Why a Target’s Shape Changes Its Limit
A plug-in allocation rule estimates a parameter and then transforms that estimate into a target allocation proportion. If the target map is smooth at the true parameter, a first-order Taylor expansion usually gives a Gaussian limit. But rules involving a maximum, minimum, absolute value, or threshold have a kink at some parameter values. At a kink, small estimation errors on different sides can be transformed in different ways. The resulting limit can be non-Gaussian, even when the estimator itself has a Gaussian limit.
The important question is not simply whether a target formula contains a maximum or absolute value. It is whether the true parameter lies at a point where the target map is nondifferentiable. Away from a kink, the map may be locally smooth and the usual delta method may still apply.
From an Estimator Limit to a Target Limit
Assume that \(\hat\theta_n\) is consistent for \(\theta_0\) and satisfies a central limit theorem, \(\sqrt{n}(\hat\theta_n-\theta_0)\xrightarrow{d} Z\), where \(Z\) has a Gaussian distribution. If \(g\) is differentiable at \(\theta_0\), the ordinary delta method gives \(\sqrt{n}\{g(\hat\theta_n)-g(\theta_0)\}\xrightarrow{d}Dg_{\theta_0}Z\), where \(Dg_{\theta_0}\) is the derivative (a linear map).
For a kink, the relevant result is the directional delta method. In the form used here, if \(g\) is Hadamard directionally differentiable at \(\theta_0\) and the estimator has the stated Gaussian limit, then \[ \sqrt{n}\{g(\hat\theta_n)-g(\theta_0)\}\xrightarrow{d}g'_{\theta_0}(Z). \] Hadamard directional differentiability requires the directional approximation to remain valid when directions are approached by convergent sequences; this regularity condition supports passing from the estimator’s limit to the transformed limit. The derivative need not be linear. Consequently, applying it to a Gaussian vector does not in general produce a Gaussian variable.
Worked Example: An Absolute-Value Target at Its Kink
Worked Example: An Absolute-Value Target at Its Kink
Suppose a hypothetical two-arm design uses the target proportion \(q(\theta)=0.4+0.1|\theta|\) for allocating to arm 1 near \(\theta_0=0\). Here \(\theta\) is a scalar response contrast. The target is locally between 0 and 1. Let \(\hat\theta_n\) be a sample mean estimator with response standard deviation 2, based on \(n\) independent observations, and suppose the usual central limit theorem holds.
At the true value, \(q(\theta_0)=0.4\), and \(\sqrt{n}\hat\theta_n\xrightarrow{d}Z\), where \(Z\sim N(0,4)\). Since the absolute-value function has directional derivative \(h\mapsto |h|\) at zero, \[ \sqrt{n}\{q(\hat\theta_n)-q(0)\} =0.1\sqrt{n}|\hat\theta_n| \xrightarrow{d}0.1|Z|. \] Writing \(Z=2W\) for \(W\sim N(0,1)\), the limit is \(0.2|W|\), a folded-normal distribution rather than a centered normal distribution.
For example, the limiting probability that this scaled target error is at most 0.1 is \(P(0.2|W|\leq 0.1)=P(|W|\leq 0.5)=2\Phi(0.5)-1\approx 0.3829\). The error is nonnegative because the target cannot fall below 0.4 in this rule. A symmetric normal approximation centered at zero would miss that feature.
Maximum Rules: Ties and Unequal Parameters
For \(g(\theta_1,\theta_2)=\max(\theta_1,\theta_2)\), the behavior depends on whether the two true coordinates tie. At a tie, \(\theta_{01}=\theta_{02}\), the directional derivative is \(g'_{\theta_0}(h_1,h_2)=\max(h_1,h_2)\). It is nonlinear, so the limit is the maximum of the Gaussian estimation errors. If instead \(\theta_{01}>\theta_{02}\), the first coordinate remains the maximum in a sufficiently small neighborhood, and the derivative is \(h_1\). In that case the limit is the first coordinate’s Gaussian error, as under an ordinary local linear approximation.
Worked Example: A Maximum at a Tie
Consider a hypothetical target \(g(\theta_1,\theta_2)=\max(\theta_1,\theta_2)\). Suppose both true parameters are zero, and two independent estimators satisfy \(\sqrt{n}(\hat\theta_{1n},\hat\theta_{2n})\xrightarrow{d}(Z_1,Z_2)\), where \(Z_1\) and \(Z_2\) are independent \(N(0,1)\) variables. The directional derivative at the tie gives \[ \sqrt{n}\{\max(\hat\theta_{1n},\hat\theta_{2n})-0\} \xrightarrow{d}\max(Z_1,Z_2). \]
For any real \(x\), independence gives \(P\{\max(Z_1,Z_2)\leq x\}=P(Z_1\leq x,Z_2\leq x)=\Phi(x)^2\). In particular, the probability that this limit is at most zero is \(\Phi(0)^2=0.5^2=0.25\). The limit is not a centered normal variable: its distribution function is \(\Phi(x)^2\), not \(\Phi(x)\), and it has a positive median (approximately 0.545). Using the derivative for just the first coordinate would incorrectly give a standard normal limit.
Allocation Fluctuations Include More Than Target Error
A plug-in target is not always the same as the realized allocation proportion. Write \(A_n\) for the proportion actually assigned to arm 1 and \(q(\hat\theta_n)\) for its estimated target. The exact decomposition \[ \sqrt{n}\{A_n-q(\theta_0)\} = \sqrt{n}\{A_n-q(\hat\theta_n)\} + \sqrt{n}\{q(\hat\theta_n)-q(\theta_0)\} \] separates allocation tracking error from plug-in target error. If the first term converges to zero in probability, the target’s directional limit also describes the realized allocation proportion. If tracking error has a nonzero limit, its joint behavior with the target error must be included. Independence should not be assumed without justification.
