Why Endpoint Inference Is Different
Partial identification does not determine one parameter value; it determines a set of values compatible with the assumptions and population information. In many applications that set is an interval, \([a,b]\). Sampling uncertainty affects both endpoints, so reporting only the estimated interval \([\hat a,\hat b]\) can give a misleading impression of precision. Inference must account for uncertainty about where the endpoints lie, while preserving the distinction between an identified set and a point estimate.
A frequent source of difficulty is that an endpoint is often the maximum or minimum of several estimated constraints. When one constraint uniquely determines the endpoint, ordinary smooth approximations may work locally. When two or more constraints tie, however, small sample fluctuations can change which constraint is binding. The resulting endpoint estimator can have a nonnormal limit, and a standard bootstrap may not reproduce it.
A common representation comes from moment inequalities. Suppose \(g_j(\theta)\geq 0\), for \(j=1,\ldots,J\), are population restrictions. Then the identified set is \(\Theta_I=\{\theta:g_j(\theta)\geq0\ \text{for every }j\}\). For simple upper and lower restrictions, this gives
provided the lower endpoint does not exceed the upper endpoint. Sample analogues replace the population bounds by estimates. The maximum and minimum operations are central to the sampling behavior of those endpoint estimates.
When Maxima and Minima Have Nonstandard Limits
Consider two estimated candidate lower bounds, \(\hat\ell_1\) and \(\hat\ell_2\), and let \(\hat a=\max(\hat\ell_1,\hat\ell_2)\). If the true lower bounds differ by a fixed positive amount, the larger one will usually remain the maximum as the sample grows. In that case, the endpoint behaves asymptotically like the estimator of that single active bound.
At a tie, both constraints determine the population endpoint. Then the sample maximum depends on which estimation error happens to be larger. The joint behavior of the candidate estimators matters: separate statements about their marginal limits do not determine the distribution of their maximum or minimum.
The joint convergence in this result is important: it specifies the dependence between the errors. The continuous mapping theorem applies to the vector and then to the maximum or minimum of its components. A normal approximation for each component separately is not enough to obtain the distribution of the endpoint.
Worked Example: A Tied Lower Endpoint
Suppose the true lower endpoint is \(a=\max(\ell_1,\ell_2)\), with \(\ell_1=\ell_2=2.0\). In repeated samples, assume \(\big(\sqrt n(\hat\ell_1-\ell_1),\sqrt n(\hat\ell_2-\ell_2)\big)\to_d(Z_1,Z_2)\), where \(Z_1\) and \(Z_2\) are independent standard normal variables. Find the limiting distribution of the scaled endpoint error.
Since both true constraints tie, the endpoint estimate is the maximum of two estimators at the same population value. Therefore \[ \sqrt n(\hat a-a)\to_d \max(Z_1,Z_2). \] For any real \(x\), independence gives \[ P\{\max(Z_1,Z_2)\leq x\} =P(Z_1\leq x,\ Z_2\leq x) =\Phi(x)^2, \] where \(\Phi\) is the standard normal cumulative distribution function. This is not the standard normal distribution: for example, at \(x=0\) its cumulative probability is \(0.5^2=0.25\), not \(0.5\).
Thus, even though each candidate estimator has a normal limit, the tied endpoint has a different limit. A usual symmetric normal interval centered on \(\hat a\) does not automatically have the advertised coverage.
The same reasoning applies to an endpoint defined by several constraints. At a tie, the local limit is the maximum or minimum of the joint Gaussian errors for the tied constraints. Away from a tie, the unique binding constraint generally determines the first-order limit. In finite samples, near ties can still matter: the data may not reveal clearly which constraint is active, even if the population constraints are not exactly equal.
Confidence Procedures for Endpoints
One practical approach is to form simultaneous confidence intervals for the candidate bounds and then transform those intervals through the maximum or minimum. This avoids requiring a normal approximation for the endpoint itself. It can be conservative, but it makes the coverage argument transparent.
Suppose simultaneous intervals \([L_k,U_k]\) cover every lower-bound quantity \(\ell_k\), and \([C_j,D_j]\) cover every upper-bound quantity \(u_j\), with probability at least approximately \(1-\alpha\). On the event that all these intervals cover, monotonicity implies \[ \max_k L_k\leq a\leq\max_k U_k, \qquad \min_j C_j\leq b\leq\min_j D_j. \] Consequently, \([\max_k L_k,\max_k U_k]\) is an interval for the lower endpoint and \([\min_j C_j,\min_j D_j]\) is an interval for the upper endpoint. If both endpoint intervals are constructed on the same simultaneous coverage event, the outer interval \([\max_k L_k,\min_j D_j]\) contains the entire identified set \([a,b]\) on that event.
Worked Example: Simultaneous Endpoint Intervals
Suppose two candidate lower bounds have estimates \(1.00\) and \(1.20\), with standard errors \(0.10\) and \(0.12\). Two candidate upper bounds have estimates \(3.00\) and \(2.80\), with standard errors \(0.15\) and \(0.20\). Assume independent sampling across observations, asymptotic normality of the four estimates with consistent standard errors, and use the Bonferroni critical value \(z_{0.99375}\approx2.498\). This gives four two-sided intervals with approximate simultaneous coverage of at least 95%.
