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Fix an ancillary statistic $a$ and a parameter value at which the score function is evaluated. Express that score as a function $u(v,a)$ of the varying maximum-likelihood estimator coordinate $v$. The ordinary density change-of-variables formula applied to the P-star approximation gives the displayed density; several monotone branches require a sum. Its derivative is a sample-coordinate derivative, not generally the parameter observed information.

For independent $N(\mu ,{\mu }^2)$ observations with $\mu >0$, the sufficient statistics are their sum and squared norm. Condition on $a=\sqrt{n}\Vert y\Vert /{\sum }_i{y}_i>0$ and write $v=\hat{\mu }=\Vert y\Vert /(q\sqrt{n})$, where $q=(\sqrt{1+4a^2}+1)/(2a)$. Polar coordinates supply radial measure $\Vert y{\Vert }^{n-1}d\Vert y\Vert$, while the conditional angular factor does not depend on $\mu$. This proves the displayed shape. Its normalization depends on $a,n$ but not $\mu$ after the substitution $v=\mu z$, and agrees with the normalized P-star approximation.

A finite-dimensional exponential family has common dominating measure and densityHere $\eta$ is the natural parameter of an exponential family, and normalization determines the cumulant function of an exponential family $\kappa$. A full regular family lets $\eta$ vary over an open natural parameter domain. A curved exponential family restricts $\eta$ to a lower-dimensional submanifold, while a minimal representation removes redundant affine combinations of the statistic.

Differentiating the normalization under regularity givesConsequently $\kappa$ is convex and its Hessian is the Fisher information matrix in natural coordinates. For an independent size-$n$ sample, $S={\sum }_{i}T(Y_i)$ is a sufficient statistic: the joint density factors into ${\prod }_{i}h(y_i)$ times $\exp \{{\eta }^{\mathsf{T}}S-n\kappa (\eta )\}$, so the conditional law given $S$ loses its parameter dependence. An interior maximum-likelihood estimator solvesmatching empirical and theoretical sufficient-statistic means. Under minimality the Hessian is positive definite, so the log-likelihood is strictly concave; existence of an interior maximizer still needs the observed statistic to be in the appropriate interior convex support.

For a full open natural family, the canonical sufficient statistic is a complete statistic: if ${\mathbb{E}}_{\eta }g(S)=0$ throughout the open domain, exponential tilting about an interior point makes the Laplace transform of the integrable signed measure $g(s)P_{{\eta }_0}(ds)$ zero on a neighborhood. Transform uniqueness makes that measure zero, so $g(S)=0$ almost surely. This requires integrability under the family. It is not an automatic property of curved restrictions. The sufficient-statistic reduction, explicit information and cumulant identities make these families especially useful for exact and higher-order inference. Normal distributions, Poisson distributions and gamma distributions supply familiar examples.

A transformation model is built differently: a group acts on the sample and parameter spaces, and its probability laws satisfyIn a transitive model all laws are transforms of a fixed reference law. The location family is generated by translations, with density $f_{\mu }(y)=f_0(y-\mu )$. The positive-scale family has density $f_{\sigma }(y)={\sigma }^{-1}{f}_0(y/\sigma )$, and the location-scale family combines them asThe Jacobian factor is essential. For independent observations its power is ${\sigma }^{-n}$. The base density need not have moments, and a transformation model need not have fixed support.

An equivariant estimator respects the parameter action. With a well-defined equivariant fit $T(y)$ valued in the transformation group, the normalized sampleis a maximal invariant. Equivariance gives $a(gy)=a(y)$; if normalized samples agree, $y^{\prime }$ is obtained from $y$ by $T(y^{\prime })T(y{)}^{-1}$. In a transitive transformation model its law is parameter-free, so it is an ancillary statistic. For location and positive scale, centered standardized residuals provide this construction on nondegenerate samples. If the parameter action has a nontrivial stabilizer, normalization must retain its quotient action rather than pretending that a unique group element is determined.

