Inference for Sensitivity Bounds: Limit Laws, Bootstrap, and Breakdown Points
The population sensitivity bounds contain minima and maxima. If two branches agree in the population, sampling noise determines which branch the plug-in estimate selects. The limit can therefore be non-Gaussian even when the inputs have Gaussian limits. (Masten, Poirier, and Zhang 2024) treats this problem with directional derivatives.
1. The estimators and their limits
Estimate the propensity score by a correctly specified parametric model and the conditional quantile function by the paper’s quantile-regression specification. Evaluate the bound formulas at those estimates and integrate to obtain mean-effect bounds.
Theorem 1. Under the paper’s sampling, specification, smoothness, overlap/moment, and nonsingularity conditions, the estimated CATE, ATE, and ATT bound endpoints have root-\(n\) limiting distributions. The limits are directional derivatives of the bound maps applied to the joint first-stage Gaussian limit, together with the empirical averaging term for marginal effects. They need not be Gaussian.
The implementation trims probability indices to \([\varepsilon,1-\varepsilon]\). With fixed trimming, the limit is centered at the corresponding trimmed population bound. Claiming a CLT centered at the untrimmed bound would require an additional argument controlling trimming bias.
2. Differentiate the active branch
For \(h(a,b)=\min(a,b)\), the directional derivative in direction \((u,v)\) is
Away from a tie, a small perturbation leaves the active branch unchanged. At a tie, the perturbation selects it. Thus a jointly Gaussian direction \((u,v)\) can yield the non-Gaussian value \(\min(u,v)\).
The quantile bounds combine branch selection with smooth changes in propensities and quantile-regression coefficients. The directional chain rule follows each of these changes through to the endpoint.
3. Averaging adds another source of noise
For an ATE endpoint of the form \(\theta=P\Gamma(W,\eta)\), where \(\eta\) denotes nuisance functions, write
Covariate averaging produces the first term; nuisance estimation produces the second. The third couples the two and must be negligible under the empirical-process and smoothness conditions. Omitting covariate sampling variation would drop a first-order term.
ATT adds the uncertainty from treated-outcome means and treatment probabilities. Those quantities must be included in the same joint expansion because they are estimated from the same observations.
The population interval describes the allowed departure from independence. Sampling uncertainty concerns its estimated endpoints. The derivative of the endpoint map determines how first-stage uncertainty enters the latter calculation.
4. A derivative calculation inside a quantile bound
In the unclipped region, write the shifted probability index as
For a propensity perturbation \(p+t h_p\),
If the estimated conditional quantile function has directional fluctuation \(h_Q(u)\) and derivative \(Q'(u)\) in its probability argument, the quantile endpoint fluctuation is
There are two effects: the quantile function itself is estimated, and the estimated propensity changes where it is evaluated. Under a positive conditional density, \(Q'(u)=1/f_Y(Q(u))\), so a low density amplifies uncertainty in the probability index. Omitting the second term would treat the estimated selection correction as known.
Clipping the index to \([\varepsilon,1-\varepsilon]\) adds another branch calculation. Below \(\varepsilon\) the derivative is zero; above it the derivative is the incoming perturbation; exactly at it the lower-clipping derivative is the positive part of that perturbation. The upper boundary gives the corresponding minimum with zero. These operations explain why a Gaussian first-stage limit need not produce a Gaussian endpoint limit.
5. A direct demonstration of ordinary bootstrap failure
Take \(h(\theta)=\min(\theta_1,\theta_2)\) at \(\theta=(0,0)\) and suppose \(\sqrt n\widehat\theta\Rightarrow(Z_1,Z_2)\) with independent standard normal coordinates. The target limit is \(\min(Z_1,Z_2)\), whose cdf at zero is \(3/4\).
An ordinary bootstrap with independent normal limiting increments instead produces, conditionally on \(Z\), the limit
If \(Z_1\leq Z_2\) and \(d=Z_2-Z_1\), its conditional cdf at zero is
This equals \(3/4\) only at \(d=0\) and approaches \(1/2\) for a large gap. The estimated root-\(n\) gap has a nondegenerate random limit, so the conditional bootstrap cdf does not approach the desired fixed cdf. This is the failure mechanism studied in the directional-delta-method theory (Fang and Santos 2018).
