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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.

A scalar M-estimator is a selected root of $n^{-1}{\sum }_i\psi (Y_i,\hat{\theta })=0$, or equivalently a minimizer of a differentiable empirical objective whose derivative is this estimating equation. Its population statistical functional $T(F)={\theta }_F$ solves $\int \psi (x,{\theta }_F)dF(x)=0$. Assume local uniqueness, consistency of the sample root, a valid differentiable local expansion, finite ${\mathbb{E}}_F{\psi }^2$, andThese conditions include domination sufficient to differentiate under the integral and the required local uniform control of the sample derivative.

For $F_{\epsilon }=(1-\epsilon )F+\epsilon {\delta }_z$, differentiate the population estimating equation at $\epsilon =0$. Since its uncontaminated integral is zero,Thus the influence function is $\mathrm{IF}(z;T,F)=-\psi (z,{\theta }_F)/A$, with a proportionality constant independent of $z$.

Expanding the sample equation at its population root givesThe summands have mean zero. The central limit theorem and Slutsky's theorem therefore yield the scalar sandwich variance of an M-estimatorThis $V$ is the limiting variance of the $\sqrt{n}$-scaled error; the estimator's variance is approximately $V/n$. In a correctly centred location family with $\psi (x,\theta )=\psi (x-\theta )$, let $m={\mathbb{E}}_F{\psi }^{\prime }(X-\theta )$. Then $A=-m$, so the influence function is $\psi (z-\theta )/m$ and $V=\mathbb{E}{\psi }^2/m^2$. The minus sign from differentiating $x-\theta$ is essential.

The clipped Huber score in the subsequent subparts requires $0<b<\infty$. The PDF only displays an upper bound: at $b=0$ the estimating function vanishes identically, and at $b<0$ it is a constant with no root. A positive clipping threshold is the regular interpretation.

For fixed-design nonparametric regression, let $x_i=i/n$ and $K_h(u)=h^{-1}K(u/h)$. The degree-$p$ local polynomial estimator takes the fitted intercept ${\hat{m}}_h(x;p)={\hat{\beta }}_0$, whereUse a nonnegative regression kernel and an invertible local polynomial Gram matrix to obtain a unique fit. For $p=0$, solving the single weighted least squares normal equation gives the local constant estimator:At the interior point $x_0$, use the standard unit-integral regression kernel convention ${\mu }_0(K)=1$. Otherwise the leading variance below is $R(K)/(nh{\mu }_0(K{)}^2)$; unlike a kernel density estimator, the local constant estimator itself is unchanged by rescaling $K$.

The local constant estimator is a linear estimator in nonparametric regression with weights $W_i=K_h(x_i-x_0)/(n{s}_{0,h})$. The independent unit-variance errors implyTo control the bias of an estimator, differentiability at this one point is enough. WriteSymmetry makes ${\mu }_1(K)=0$, and the supplied moment approximation gives $s_{1,h}=o(h)$. Therefore the linear contribution to the bias of an estimator is $m^{\prime }(x_0)s_{1,h}/s_{0,h}=o(h)$. The remaining contribution has absolute value at most $\eta (h)h$, because only $|x_i-x_0|\le h$ receive positive weight. This is the interior first-order bias cancellation for local constant regression, and gives squared bias of an estimator $o(h^2)$. Combining with the variance provesFor fixed $\epsilon >0$, take $h_{\epsilon }={\epsilon }^{-1/3}{n}^{-1/3}$. It satisfies both asymptotic smoothing bandwidth conditions, and the displayed remainder, multiplied by $n^{2/3}$, tends to zero for this fixed $\epsilon$. ThusLetting $\epsilon \downarrow 0$ proves the optimized scaled pointwise error tends to zero. The order of these limits matters: no remainder uniform in $\epsilon$ has been assumed.

For the lower bound over the whole function class, the absence of a uniform bound on derivatives permits a particularly direct argument. It also avoids assuming a normal distribution for errors when the model only supplies their first two moments. Fix $n$ and $x_0$. Choose a smooth bump function $b$ on the interval, with $b(x_0)=1$, whose support contains no design point except possibly $x_0$. Such a bump exists because the design is finite; at an endpoint use the restriction of a smooth bump on the real line. Every $m_a=ab$, $a\in \mathbb{R}$, belongs to the differentiability class.

If $x_0$ is not a design point, all observation means are zero for every $a$. The Le Cam two-point lemma applied to $a=\pm A$ has total variation distance zero and gives a lower bound tending to infinity with $A$. In fact, the supremum of the mean squared error is infinite at every such $x_0$.

If $x_0$ is a design point, only its response depends on $a$. The resulting location family is the single observation $Y=a+ϵ$; all remaining observations are independent of $a$ and give no extra information. The single-observation location minimax bound isHere is a proof covering discrete as well as continuous errors. Give $a$ the prior distribution $U[-A,A]$, independently of the noise. Put $p_M=\mathbb{P}(|ϵ|\le M)$ and $v_M=\mathrm{Var}(ϵ\mid |ϵ|\le M)$. Consider an easier experiment in which an oracle also reveals whether $|ϵ|\le M$, and reveals $a$ itself on the complementary event. On the event $|ϵ|\le M$ and $|Y|\le A-M$, every possible truncated noise value places $a=Y-ϵ$ within $[-A,A]$. The flat prior distribution then makes the Bayesian posterior noise law exactly its truncated original law. The smallest conditional squared-error loss is its conditional variance $v_M$, attained by the posterior mean.

The probability of that event is at least $p_M(1-2M/A)$ for $A>2M$, by restricting further to $|a|\le A-2M$. Thus every rule's supremum risk function is at least the original Bayes risk, which is at least the easier experiment's Bayes risk, and therefore at least $p_M(1-2M/A)v_M$. Let $A\to \infty$ and then $M\to \infty$. Finite second moment and zero mean give $p_M{v}_M\to 1$. The rule $T(Y)=Y$ has constant mean squared error one, proving the equality.

Consequently, at every $x_0\in [0,1]$, the unrestricted differentiability class satisfies the stronger conclusionThus $c=1$ works for every $n\ge 1$. There is no conflict with the preceding fixed-function limit: the worst functions, including arbitrarily narrow bumps, can vary with $n$. This is the pointwise versus uniform risk distinction. A smoothness ball with a common derivative bound would pose a different minimax risk problem.

Write $X=\sigma Z$ with $Z\sim F_1$. Thenwhere the distribution of $W=\log Z^2$ does not depend on $\sigma$. Consequentlywhich is a location family. Since $\sigma >0$, the transformed parameter $\theta =\log {\sigma }^2$ ranges over all of $\mathbb{R}$; the paper's restriction $\theta >0$ is unnecessary.