Asymptotic Normality of Doubly Adaptive Biased Coin Designs

Why a Gaussian Limit Has Two Sources

A doubly adaptive biased coin design (DBCD) uses two kinds of information: the allocation history and an estimate of a response-based target. Its allocation proportions fluctuate because each assignment is randomized, but they can also fluctuate because the estimated target itself changes as responses arrive. Consequently, a Gaussian limit for the allocation proportion generally contains both allocation noise and response-estimation feedback.

The convergence of the allocation proportion to an interior target is not enough to determine its fluctuations. The asymptotic normality result below describes the scale and distribution of those fluctuations, and makes explicit where the response estimator enters the covariance. The result is stated for two arms, a smooth target, and regular estimators; more complicated allocation settings require additional structure.

Design and Regularity Conditions

Let $N_{1,n}$ be the number of participants assigned to arm 1 among the first $n$ participants, and let $X_n=N_{1,n}/n$. The target proportion for arm 1 is $\rho=\rho(\theta)$, where $\theta$ denotes the response parameter and $0<\rho<1$. At stage $n+1$, write $I_{n+1}=1$ if the new participant is assigned to arm 1 and $I_{n+1}=0$ otherwise. The design uses the current estimate $\hat\rho_n=\rho(\hat\theta_n)$.

For a fixed bias parameter $\gamma\geq 0$, a two-arm DBCD can use the assignment probability

p_n=P(I_{n+1}=1\mid\mathcal F_n) = \frac{\hat\rho_n^{\,1+\gamma}X_n^{-\gamma}} {\hat\rho_n^{\,1+\gamma}X_n^{-\gamma} +(1-\hat\rho_n)^{\,1+\gamma}(1-X_n)^{-\gamma}},

where $\mathcal F_n$ contains the allocation and response history through stage $n$. This formula is used when both current proportions and estimated targets are interior. In practice, initial allocations and suitable safeguards are needed to keep the procedure well-defined if an arm has no observations or an estimate reaches a boundary.

For the limit calculation, assume the true target is interior, the target function is differentiable near the true response parameter, and the response estimator is consistent and admits a suitable asymptotic linear expansion. Assume also that responses have finite second moments, the assignment mechanism is implemented as specified, and the design and estimator satisfy a joint martingale central limit theorem. More specifically, suppose the target-estimation error can be represented, to the order needed for the limit, as

\hat\rho_n-\rho = \frac{1}{n}\sum_{j=1}^{n}\psi_j+o_p(n^{-1/2}),

where $\psi_j$ is the target-scale influence contribution from observation $j$. Assume the paired increments $(I_j-p_{j-1},\psi_j)$ have limiting covariance matrix with entries $\sigma_\varepsilon^2$, $\sigma_\psi^2$, and $\sigma_{\varepsilon\psi}$, respectively, and that the weighted remainder in the expansion is negligible. These regularity conditions make precise the requirement that the estimator’s influence process and allocation noise have a joint Gaussian limit. They are stronger than consistency alone.

Definition: The centered allocation proportion is $X_n-\rho$. Its asymptotic fluctuation is the scaled quantity $\sqrt n(X_n-\rho)$. Here $\sigma_\varepsilon^2$ is the limiting variance of the allocation innovation $I_{n+1}-p_n$, $\sigma_\psi^2$ is the limiting variance of the target-estimation influence contribution, and $\sigma_{\varepsilon\psi}$ is their limiting covariance.

Linearizing the Allocation Rule

The key step is to see how the biased coin responds to small deviations in the current allocation and the estimated target. Let $g(x,q)$ denote the displayed allocation probability when $X_n=x$ and $\hat\rho_n=q$. Its log odds are

\log\!\left(\frac{g(x,q)}{1-g(x,q)}\right) =(1+\gamma)\log\!\left(\frac{q}{1-q}\right) -\gamma\log\!\left(\frac{x}{1-x}\right).

At $x=q=\rho$, $g(\rho,\rho)=\rho$. Differentiating the log odds and then converting from log odds to probability gives $g_x(\rho,\rho)=-\gamma$ and $g_q(\rho,\rho)=1+\gamma$. Thus, with $c=1+\gamma$, a first-order expansion is

p_n-\rho = -\gamma(X_n-\rho)+c(\hat\rho_n-\rho) +o_p\!\left(|X_n-\rho|+|\hat\rho_n-\rho|\right).