Worked Example: Adding Root-\(n\) Allocation Noise
Continue with a hypothetical rule \(q(\theta)=0.4+0.1\max(\theta,0)\) at \(\theta_0=0\). Suppose \(\sqrt{n}\hat\theta_n\xrightarrow{d}Z\sim N(0,1)\), and assume the scaled tracking error \(\sqrt{n}\{A_n-q(\hat\theta_n)\}\) converges jointly with \(Z\) to an independent \(V\sim N(0,0.04)\). These are explicit assumptions about this hypothetical design, not consequences of the target formula alone.
The target-error component converges to \(0.1\max(Z,0)\), which equals zero with probability \(0.5\) and is positive otherwise. Therefore the realized allocation limit, centered at 0.4, is \(V+0.1\max(Z,0)\). Its mean is \(0.1E\{\max(Z,0)\}=0.1/\sqrt{2\pi}\approx 0.03989\). Its variance, using independence, is \[ 0.04+0.01\operatorname{Var}\{\max(Z,0)\} =0.04+0.01\left(\frac{1}{2}-\frac{1}{2\pi}\right) \approx 0.04341. \] The target component has an atom at zero, while adding the independent normal tracking error removes that atom from the sum. The resulting limit is still not generally normal: adding allocation noise does not make the target’s directional transformation linear.
Justification at a Kink
The limit for the absolute-value example follows directly from the definition of the directional derivative. For \(t>0\), \[ \frac{|0+th|-|0|}{t}=\frac{|th|}{t}=|h|. \] Thus the directional derivative at zero is \(h\mapsto |h|\), not a linear function of \(h\). For the maximum at a tie, \[ \frac{\max(\theta_{01}+th_1,\theta_{02}+th_2)-\theta_{01}}{t} =\frac{\max(th_1,th_2)}{t} =\max(h_1,h_2), \] because \(\theta_{01}=\theta_{02}\) and \(t>0\). Substituting the Gaussian estimator limit into these derivatives gives the stated nonstandard limits. By contrast, when \(\theta_{01}>\theta_{02}\), sufficiently small perturbations leave the first coordinate as the maximum, so the derivative is \(h_1\). This explains why a tie, rather than the mere presence of the word “maximum,” is central.
Diagnostics, Alternatives, and Limitations
A first diagnostic is to locate the true parameter relative to the kink. In applications it is unknown, so examine whether plausible values include a tie or threshold and whether the target changes formula across that region. Also check the estimator’s assumptions: the estimator must have the asserted limit, the target must have the required directional regularity, and any claimed allocation tracking limit must be supported by the design’s behavior. A Gaussian estimator limit does not by itself justify a Gaussian target limit.
| Approach | When it applies | What it gives or risks |
|---|---|---|
| Ordinary delta method | The target is differentiable at the true parameter. | A linear transformation of the estimator’s Gaussian limit. |
| Directional delta method | The target is directionally differentiable, including at a kink. | The directional derivative applied to the Gaussian limit; this may be non-Gaussian. |
| Simulation from the limiting estimator law | The estimator limit and directional map are specified adequately. | An approximation to the transformed limiting distribution; it can expose asymmetry and point masses. |
A common mistake is to choose one side of a kink, calculate its derivative, and report a symmetric normal interval as if that derivative applied in every direction. At a maximum tie, for example, this replaces \(\max(Z_1,Z_2)\) with \(Z_1\); the two distributions are different. The correction is to derive the directional derivative at the actual parameter configuration and transform the full estimator limit through it.
At a kink, the usual \(n\)-out-of-\(n\) bootstrap need not consistently estimate the transformed limit, because resampling around the estimated parameter may not reproduce the true kink geometry. Directional-derivative-based bootstrap methods or simulation from a justified limiting law may be appropriate, but their validity depends on the setting. The directional limit also does not resolve finite-sample uncertainty about whether the true parameter is exactly at a tie. When the target is close to a kink, approximations can be sensitive to small parameter changes; report the target rule, the relevant uncertainty, and the assumptions behind the approximation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Let \(g(\theta)=|\theta|\) and \(\sqrt{n}\hat\theta_n\xrightarrow{d}Z\sim N(0,1)\), with \(\theta_0=0\). What is the limit of \(\sqrt{n}\{g(\hat\theta_n)-g(0)\}\)?
- For \(g(\theta_1,\theta_2)=\max(\theta_1,\theta_2)\), what is the directional derivative when \(\theta_{01}=\theta_{02}\)?
- If \(\theta_{01}>\theta_{02}\), which estimator’s first-order error determines the limit of the maximum target?
- What additional term must be considered when the realized allocation proportion does not track its plug-in target to an error smaller than \(n^{-1/2}\)?
Solutions. 1. The directional derivative is \(|h|\), so the limit is \(|Z|\), a folded-normal variable. 2. It is \(g'_{\theta_0}(h_1,h_2)=\max(h_1,h_2)\), which is nonlinear. 3. The first coordinate determines it: locally the first parameter remains larger, so the limit is the first coordinate’s Gaussian estimation error. 4. Include the scaled tracking error \(\sqrt{n}\{A_n-q(\hat\theta_n)\}\), and derive its joint limit with the target-error term before combining them.