For the lower-bound candidates, the intervals are \[ 1.00\pm2.498(0.10)=[0.750,1.250], \qquad 1.20\pm2.498(0.12)=[0.900,1.500]. \] The estimated lower endpoint is \(\hat a=\max(1.00,1.20)=1.20\). Transforming the simultaneous intervals through the maximum gives \[ a\in[\max(0.750,0.900),\max(1.250,1.500)] =[0.900,1.500]. \]
For the upper-bound candidates, \[ 3.00\pm2.498(0.15)=[2.625,3.375], \qquad 2.80\pm2.498(0.20)=[2.300,3.300]. \] The estimated upper endpoint is \(\hat b=\min(3.00,2.80)=2.80\), and its transformed interval is \[ b\in[\min(2.625,2.300),\min(3.375,3.300)] =[2.300,3.300]. \]
The estimated identified set is \([1.20,2.80]\). The outer interval \([0.900,3.300]\) is a conservative 95% confidence envelope for the entire population identified set under the stated asymptotic and simultaneous-coverage assumptions. It is not a confidence interval asserting that the parameter is point identified.
Bonferroni simultaneous intervals do not require the four estimates to be independent of one another; the union bound controls the probability that any component interval misses. They do require valid individual interval approximations. More refined procedures can use the joint covariance structure or exploit the moment-inequality formulation, but their critical values must account for which inequalities may be binding.
Inference from Moment Inequalities
Moment inequalities can also be used to assess whether a proposed value of \(\theta\) is compatible with the restrictions. For a fixed candidate, estimate each moment \(g_j(\theta)\) and its standard error. A negative estimated moment is evidence against the restriction, but sampling variation means that a small negative value need not imply incompatibility.
A simple conservative test rejects the null that all \(J\) inequalities hold if at least one estimated moment is significantly negative. If the standardized estimated moments are approximately normal, a Bonferroni one-sided critical value \(z_{1-\alpha/J}\) controls the familywise rejection probability asymptotically. This test may have low power, but it avoids treating each inequality as though it were the only one examined.
Worked Example: Checking a Candidate Against Two Inequalities
Suppose a candidate value \(\theta=3.8\) must satisfy two upper-bound restrictions, with population moments \(g_1(\theta)=u_1-\theta\geq0\) and \(g_2(\theta)=u_2-\theta\geq0\). In a hypothetical sample, the estimated moments are \(0.20\) and \(-0.10\), with standard errors \(0.20\) and \(0.25\). Use a 5% Bonferroni test.
The estimated standardized moments are \[ \frac{\hat g_1}{SE_1}=\frac{0.20}{0.20}=1.00, \qquad \frac{\hat g_2}{SE_2}=\frac{-0.10}{0.25}=-0.400. \] For two one-sided restrictions, the Bonferroni lower-tail critical value is \(z_{1-0.05/2}\approx1.960\). Reject only if at least one standardized moment is less than \(-1.960\). Neither \(1.00\) nor \(-0.400\) crosses that threshold, so this test does not reject the candidate at the 5% level.
This conclusion means the sample does not provide sufficient evidence against the candidate under this test. It does not prove that \(3.8\) belongs to the population identified set; nor does it establish that the candidate is the true parameter value.
Why the Ordinary Bootstrap Can Fail
The ordinary \(n\)-out-of-\(n\) bootstrap often works for estimators whose first-order behavior is a smooth, linear transformation of an asymptotically normal estimator. A maximum or minimum at a tie is different: its first-order transformation is nonlinear. At a tie, the centered endpoint error has a limit such as \(\max(Z_1,Z_2)\), while the bootstrap calculation is centered around the sample-selected maximum. The sample may select one constraint because of noise, and the conditional bootstrap distribution can then behave as though that selected constraint were uniquely active. It may fail to reproduce the tied limiting distribution.
This failure is not a claim that the bootstrap is always invalid for endpoints. If a single constraint is separated from the others by a fixed gap and regularity conditions hold, the endpoint can behave smoothly in a neighborhood of the population value, and ordinary bootstrap inference may be justified. The concern is nonuniform behavior near ties, where a procedure that works under a fixed separation can give poor coverage as the gap shrinks.
Alternatives include conservative simultaneous intervals for the underlying bounds, subsampling or an \(m\)-out-of-\(n\) bootstrap under suitable conditions, and methods designed to approximate the relevant directional limit. For moment inequalities, critical values must also reflect the possibility that some restrictions bind while others are slack. No one method is automatically valid in every application; the data structure, dependence, and assumptions behind the moment estimates matter.
Common Mistakes and Practical Checks
- Reporting only \([\hat a,\hat b]\) as if its endpoints were known. The estimated interval does not convey sampling uncertainty. Report endpoint uncertainty or a confidence envelope as well.
- Using marginal limits to infer a maximum's limit. Marginal normal limits do not determine the distribution of a maximum or minimum. State joint convergence of the vector and its dependence structure.
- Assuming every endpoint estimate is asymptotically normal. A tied maximum or minimum can have a nonnormal limit. Check whether constraints may tie or be close enough to tie to matter.
- Reading non-rejection as proof of membership. Failure to reject a candidate moment restriction is not proof that the candidate belongs to the identified set.
- Bootstrapping without checking smoothness. The ordinary bootstrap can fail at a tie even when each underlying sample moment has a regular normal limit.
Check Your Understanding
Use these questions to check whether you can distinguish endpoint uncertainty from point identification.
- For \(\Theta_I=[\max(\ell_1,\ell_2),\min(u_1,u_2)]\), which candidate lower and upper constraints determine the endpoints?
- Why is joint convergence of two estimator errors needed to derive the limit distribution of their maximum?
- If two independent standard normal errors determine a tied maximum, what is the cumulative distribution function of that maximum at \(x\)?
- What does a simultaneous confidence envelope for \([a,b]\) cover, and how is that different from claiming a single parameter value has been identified?
- Why can the ordinary \(n\)-out-of-\(n\) bootstrap fail when two constraints tie?