Since $X_{n+1}=X_n+(I_{n+1}-X_n)/(n+1)$, its conditional drift is governed by $p_n-X_n$. Substituting the expansion shows that this drift is approximately $-c(X_n-\rho)+c(\hat\rho_n-\rho)$. The first term pulls allocation proportions toward the target. The second term transmits estimation error into the allocation path.

Limiting Distribution and Justification

Theorem: Under the interior-target, smoothness, moment, joint central limit, and negligible-remainder conditions stated above, the centered allocation proportion has the limit $$ \sqrt n(X_n-\rho)\ \xrightarrow{d}\ N(0,V), $$ where $c=1+\gamma$, $$ V=\sigma_\varepsilon^2\int_0^1 t^{2c-2}\,dt +\sigma_\psi^2\int_0^1 K_c(t)^2\,dt +2\sigma_{\varepsilon\psi}\int_0^1 t^{c-1}K_c(t)\,dt. $$ Here $$ K_c(t)= \begin{cases} \dfrac{c}{c-1}\left(1-t^{c-1}\right),&c>1,\\[6pt] -\log t,&c=1. \end{cases} $$ The first term is allocation-randomization variation; the second is response-estimation feedback; the third accounts for covariance between their influence increments.

A short derivation explains the two kernels. Write $D_n=X_n-\rho$ and $\varepsilon_{n+1}=I_{n+1}-p_n$. Ignoring terms that are negligible after scaling, the allocation recursion is

D_{n+1} \approx \left(1-\frac{c}{n+1}\right)D_n +\frac{\varepsilon_{n+1}+c(\hat\rho_n-\rho)}{n+1}.

Iterating this recursion gives a weight proportional to $n^{-c}k^{c-1}$ for the innovation at stage $k$. After multiplication by $\sqrt n$, the allocation innovations therefore have limiting weight $t^{c-1}$, where $t=k/n$. For the estimator contribution, substitute $\hat\rho_k-\rho\approx k^{-1}\sum_{j\leq k}\psi_j$ into the recursion and collect the coefficient of each $\psi_j$. For $c>1$, its limiting weight is

\frac{c}{c-1}(1-t^{c-1})=K_c(t).

When $c=1$, the corresponding sum has a logarithmic weight and its limit is $-\log t$. Applying the joint martingale central limit theorem to these weighted sums gives a centered normal limit. The variance is the sum of the allocation-innovation variance, the estimator-influence variance, and twice their covariance, each integrated against the appropriate pair of weights. The initial allocation contributes no term at the $\sqrt n$ scale under these conditions.

Worked Examples

Worked Example: Known Target and Randomization Noise

Suppose the target is known to be $\rho=0.30$, and the DBCD uses $\gamma=1$. There is no target-estimation error, so $\sigma_\psi^2=0$ and $\sigma_{\varepsilon\psi}=0$. The allocation innovation is Bernoulli with limiting probability $0.30$, giving $\sigma_\varepsilon^2=0.30(0.70)=0.2100$. Here $c=2$, and

V=0.2100\int_0^1 t^2\,dt =0.2100\left(\frac{1}{3}\right) =0.07000.

Thus $\sqrt n(X_n-0.30)$ converges to a normal distribution with mean 0 and variance $0.07000$, or standard deviation $\sqrt{0.07000}=0.2646$. This standard deviation is on the scaled proportion scale; for large $n$, the approximate standard deviation of $X_n$ is $0.2646/\sqrt n$.

Worked Example: An Estimated Target Adds Variability

Consider the same target and bias parameter, but suppose the target is estimated from responses. For illustration, assume the limiting target-influence variance is $\sigma_\psi^2=0.1600$, and the allocation and target influence increments have zero limiting covariance. These are hypothetical asymptotic inputs, not data from a real trial. The randomization contribution remains $0.07000$. With $c=2$, $K_c(t)=2(1-t)$, so the estimator-feedback contribution is

0.1600\int_0^1 [2(1-t)]^2\,dt =0.1600\cdot4\left(\frac{1}{3}\right) =0.2133.

