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On a bounded parameter set, $\log f_{\theta }(x)=\theta x-K(\theta )+\log f_0(x)$ has an integrable envelope of a function class whenever $K$ is bounded and $|X|+|\log f_0(X)|$ is integrable under the sampling law. A compact subset of the interior natural-parameter domain bounds the cumulant function of an exponential family and provides moments under model sampling; log-base-density integrability is an additional condition.

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.

Write $Ph=\mathbb{E}h(X)$ and $P_{n}h=n^{-1}{\sum }_{i}h(X_i)$ for the expectation and empirical measure. A sufficient bracketing of a function class condition is that, for every $\epsilon >0$, finitely many brackets $[l_j,u_j]$ cover $\mathcal{H}$, with measurable integrable endpoints and $P(u_j-l_j)<\epsilon$. The uniform strong law from finite L1 bracketing then givesHere the observations are independent and identically distributed. For an uncountable class, one either assumes a measurable supremum, for example through a pointwise separable function class, or formulates the conclusion as a pathwise bound on a common probability-one event.

For the parameterized class, the Heine-Borel theorem makes $\Theta$ compact. Put $M(x)={\sup }_{\theta \in \Theta }|q(\theta ,x)|$. A countable dense subset of $\Theta$ gives the same supremum because of continuity, so $M$ is measurable. Define the modulusThe supremum is measurable by taking a countable dense subset of the compact set of admissible pairs. For each $x$, uniform continuity on $\Theta$ gives $w_{\delta }(x)\to 0$. Also $w_{\delta }\le 2M$, and $PM<\infty$. Thus the dominated convergence theorem gives $P{w}_{\delta }\to 0$.

Choose a finite net of radius $\delta$ ${\theta }_1,\dots ,{\theta }_N$ in $\Theta$. The lower and upper envelopes of $q(\theta ,\cdot )$ over each closed ball $\Theta \cap \overline{B}({\theta }_j,\delta )$ are measurable integrable function brackets; the same countable-dense-set argument applies within each such compact ball. Their widths are at most $w_{2\delta }$. They cover the whole class, so its bracketing of a function class condition follows. Moreover, dominated convergence shows that $\theta \mapsto Pq(\theta ,\cdot )$ is continuous. Both empirical and population functions therefore have a supremum over a common countable dense parameter set. Applying the uniform strong law from finite L1 bracketing proves uniform almost-sure convergence over the entire compact parameter set, rather than merely convergence at each fixed parameter.

For the exponential family,where $A={\sup }_{\Theta }|\theta |$ and $B={\sup }_{\Theta }|K(\theta )|$. Thus a weak sufficient condition is bounded $K$ on $\Theta$, $f_0(X)>0$ almost surely, and integrability of $|X|+|\log f_0(X)|$ under the sampling law. No positive lower bound on $f_0$ over the whole real line is required. Zeros off the sampling support are harmless: choose arbitrary finite versions of the log-densities on that common null set when applying the function-class theorem.

If the sampling law is $f({\theta }_0,\cdot )$, one convenient assumption is that $\Theta$ is a compact subset of the interior of the finite domain of the cumulant function of an exponential family, andOn that interior, $K$ is continuous, hence bounded on $\Theta$. Finiteness of $K$ at ${\theta }_0\pm a$ for some $a>0$ implies ${\mathbb{E}}_{{\theta }_0}{e}^{a|X|}<\infty$, hence ${\mathbb{E}}_{{\theta }_0}|X|<\infty$. The displayed integral gives the remaining integrability. If the observations come from an arbitrary unrelated law, assumptions on $K$ and $f_0$ alone cannot control that law's tails; the sampling-law integrability must then be stated explicitly.

Write the exponential dispersion family in the conventionAssume that the support does not depend on the parameters, that $\theta$ is in the interior of the natural parameter space, and that differentiation can pass under the normalizing integral or sum. Differentiating $\int f=1$ at fixed dispersion parameter givesso the expectation is $b^{\prime }(\theta )$. A second differentiation givesConsequentlyHere $b$ is the cumulant function of an exponential family; the same identities follow by differentiating $\log \mathbb{E}{e}^{tZ}=[b(\theta +\varphi t)-b(\theta )]/\varphi$. For a known observation weight $w$, replace $\varphi$ by $\varphi /w$. These regularity assumptions for differentiation under the integral sign matter: differentiation of a parameter-dependent support can introduce additional boundary terms.

For fixed known $r>0$, take the natural parameter of an exponential familyThe negative binomial distribution has log-likelihoodThus this is a negative binomial exponential family, with dispersion parameter $\varphi =1$ in the convention of part (a). The carrier contains the Gamma function and depends on the fixed size $r$, which need not be an integer. Differentiating the cumulant function of an exponential family givesThereforeThe size $r$ controls overdispersion relative to a Poisson distribution; it is different from the unit dispersion parameter used by this fixed-size generalized linear model. Another common convention applies to the scaled response $Z=Y/r$: then $b(\theta )=-\log (1-e^{\theta })$ and $\varphi =1/r$. Keeping track of the response scale prevents confusing these two representations. At $\lambda =0$ the law is concentrated at zero and is obtained as a boundary limit, rather than by a finite natural parameter of an exponential family.

Multiplying the exponential family likelihood by its natural conjugate prior givesThus the posterior remains in the same family, with updated hyperparametersUnder quadratic loss, the Bayes estimator under squared error loss is the posterior mean. Differentiating the log-partition function that normalizes the conjugate prior gives

A one-parameter exponential family with natural statistic $Y$ has density or probability mass functionwhere $\theta$ is the natural parameter of an exponential family and $K$ is the cumulant function of an exponential family. Its mean parameter is

The exponential-family deviance from ${\theta }_1$ to ${\theta }_2$ is twice the Kullback-Leibler divergence:The carrier $h(Y)$ cancels from the likelihood ratio, so, with ${\mu }_1={\mathbb{E}}_{{\theta }_1}Y$,

For the Poisson distribution with mean $\mu$,ThereforeWriting ${\mu }_2={\mu }_1+\delta$ and applying the Taylor series of $\log (1+\delta /{\mu }_1)$ givesso the second-order approximation is