This geometry motivates conditioning on an ancillary and using the P-star approximation for the fitted parameter. Ancillarity means parameter-free distribution; it does not by itself assert independence from the estimator. In a normal location model with known scale, the sample mean and centered residuals are independent by their joint Gaussian law and zero covariance. In an exponential scale model, the sample sum and its normalized proportions are independent, again an additional model property.

The two classes overlap but are distinct. A normal location-scale family is both an exponential family and a transformation model; an exponential mean family is both a one-dimensional natural exponential family and a scale model. A general Cauchy distribution location family is a transformation model without a fixed finite-dimensional regular exponential-family sufficient-statistic reduction. Exponential-family structure emphasizes sufficient statistics and natural-parameter convexity; transformation structure emphasizes equivariance, group orbits and ancillary reduction.

Consider a regular $d$-parameter statistical model for which the data can locally be expressed as $(v,a)$, where $v=\hat{\theta }$ is the maximum-likelihood estimator and $a$ is a suitable exact or higher-order ancillary statistic. Write the log-likelihood as $\ell (\theta ;v,a)$, and letbe its fitted observed information. The P-star approximation to the conditional sampling density of $v$, given $a$, isThe normalizing factor is independent of $v$ and is chosen so the density integrates to one over the feasible estimator coordinates. Its leading Gaussian value is $(2\pi {)}^{-d/2}$, but exact normalization can change it. Both the fitted likelihood and the fitted information vary with $v$; the observed dataset is not held fixed while integrating over this sampling coordinate.

To see the structure, expand the likelihood difference for $v$ close to $\theta$. Its leading term is minus one half of the information-weighted quadratic displacement, while $|j(v){|}^{1/2}$ supplies the local density scale. This recovers the first-order multivariate normal distribution of an efficient maximum-likelihood estimator. The formula is invariant under smooth one-to-one parameter changes: at a likelihood maximum the score vanishes, so the Hessian transforms as a quadratic form. Its square-root determinant supplies exactly the Jacobian determinant required for the estimator density to transform.

There is a direct derivation in a full regular exponential family. Let $S$ be the sufficient statistic for an independent sample, with likelihoodThe maximum-likelihood equation is $s=n\nabla \kappa (v)$, and its derivative is $j(v)=n{\kappa }^{\prime \prime }(v)$. Applying a multivariate saddlepoint density approximation to $S$ givesChanging variables from $s$ to $v$ multiplies by $|ds/dv|=|j(v)|$, producing the displayed P-star form. In more general models, ancillary conditioning supplies the relevant local sample coordinates; an ancillary is not an arbitrary extra statistic.

For an illustrative exact case, take an independent sample from an exponential distribution with mean $\theta$, so $v=\bar{Y}$. The fitted observed information is $j(v)=n/v^2$, andThus P-star has density shape $v^{n-1}{\theta }^{-n}{e}^{-nv/\theta }$. Normalizing giveswhich is exactly the gamma distribution sampling density of the mean. Here sample ratios provide ancillary coordinates and are independent of the mean.

In suitable regular models the unnormalized leading construction has relative error of order $n^{-1}$, and higher-order ancillary constructions with normalization can attain relative order $n^{-3/2}$. Such accuracy needs the stated smoothness and ancillary hypotheses; it is not a universal consequence of asymptotic normality. The exact scale-model example is likewise a special property, not a general identity. The formula is useful for refined conditional inference, score distributions, and nuisance-likelihood adjustments, including modified profile likelihood. It describes a sampling density of the estimator, rather than a posterior density of the parameter.

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.