The same minimum also changes first-order centering. At the tie,
Since \(Z_1-Z_2\sim \mathcal{N}(0,2)\) and a centered normal with standard deviation \(\sigma\) has mean absolute value \(\sigma\sqrt{2/\pi}\),
Unbiased first-stage coordinates can thus produce an endpoint whose root-\(n\) limit has a nonzero mean. This is not an omitted smooth Taylor term: choosing the smaller noisy branch creates an effect at the leading stochastic order. Likewise, a lower-clipping derivative \(h\mapsto\max(h,0)\) produces an atom at zero when \(h\) has a continuous symmetric distribution.
The numerical derivative’s tolerance separates population ties from sampling noise. If \(t_n=n^{-\gamma}\) with \(0<\gamma<1/2\), then \(t_n\to0\) but \(n^{-1/2}/t_n\to0\). A fixed branch gap is eventually much larger than \(t_n\); a root-\(n\) estimated gap at a true tie is much smaller. Choosing \(t_n\) of the same order as \(n^{-1/2}\) would retain the random gap in the limiting derivative calculation and reproduce the bootstrap problem.
The derivative calculation retains all branches tied at the population value. Selecting only the branch with the smaller estimated value would preserve sampling noise in the limiting derivative and reproduce the bootstrap failure. The full expansion also retains covariance with empirical averaging.
6. Estimate the directional derivative before applying it
Theorem 2. Under the paper’s estimation and derivative-approximation conditions, bootstrapping the joint first-stage fluctuation and applying a consistent estimate of the directional derivative consistently estimates the endpoint limit law. For fixed \(c\), the ordinary bootstrap for ATE endpoints is also valid under the additional condition \(\Pr\{p(W)\in\{c,1-c\}\}=0\). The corresponding ATT result uses \(\Pr\{p(W)=c\}=0\) and its additional moment conditions.
Use a tolerance \(t_n\) satisfying \(t_n\to0\) and \(\sqrt n\,t_n\to\infty\). A population tie creates an estimated gap of order \(n^{-1/2}\), smaller than the tolerance with high probability; a fixed nonzero population gap eventually exceeds it. This distinguishes tied branches from separated ones without treating the sample winner as known.
For the minimum example, a numerical derivative is
At the true tie, \(\widehat\theta/t_n=o_p(1)\). Positive homogeneity of the minimum then makes this expression approach \(\min(Z_1^*,Z_2^*)\), the correct limit. For the full sensitivity functional, the same idea requires directional differentiability in the relevant function space and uniform control of the numerical approximation. The step-size restriction alone is not a complete proof.
For an ATE endpoint, bootstrap covariate averaging jointly with first-stage estimation, since both use the same observations. Independent noise would discard their covariance. The ATT expansion includes treated means and propensity normalization in this joint calculation as well.
7. Confidence bands and breakdown points
Let \(L(c)\) be the lower ATE endpoint and define the robustness threshold for a positive-effect claim by
A simultaneous lower confidence band \(\ell(c)\) satisfies \(\ell(c)\leq L(c)\) throughout the specified range with probability at least \(1-\alpha\). On that event,
Taking suprema gives a conservative certified threshold. If the first set is empty, no positive range is certified. No derivative at the crossing is needed for this argument.
A band can be calibrated from the bootstrap supremum of a properly standardized endpoint process, provided process-level convergence and bootstrap validity hold over the whole range. Pointwise critical values at several chosen \(c\) values do not automatically provide this guarantee.
For comparison, a regular crossing with \(L(c^*)=0\) and \(L'(c^*)<0\) permits a root expansion
This expansion requires local uniform control and a unique regular crossing. At a flat crossing, a shape such as \(L(c)=-(c-c^*)^3\) converts bound error of order \(n^{-1/2}\) into crossing error of order \(n^{-1/6}\). Inverting a simultaneous band avoids assuming a regular normal limit for every such threshold.