Kinks, Binding Inequalities, and Simulation Inference

Conceptual map

II-08.01Nonsmooth criteria
II-08.02Mean and centered components
II-08.03Probability mass at the kink
II-08.04Limits indexed by kappa
II-08.05Separate bias rates

1. A binding inequality exposes the kink

Centered simulation variation and binding-point mean shift are separate. A covariance correction addresses the first and leaves the second unchanged.

Let the population moment be $m(\theta)=\theta$ and use squared violation $Q(\theta)=[m(\theta )]_{+}^{2}$ , where $[x]_{+}=\max\{0,x\}$ . Approximate the moment at $\theta _{0}=0$ with $\hat{m}_{s}(0)=n_{s}^{-1}\sum _{j=1}^{n_{s}}Z_{j}$ , where the $Z_{j}$ are independent Rademacher variables.

The simulated score at the binding point is $S_{s}=2[\hat{m}_{s}(0)]_{+}$ . Symmetry gives a positive mean whenever the simulated average can be nonzero.

2. A two-draw score has an exact mean shift

For $n_{s}=2$ , the simulated average is $$-1$$ ,0,1 with probabilities 1/4,1/2,1/4. Therefore $S_{s}$ is zero with probability 3/4 and two with probability 1/4:

\[E[S_{s}]=\frac{1}{2}, \quad \operatorname{Var}(S_{s})=\frac{3}{4}.\](1)

Subtracting the mean yields a centered variable but does not make the original score unbiased. A variance estimator alone records 3/4 and misses the shift 1/2.

3. The relative simulation rate determines the first-order term

Proposition 1 · Binding-score rate

As $n_{s}\to \infty$ ,

\[\begin{aligned}\sqrt{n_{s}}E[S_{s}]\to \sqrt{2/\pi }, \\ \sqrt{n_{s}}(S_{s}-ES_{s})\Rightarrow 2[Z_{+}-E(Z_{+})], \quad Z\sim N(0,1).\end{aligned}\](2)

Let $\kappa _{n}=\sqrt{n/n_{s}}$ . At root- $$n$$ scale the contribution vanishes if $\kappa _{n}\to0$ , has a finite positive mean shift and non-Gaussian centered limit if $\kappa _{n}\to\kappa\in(0,\infty)$ , and is not tight on that scale if $\kappa _{n}\to \infty$ .

Proof sketch. The central limit theorem gives $\sqrt{n_{s}}\hat{m}_{s}(0)\Rightarrow Z$ . The positive-part map is continuous, and bounded second moments give uniform integrability for the first two moments. Multiply the resulting limit by $\kappa _{n}$ . ∎

4. Slack moments and constraint boundaries behave differently

If $m(\theta _{0})=-a<0$ with bounded simulation draws, the probability that simulation crosses the kink decreases exponentially in $n_{s}$ by a concentration bound. Binding moments have no such fixed margin. Local-to-boundary sequences with $$a$$ shrinking at the simulation standard-error rate interpolate between these cases.

Failure case · Centered covariance correction used for kink bias

Correcting $\operatorname{Var}(S_{s})$ leaves E[ $S_{s}$ ] unchanged. At finite positive $\kappa$ , the binding-point shift remains first order.

Other nonsmooth losses produce different maps and constants. Multiple binding inequalities introduce a multivariate cone projection and dependence across simulated moments. The result in (2) belongs to the declared scalar positive-part criterion.

5. Implementation, exercises, and sources

Classify moments as slack, binding, or local-to-binding under a documented tolerance. Estimate centered covariance and mean shift separately across independent top-level seeds. Report $\kappa _{n}$ , the simulation draw law, common-random-number structure, and how every generated stage is recomputed.

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Exercises

Verify the complete two-draw distribution and (1).
Compute the exact binding-score mean for one and three Rademacher draws.
Let $n_{s}=n^{c}$ and classify the $\kappa$ regimes.

Partial solutions

1. Only the two-positive-draw outcome generates score two. 2. For one draw the score is zero or two equally, so its mean is one. For three draws the positive averages are 1/3 and 1 with probabilities 3/8 and 1/8, giving score mean 1/2. 3. $\kappa _{n}=n^{(1-c)/2}$ ; it vanishes for $$c>1$$ , is one for $$c=1$$ , and diverges for $$c<1$$ .

A. Shapiro (1991), “Asymptotic Analysis of Stochastic Programs,” Annals of Operations Research 30, 169–186.Nonsmooth stochastic-program limits.
Donald W. K. Andrews and Gustavo Soares (2010), “Inference for Parameters Defined by Moment Inequalities,” Econometrica 78, 119–157.Binding and slack moment regimes.
A. W. van der Vaart, Asymptotic Statistics, Chapter 3.Continuous mapping and delta-method boundaries.

6. Audit checkpoint

State the loss and directional map, population slack, binding and local sequences, simulation law, draw count, $\kappa$ sequence, score mean, centered covariance, dependence, tolerance, parameter boundary, and which stages are rerun.

7. Scope boundary

The formal calculation covers one scalar squared-violation score with bounded simulation draws. Other nonsmooth estimators require their own directional derivatives and resampling theory.