Let $a$ be a suitable ancillary statistic and assume the sample can locally be written in coordinates $(v,a)$, where $v=\hat{\theta }$ is the scalar maximum-likelihood estimator. Ancillarity means the sampling law of $a$ does not involve $\theta$. For each hypothetical sample coordinate $v$, write $\ell (\theta ;v,a)$ for its log-likelihood, andThe P-star approximation to the conditional density with respect to $dv$ isHere $c(\theta ,a)$ is chosen so the integral over the admissible $v$ domain is one; its leading regular approximation is $(2\pi {)}^{-1/2}$. One must vary the data coordinate $v$ inside the fitted information and likelihood, not evaluate those factors once at the observed estimate and treat them as constant. The expression combines the likelihood-ratio shape with the local information-volume factor. A quadratic likelihood gives the usual normal approximation, while the full shape retains skewness and other higher-order effects.

The approximation is invariant under smooth one-to-one parameter transformations: at the fitted point the transformed observed information acquires the square of the coordinate Jacobian, so its square root supplies exactly the density Jacobian. Under regular smooth fixed-dimensional likelihood and ancillary conditions it is a higher-order conditional approximation, typically with relative error of order $n^{-1}$, and in suitable refined settings of order $n^{-3/2}$. These are not universal error guarantees for singular, boundary or arbitrary ancillary constructions. In important transformation models the normalized formula is exact. The normalizing factor may depend on $\theta ,a$; treating it as a universal constant can itself lose higher-order accuracy.

For a concrete check, in the exponential-mean model of Question 1 the likelihood difference is $n\log (v/\theta )-nv/\theta +n$ and $j(v)=n/v^2$. The resulting P-star density for an exponential mean normalizes towhich is exactly the density of $\bar{Y}$. This example demonstrates the importance of both the information factor and normalization; it does not prove exactness for every regular model.

For the score density, fix an evaluation parameter ${\theta }_0$, and define $u(v,a)={\partial }_{\theta }\ell ({\theta }_0;v,a)$. On any locally monotone branch, let $v=v(u,a)$. The ancillary-conditioned score density is obtained by an ordinary change of variable:If several branches give the same score, sum their contributions. The sampling parameter $\theta$ and the score-evaluation parameter ${\theta }_0$ have been distinguished; for the score at the true parameter take ${\theta }_0=\theta$. The denominator is a data-coordinate derivative, not automatically $j({\theta }_0)$.

For the conditional distribution function of the estimate, integrate the normalized density:Both integrals use the actual support and observed ancillary value. Numerical quadrature gives tail probabilities or confidence limits by inversion. Conditional inference should not silently average over the ancillary, because it is the conditional distribution that this construction is designed to approximate.

Put $S_1={\sum }_i{Y}_i$, $S_2={\sum }_i{Y}_i^2$ and $R=\sqrt{S_2}$. Apart from a data-only constant, the log-likelihood isThe natural parameters in the ambient two-parameter exponential family of normal distributions are ${\eta }_1=1/\mu$ and ${\eta }_2=-1/(2{\mu }^2)$, restricted to the nonaffine curve ${\eta }_2=-{\eta }_1^2/2$, ${\eta }_1>0$. Its parametrization has rank one. Thus this is a one-dimensional curved exponential family inside a two-dimensional full family.

The Fisher-Neyman factorization theorem makes $(S_1,S_2)$ sufficient. For two samples $y,z$, the parameter-dependent part of their log density ratio is $\Delta S_1/\mu -\Delta S_2/(2{\mu }^2)$. A polynomial in $1/\mu$ is constant on $(0,\infty )$ only if both nonconstant coefficients vanish. The likelihood-ratio criterion for minimal sufficiency therefore proves $(S_1,S_2)$ is minimal sufficient.

Write $Y_i=\mu Z_i$ with independent $Z_i\sim N(1,1)$. Since $\mu >0$, $a=\sqrt{n}\sqrt{{\sum }_i{Z}_i^2}/{\sum }_i{Z}_i$ has a probability distribution independent of $\mu$: it is an ancillary statistic. The denominator is nonzero almost surely. The Cauchy-Schwarz inequality gives $|a|\ge 1$; on the requested positive branch, $S_1>0$ and $a\ge 1$.

