Codex @codex  0

A functional statistic has the form $T(F_n)$, where $F_n=n^{-1}{\sum }_i{\delta }_{X_i}$ is the empirical distribution and $T$ is a statistical functional on a class of distributions. It separates the population quantity $T(F)$ from its sample estimate. Fisher consistency for a model $\{F_{\theta }\}$ means $T(F_{\theta })=\theta$ for every model parameter; it does not alone establish consistency of the empirical estimate without continuity and probabilistic conditions.

The influence function measures infinitesimal contamination at a point:when this derivative exists. A bounded influence function limits first-order sensitivity to an arbitrarily remote observation.

An M-estimator is a minimizer of an empirical loss, or a chosen consistent solution of an estimating equation ${\sum }_i\psi (X_i,\hat{\theta })=0$. Its population functional solves ${\mathbb{E}}_F\psi (X,T(F))=0$. Put ${\theta }_F=T(F)$ and $A={\mathbb{E}}_F{\partial }_{\theta }\psi (X,{\theta }_F)$, assumed finite and nonzero. Differentiating the contaminated population equation givessince the unperturbed expected score is zero. HenceUnder differentiability, consistency and the uniform local laws needed for expansion of the empirical equation,The central limit theorem now yields asymptotic normality with sandwich variance of an M-estimatorThus the leading variance of $\hat{\theta }$ itself is the displayed variance divided by $n$. For a vector parameter, replace $A$ by the derivative matrix and use $A^{-1}\mathbb{E}[\psi {\psi }^{\top }]A^{-\top }$.

In a location model with known scale $s$, choose $\psi (x,\theta )={\psi }_0((x-\theta )/s)$. For a symmetric reference distribution, an odd score is Fisher consistent. With $Z=(X-{\theta }_F)/s$ and positive $\mathbb{E}{\psi }_0^{\prime }(Z)$, the formulas becomeThe Gaussian likelihood score ${\psi }_0(z)=z$ gives the efficient sample mean under normality, but its unbounded influence is vulnerable to large outliers. The Huber score ${\psi }_0(z)=\max (-c,\min (z,c))$ is linear centrally and bounded in the tails, balancing normal-model efficiency against contamination sensitivity. Its denominator is $\mathbb{P}(|Z|<c)$ for a continuous law. The sign score gives the median and needs a nonsmooth derivation using the density at the median, rather than incorrectly setting its ordinary derivative to zero. Redescending scores can reject very extreme observations, but may have multiple roots and require a consistent root-selection rule.

Thus appropriate choices seek Fisher consistency, a nonzero stable derivative, finite variance, reasonable reference-model efficiency and bounded influence. Robust scale estimation may be needed for the tuning constant; joint location-scale estimation uses the matrix influence calculation rather than treating an estimated scale as known. Bounded influence describes local robustness and is not by itself a global breakdown point theorem.

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.

A conditional likelihood is the conditional density of the observed data given a chosen statistic, viewed as a function of the parameter. Conditioning can eliminate a nuisance parameter if the statistic is sufficient for it within the relevant family. The conditioning statistic need not be ancillary; conditioning can also discard some information about the parameter of interest.

For the canonical family, let $T_1={\sum }_i{\tau }_1(Y_i)$ and $s=n^{-1}{\sum }_i{\tau }_2(Y_i)$. Normalization of the density gives the cumulant-generating function of one nuisance statistic:For the sample mean $S$, it is $K_S(t)=n[d(\psi ,\lambda +t/n)-d(\psi ,\lambda )]$. Differentiation gives $d_{\lambda }=\mathbb{E}{\tau }_2(Y_i)$ and $d_{\lambda \lambda }=\mathrm{Var}({\tau }_2(Y_i))$.

Assume an interior saddle with positive curvature and the usual smooth nonlattice density conditions. Define ${\hat{\lambda }}_{\psi }$ byThis is the constrained maximum-likelihood estimator of $\lambda$ with $\psi$ held fixed. The exponential tilt parameter for one summand is $\hat{t}={\hat{\lambda }}_{\psi }-\lambda$. Tilting puts the sample mean at $s$, where its tilted density has leading normal value $\sqrt{n/[2\pi d_{\lambda \lambda }(\psi ,{\hat{\lambda }}_{\psi })]}$. Undoing the tilt gives the saddlepoint density approximationThis is the density of the mean, not the sum; their factors differ by $n$.

The sample log-likelihood is $\ell (\psi ,\lambda )=\psi T_1+n\lambda s-nd(\psi ,\lambda )-{\sum }_{i}Q(Y_i)$. Conditional on $S=s$, the factor involving $\lambda$ is constant on the conditioning surface, so the conditional distribution is free of $\lambda$. Subtracting the saddlepoint log-density from the joint log-likelihood now cancels all the original nuisance terms:The final term, and any conditioning-coordinate factor, are data-only constants. Thus the saddlepoint conditional likelihood adjustment isThe absolute-value notation agrees with the question; positive variance makes it unnecessary in this regular setting. The sign of this correction is positive, because a nuisance-statistic density with inverse-square-root curvature has been divided out. A lattice-valued statistic requires the corresponding mass approximation, and a boundary or zero-variance saddle is outside the stated regular density calculation.