Binding function and indirect-inference target

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment and let $S:\mathcal{X}\to {\mathbb{R}}^d$ be a statistic, meaning a measurable map to its declared target as in Statistics and estimators, with finite second moment under each $P_{\theta }$. Expectations are integrated against the law as in Law or distribution of a random element and Expectation of a nonnegative or integrable random variable, and covariance uses the centered second moment as in Moments, variance, and covariance on a probability space. The moment binding function for this page is $b:\mathrm{\Theta }\to {\mathbb{R}}^d,b(\theta ):={\mathbb{E}}_{\theta }[S]\mathrm{.}$ The name refers here to the mean of the declared summary vector; classical indirect inference also uses bindings defined by auxiliary-model estimators, which are a different interface.

The summary-law map is itself a parametric model in the sense of Parametric statistical models: $Q:\mathrm{\Theta }\to \mathcal{P}({\mathbb{R}}^d),Q_{\theta }(B):=P_{\theta }(\{x:S(x)\in B\})$ for each Borel set $B\subseteq {\mathbb{R}}^d$. It records the full law of the summary, whereas $b$ records only its mean.

For the finite-dimensional minimum-distance target, assume $p\ge 1$, a nonempty compact parameter set $\mathrm{\Theta }\subseteq {\mathbb{R}}^p$ with its Euclidean subspace metric (Open cover, subcover, compact metric space, and compact subset of a metric space Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric), a binding map $b$ continuous from this metric to the Euclidean metric on ${\mathbb{R}}^d$ (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form), and a fixed summary vector $s\in {\mathbb{R}}^d$ (either the observed summary or a specified deterministic probability limit). Define $q_s(\theta ):=\mathrm{\parallel }b(\theta )-s{\mathrm{\parallel }}_2^2\mathrm{.}$ Here $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ is the Euclidean norm. Require $q_s:\mathrm{\Theta }\to \mathbb{R}$ to be continuous as part of this target’s admissible input contract. Its minimizer set is $M_s:={\mathrm{arg\hspace{0.17em}min}}_{\theta \in \mathrm{\Theta }}{q}_s(\theta )=\{\theta \in \mathrm{\Theta }:q_s(\theta )={\inf }_{\eta \in \mathrm{\Theta }}{q}_s(\eta )\}\mathrm{.}$ The indirect-inference target ${\theta }^{\mathrm{I}\mathrm{I}}(s)$ is the lexicographically least member of $M_s$: among all minimizers, minimize the first coordinate; among those, minimize the second; continue through coordinate $p$. The target is well-defined under exactly the stated compactness and continuity contract by The compact binding target exists and has a unique lexicographic tie-break ↗.

For moment-based parameter recovery, the proposed limiting summary must satisfy $s=b({\theta }_0)$ for some ${\theta }_0\in \mathrm{\Theta }$. The moment-identification condition at ${\theta }_0$ is $\{\theta \in \mathrm{\Theta }:b(\theta )=b({\theta }_0)\}=\{{\theta }_0\}\mathrm{.}$ Global moment identification means that $b$ is injective. Identifiability of $b$ as an estimand in Identifiability of an estimand is weaker: equality of sampling laws already implies equality of these means and does not ensure parameter recovery.

If the method instead compares the full summary laws, parameter identification requires injectivity of $\theta \mapsto Q_{\theta }$. By Identifiability of an estimand and Identifiability is constancy on observational classes, this is exactly identifiability of the identity estimand in the summary experiment. These identification conditions are additional assumptions for parameter recovery; the minimum-distance target and its tie rule do not require them.

No consistency theorem for an estimated minimizer follows from this target definition. Such a theorem needs additional hypotheses controlling sampling, simulation error, separation of the population target and the optimizer; the plan does not assert those hypotheses here.

The BCS-20 prerequisite page introduces numerical optimization procedures. Their iterates are computational outputs; this target remains the exact argmin with the tie rule specified above. No convergence of a numerical optimization routine to this target is claimed.