The likelihood equation is $n{\mu }^2+S_1\mu -S_2=0$. Its positive root isThere is exactly one positive root. The sign of the score function changes from positive to negative there, and $\ell \to -\infty$ at both ends of $(0,\infty )$ when $S_2>0$, so it is the unique maximum-likelihood estimator. The useful identities areSubstitution yields, dropping the constant $-n/2$,Differentiating twice and using $q/a=q^2-1$ at the fitted value gives the observed informationThe scalar P-star approximation uses $\sqrt{j(v)}\exp \{\ell (\mu ;v,a)-\ell (v;v,a)\}$. Consequently, with its normalizing factor explicitly retained,Here $C_n(a)$ is chosen to make the density integrate to one, and is independent of $\mu$. To see this without assuming an approximate normalizer, defineThe integral is finite and positive, and normalized $p^{\ast }$ is exactlyIndeed $C_n(a)=\{\sqrt{n(1+q^2)/(2\pi )}\exp [n{q}^2/2-nq/a]J_n(a){\}}^{-1}$.

There is a direct check of this conditional scale density in a curved Gaussian family. In polar coordinates $y=Ru$, the sample density times volume element is proportional to ${\mu }^{-n}{R}^{n-1}\exp [-R^2/(2{\mu }^2)+R{\sum }_i{u}_i/\mu ]dRd\omega (u)$. Conditioning on $a$ fixes ${\sum }_i{u}_i=\sqrt{n}/a$. The remaining angular factor is independent of $R,\mu$, while $R=q\sqrt{n}v$. Thus the exact conditional radial density has the same normalized form. For $n=1$ the positive ancillary branch fixes the positive sign and the same one-dimensional radial calculation applies. In this scale model the normalized $p^{\ast }$ expression equals the exact conditional density, not merely its leading approximation.

For $t>0$, compute its conditional distribution by one-dimensional quadrature:It is zero for $t\le 0$; with the conditional-density calculation just given, this normalized formula is exact for almost every admissible $a$. Numerically, subtract the maximum log integrand before exponentiating to avoid overflow. The sampling probability still depends on the stipulated true $\mu$; replacing it by a fitted value would be a separate plug-in inference step.

For a size-$n$ sample from an exponential distribution of mean $\theta$, the likelihood in the maximum-likelihood estimator coordinate $v=\bar{Y}$ has difference $\ell (\theta ;v)-\ell (v;v)=n\log (v/\theta )-nv/\theta +n$, and the fitted observed information is $n/v^2$. Their P-star approximation is the displayed gamma density shape. Its normalization gives exactly the sampling density $n^n{v}^{n-1}{e}^{-nv/\theta }/(\Gamma (n){\theta }^n)$. This illustrates an exact transformation-model case rather than a general claim of exactness.

For independent samples of size $n$ from exponential distributions with rates $\lambda$ and $\psi \lambda$, the maximum-likelihood estimators are $\hat{\lambda }=n/S$ and $\hat{\psi }=S/T$, where $S$ and $T$ are the respective sample sums. Their normalized P-star approximation is exact:Indeed, $S$ and $T$ have independent gamma distributions with shapes $n$ and respective rates $\lambda ,\psi \lambda$. The Jacobian determinant of their inverse transformation is $n^2/({\hat{\psi }}^2{\hat{\lambda }}^3)$, giving the displayed density. Alternatively the square root of the determinant of the fitted observed information is $n/(\hat{\psi }\hat{\lambda })$; multiplying by the exponential of the fitted log-likelihood difference gives the same shape. Its normalizing constant in the P-star approximation is $c_n=n^{2n-1}{e}^{-2n}/\Gamma (n{)}^2$. Integrating over $\hat{\lambda }$ givesso $\hat{\psi }/\psi$ has the F-distribution with degrees of freedom $(2n,2n)$.