Consequently $V=0.07000+0.2133=0.2833$, and the limiting standard deviation of $\sqrt n(X_n-\rho)$ is $\sqrt{0.2833}=0.5323$. Ignoring target estimation would understate the asymptotic allocation variability in this example.

Worked Example: Nonzero Covariance Between Influences

Retain the preceding hypothetical values, but now suppose the limiting covariance is $\sigma_{\varepsilon\psi}=0.03000$. The covariance matrix is admissible: its determinant is $0.2100(0.1600)-(0.03000)^2=0.03360-0.0009000=0.03270>0$. For $c=2$, the cross-term integral is

2(0.03000)\int_0^1 t\,[2(1-t)]\,dt =0.06000\cdot 2\left(\frac{1}{2}-\frac{1}{3}\right) =0.02000.

The total limiting variance is $V=0.07000+0.2133+0.02000=0.3033$, with standard deviation $\sqrt{0.3033}=0.5507$ on the scaled proportion scale. A positive covariance increases the variance here; a negative covariance would reduce it, provided the full covariance matrix remains valid.

Comparison, Diagnostics, and Limitations

With a known target, the limit contains only allocation noise. With a response-estimated target, the influence of the estimator must also be included, along with any covariance between the two sources. This distinguishes DBCD from a design that randomizes independently around a fixed target. Increasing $\gamma$ strengthens the local pull back toward the current target, changing the weights on both sources; it does not make response-estimation uncertainty disappear. The Gaussian limit also differs from random-limit behavior in classical urn schemes, where the limiting proportion need not be a fixed interior target.

The assumptions should be checked against the design and estimator, not inferred from a plausible numerical fit. Relevant diagnostics include whether both arms continue to receive enough participants, whether estimated targets remain interior, whether the target map is smooth near the true parameter, and whether response residuals have finite and reasonably stable second moments. A simulation can help assess finite-sample approximation, but it does not establish the required joint central limit theorem or negligible remainder.

The result may fail or require a different normalization if the target is on the boundary, the estimator is nonregular, response tails invalidate the moment assumptions, or allocation probabilities become degenerate. A nonsmooth target can also invalidate the first-order Taylor expansion. Even when the theorem applies, it is a large-sample approximation, not a claim that the finite-sample allocation proportion is exactly normal.

Common mistake: using only $\sigma_\varepsilon^2\int_0^1t^{2c-2}\,dt$ because the assignments are randomized. Correction: if the target is estimated, include the estimator-feedback term and, when present, the cross-covariance term.
Common mistake: substituting the variance of $\hat\theta_n$ directly for $\sigma_\psi^2$. Correction: the influence variance must be on the target scale, including the derivative of $\rho(\theta)$, and its time-weighted effect is given by $K_c$.
Common mistake: treating convergence $X_n\to\rho$ as proof of a $\sqrt n$ Gaussian limit. Correction: convergence alone gives neither the rate nor the distribution; the linearization and joint limit conditions are essential.

Key takeaway: The limiting allocation variance is determined by weighted allocation innovations and weighted target-estimation influence increments. Response estimation enters through its influence process, not merely through the target value to which the design converges.

Check Your Understanding

Answer each question, then compare with the brief solutions.

For a known target and $\gamma=0$, what is the asymptotic allocation variance in terms of $\sigma_\varepsilon^2$?
Why can the response-estimation contribution not generally be found by adding $\sigma_\psi^2$ directly to the allocation variance?
For $c=2$, evaluate $\int_0^1K_c(t)^2\,dt$.
Name two conditions whose failure can make the stated Gaussian limit unreliable.

Solutions. (1) Here $c=1$, so $\int_0^1t^0\,dt=1$ and the variance is $\sigma_\varepsilon^2$. (2) The estimator influence is accumulated over stages with weight $K_c(t)$, so its contribution is $\sigma_\psi^2\int_0^1K_c(t)^2\,dt$, with a possible cross term. (3) Since $K_2(t)=2(1-t)$, the integral is $4\int_0^1(1-t)^2dt=4/3$. (4) Examples include a boundary target, a nonsmooth target map, inadequate arm sampling, or response moments too heavy for the assumed joint central limit theorem.