit of a group action and have probability zero under an independent continuous base distribution.

Finally, in a transitive transformation model every invariant statistic is ancillary. Choose $g$ taking ${\theta }_0$ to $\theta$ and use $X_{\theta }\stackrel{d}{=}g{X}_{{\theta }_0}$; invariance gives $A(X_{\theta })\stackrel{d}{=}A(X_{{\theta }_0})$. Thus the standardized residual configuration provides a natural conditioning variable, while the equivariant fit contains the location and scale information.

A statistical functional is a map $T$ defined on a class of probability distributions. Its associated functional statistic is $T(F_n)$, where $F_n=n^{-1}{\sum }_i{\delta }_{X_i}$ is the empirical distribution and each ${\delta }_{X_i}$ is a Dirac measure. For example, $T(F)=\int xdF(x)$ yields the sample mean. The influence function records the derivative in the direction of point contamination:It is a local, infinitesimal robustness diagnostic, not a description of arbitrary large contamination.

Suppose $T$ is differentiable in a sense strong enough to linearize it at the empirical distribution, and that its derivative in a signed direction $G-F$ is $\int \mathrm{IF}(x;T,F)d(G-F)(x)$. Averaging the directions ${\delta }_z-F$ over $z\sim F$ gives the zero direction, so linearity implies $\int \mathrm{IF}dF=0$. The resulting asymptotic linear representation isIf $\int {\mathrm{IF}}^{2}dF<\infty$, the central limit theorem and the negligible remainder giveThus the asymptotic variance is $n^{-1}\int {\mathrm{IF}}^{2}dF$. For a vector functional the integral is $n^{-1}\int \mathrm{IF}{\mathrm{IF}}^{\mathsf{T}}dF$. Convergence of the actual finite-sample variance to this leading expression additionally requires control of second moments of the remainder, or an appropriate uniform integrability condition; convergence in distribution alone does not provide that. Existence of a contamination derivative by itself also does not guarantee the required empirical linearization.

Several complementary robustness measures follow. Gross-error sensitivity is ${\gamma }^{\ast }={\sup }_z|\mathrm{IF}(z)|$. Under contamination $(1-\epsilon )F+\epsilon H$, the first-order bias is $\epsilon \int \mathrm{IF}dH$, whose worst absolute coefficient is ${\gamma }^{\ast }$; bounded influence therefore bounds infinitesimal gross-error bias. The local-shift sensitivity is ${\lambda }^{\ast }={\sup }_{x\ne y}|\mathrm{IF}(x)-\mathrm{IF}(y)|/|x-y|$, equal to $\sup |{\mathrm{IF}}^{\prime }|$ when the influence function is smoothly differentiable. It measures the effect of moving a contaminant a small distance. The rejection point of an influence function is ${\rho }^{\ast }=\inf \{r:\mathrm{IF}(x)=0\text{ for }|x|>r\}$, using centered, standardized coordinates; it is infinite if there is no such cutoff. A finite cutoff completely rejects sufficiently extreme observations to first order. These measures address different aspects of robustness. The integral of squared influence controls asymptotic precision and efficiency, while breakdown point concerns finite amounts of contamination and is not determined by bounded influence alone.

For the quantile, assume $0<p<1$, a unique solution $q=q_p(F)$, and a density continuous and positive at $q$. Define quantiles of the contaminated distribution by generalized inversion, since a distribution with a point mass need not attain $p$ as an equality. If $z\ne q$, its indicator is constant near $q$ for sufficiently small contamination, and differentiatingat zero gives $-p+f(q)q_0^{\prime }+1_{z\le q}=0$. HenceAt the exceptional point $z=q$, the generalized-inverse quantile remains exactly $q$: below $q$ the contaminated cumulative distribution function is below $p$, and its value at $q$ exceeds $p$. Its actual derivative there is zero. The usual step-function expression is an almost-everywhere version, which is all that the asymptotic variance needs. Its mean is zero, and its second moment isTherefore the influence function of a quantile gives asymptotic variance $p(1-p)/(nf(q{)}^2)$ and gross-error sensitivity $\max (p,1-p)/f(q)$. Its jump makes local-shift sensitivity infinite, and its nonzero constant tails make its rejection point infinite. If $f(q)=0$ or the quantile is not unique, this regular derivation does not apply.

Discarding zero observations and resolving ties is unnecessary with probability one under the stated independent continuous model. Rank the absolute observations from $1$ to $n$, let $R_i$ be the rank of $|X_i|$, and form the positive Wilcoxon signed-rank statisticReject the null hypothesis for large $W^{+}$. Under symmetry about zero, each sign is a fair independent binary variable, independent of all the absolute values. To see this independence, for every measurable positive set $B$, symmetry gives $\mathrm{Pr}(X\in B)=\mathrm{Pr}(X\in -B)=\frac{1}{2}\mathrm{Pr}(|X|\in B)$. Independence of observations then gives independence of the whole sign vector from the absolute-value vector. Conditional on those absolute values, relabeling by rank therefore givesThe null distribution is consequently independent of the unknown symmetric shape. Its probability generating function is $2^{-n}{\prod }_{r=1}^n(1+z^r)$, so the coefficient of $z^w$ gives the exact probability of $W^{+}=w$. Choose an upper critical value with null tail probability at most the desired level, or randomize at the boundary to obtain that level exactly. The observed upper-tail probability is an exact one-sided p-value. A positive translation raises each Walsh average, as established below, so this upper-tail rule has the correct direction against positive centers for every fixed symmetric shape.

The independent-sign representation provesThe asymptotic null distribution, stated without proof, isThis supplies the usual upper normal distribution critical value when exact calculation is inconvenient; for the discrete upper tail $\mathrm{Pr}(W^{+}\ge c)$, use $c-1/2$ in the continuity correction.

For exact equivalence with the proposed statistic, reorder the observations so $|X_{(1)}|<\cdots <|X_{(n)}|$. Partition the unordered pairs, including diagonals, by the member with larger absolute value. The block belonging to $X_{(r)}$ consists of the $r-1$ pairs with an earlier member and its diagonal pair. For an earlier member $X_{(j)}$, the strict inequality $|X_{(j)}|<|X_{(r)}|$ means the sign of $X_{(j)}+X_{(r)}$ is the sign of $X_{(r)}$; the diagonal sum has that same sign. Thus the entire block supplies exactly $r$ positive averages when $X_{(r)}>0$, and none otherwise. Summing the blocks provesThe two tests therefore have exactly the same statistic, critical values and p-values, not merely the same limiting distribution. The Walsh averages are dependent, so their positivity indicators must not instead be treated as an independent binomial sample. For a translation $X_i=Z_i+\theta$ of any null-symmetric sample, every Walsh average increases by $\theta$; their positive count is nondecreasing, which also verifies the direction of the one-sided test.

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.