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The compact binding target exists and has a unique lexicographic tie-break

Statement

Let $Θ \subseteq R^{p}$ be the nonempty compact parameter set of Binding function and indirect-inference target, and fix an admissible $s$ for which its objective $q_{s} : Θ \to R$ is continuous. Then $M_{s} : = {arg min}_{θ \in Θ} q_{s} (θ)$ is a nonempty compact set and has a unique lexicographically least element. Thus the target selector specified in Binding function and indirect-inference target exists.

Facts & Assumptions

Given: A nonempty compact metric space $Θ$ with its Euclidean metric, and a continuous real-valued objective $q_{s}$ .

[F1]

The binding-target definition specifies a nonempty compact $Θ \subseteq R^{p}$ with its Euclidean subspace metric, continuity of $q_{s}$ , and lexicographic coordinate tie-breaking. Binding function and indirect-inference target

[F2]

A continuous real-valued function on a nonempty compact metric space attains its minimum. A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value

[F3]

A closed subset of a compact metric space is compact. A closed subset of a compact metric space is compact

[F4]

Continuity at $θ$ means that for every $ε > 0$ there is $δ > 0$ such that $d (η, θ) < δ$ implies $∣ q_{s} (η) - q_{s} (θ) ∣ < ε$ . Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form

[F5]

The compactness hypothesis is open-cover compactness in the induced metric; the Euclidean distance is a 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

Proof

technique · direct

1.1F1F2given

By [F2], $q_{s}$ attains a minimum $c$ on $Θ$ . Hence $M_{s} = {θ \in Θ : q_{s} (θ) = c}$ is nonempty.

2.1F1F4step 1.1algebra

The set $M_{s}$ is closed in $Θ$ . Indeed, if $θ \notin M_{s}$ , then $q_{s} (θ) > c$ ; take $ε = (q_{s} (θ) - c) / 2 > 0$ . By [F4], some $δ > 0$ makes $d (η, θ) < δ$ imply $∣ q_{s} (η) - q_{s} (θ) ∣ < ε$ , and therefore $q_{s} (η) > c$ . This ball misses $M_{s}$ , so every point of its complement has a neighborhood in that complement.

3.1F1F3step 2.1

By [F3], $M_{s}$ is compact.

4.1F2F3F5step 3.1givenconstructinduction

Set $M_{0} : = M_{s}$ . For $k = 1, \dots, p$ , suppose $M_{k - 1}$ is nonempty compact. The coordinate map $π_{k} (x) = x_{k}$ is continuous because $∣ π_{k} (x) - π_{k} (y) ∣ \leq ∥ x - y ∥_{2}$ . By [F2], it attains a minimum $a_{k}$ on $M_{k - 1}$ . Define $M_{k} : = {x \in M_{k - 1} : x_{k} = a_{k}}$ ; this set is nonempty. It is closed in $M_{k - 1}$ : for $x_{k} \neq a_{k}$ , a ball of radius $∣ x_{k} - a_{k} ∣ / 2$ around $x$ misses $M_{k}$ by the same coordinate inequality. Hence [F3] makes $M_{k}$ compact. Finite induction constructs nonempty compact $M_{1}, \dots, M_{p}$ .

5.1F1step 4.1algebra∎

The nonempty set $M_{p}$ contains a vector whose $k$ th coordinate is $a_{k}$ for each $k$ , so it contains at most one vector and is a singleton. For any $x \in M_{s}$ , take the first coordinate $k$ at which $x_{k} \neq a_{k}$ , if one exists. Then $x \in M_{k - 1}$ and the definition of $a_{k}$ gives $x_{k} > a_{k}$ . Thus the unique element of $M_{p}$ is lexicographically no larger than every member of $M_{s}$ , and is the unique lexicographically least minimizer.

Depends on

Used by

A non-injective binding map cannot recover a sign Example

Cited to discharge well-definedness by Binding function and indirect-inference target.

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christian Gouriéroux, Alain Monfort, and Eric Renault, Indirect Inference, Journal of Applied Econometrics 8, Supplement, S85–S118 (1993) (standard reference, not scraped)
Thomas J. Sargent and John Stachurski, Intermediate Quantitative Economics with Python, Chapter 38: Which Moments to Match? (standard reference, not scraped)