Estimating the Variance Function in a Stabilizing Transformation

Why Estimate the Variance Function?

A variance-stabilizing transformation is built from the asymptotic variance of an estimator. In a simple model that variance may depend only on the parameter of interest. In many applications it also depends on nuisance quantities, such as a dispersion parameter, a variance, or a within-cluster correlation. Those quantities are unknown, so the transformation used in practice must be estimated as well.

The first-order idea is direct: estimate the nuisance quantities consistently, substitute them into the variance function, and use the resulting derivative. Continuity makes the estimated derivative converge to the derivative that would have been used if the nuisance parameters were known. One important qualification is that this local stabilization does not, by itself, remove uncertainty in a transformed estimate whose transformation is itself random.

The derivative, rather than the particular value of the transformation, is what stabilizes the variance. If \(\lambda\) were fixed and known, the delta method would give asymptotic variance \(g'_\lambda(\theta)^2V(\theta,\lambda)=1\) for \(\sqrt n\{g_\lambda(\hat{\theta})-g_\lambda(\theta)\}\). With unknown \(\lambda\), consistency can make the plug-in derivative behave the same way locally.

The First-Order Plug-In Result

Here is a precise version of that claim. Assume \(\theta\) is an interior point of its parameter space, \(\hat{\theta}\xrightarrow{p}\theta\), and \(\sqrt n(\hat{\theta}-\theta)\xrightarrow{d}N(0,V(\theta,\lambda))\), with \(0<V(\theta,\lambda)<\infty\). Assume also that \(\hat{\lambda}\xrightarrow{p}\lambda\), and that \(V(x,\eta)\) is positive and continuous for \((x,\eta)\) in a neighborhood of \((\theta,\lambda)\). Define \(g_\eta(x)=\int_{\theta_0}^{x}V(u,\eta)^{-1/2}\,du\), where \(\theta_0\) is any fixed point in the relevant interval. Then:

Proof. Let \(h(x,\eta)=V(x,\eta)^{-1/2}\). By continuity and positivity, \(h\) is continuous near \((\theta,\lambda)\). The integral definition gives \(g'_\eta(x)=h(x,\eta)\). The fundamental theorem of calculus yields \[ g_{\hat{\lambda}}(\hat{\theta})-g_{\hat{\lambda}}(\theta) =(\hat{\theta}-\theta)\int_0^1 h\bigl(\theta+t(\hat{\theta}-\theta),\hat{\lambda}\bigr)\,dt. \] Consistency of both estimates implies that, for every \(t\in[0,1]\), the arguments of \(h\) approach \((\theta,\lambda)\) uniformly in \(t\) in probability: the first coordinate differs from \(\theta\) by at most \(|\hat{\theta}-\theta|\), and the second is \(\hat{\lambda}\). Continuity therefore makes the integral converge in probability to \(h(\theta,\lambda)=1/\sqrt{V(\theta,\lambda)}\). Multiplying by \(\sqrt n(\hat{\theta}-\theta)\), which converges in distribution and is thus bounded in probability, and applying Slutsky’s theorem gives a limiting normal distribution with variance \(V(\theta,\lambda)\{1/\sqrt{V(\theta,\lambda)}\}^2=1\). \(\square\)

The theorem requires consistency of \(\hat{\lambda}\), not a root-\(n\) rate, because the nuisance estimate appears in the local derivative and the centering uses the same estimated transformation. For practical calculations, the plug-in variance estimate also satisfies \(V(\hat{\theta},\hat{\lambda})\xrightarrow{p}V(\theta,\lambda)\) by continuity. That consistency supports estimating the local derivative and variance; it is not a claim that every inferential quantity involving \(\hat{\lambda}\) has negligible uncertainty.

Worked Plug-In Transformations

When Nuisance Estimation Also Affects the Target

The theorem deliberately centers at \(g_{\hat{\lambda}}(\theta)\). If the scientific target is instead the fixed quantity \(g_\lambda(\theta)\), the difference between those centers can be random at first order. For example, suppose \((\hat{\theta},\hat{\lambda})\) has a joint asymptotic normal limit at rate \(\sqrt n\), and \(g_\lambda(\theta)\) is differentiable in both arguments. A first-order expansion about the fixed target has two terms: \[ \sqrt n\{g_{\hat{\lambda}}(\hat{\theta})-g_\lambda(\theta)\} = \frac{\partial g_\lambda(\theta)}{\partial\theta}\sqrt n(\hat{\theta}-\theta) + \frac{\partial g_\lambda(\theta)}{\partial\lambda}\sqrt n(\hat{\lambda}-\lambda) +o_p(1). \] The nuisance term and its covariance with the first term contribute to the limit variance. Consistency alone does not make that term negligible at the \(\sqrt n\) scale.

This distinction matters when the transformation itself is the reported estimand or when uncertainty is calculated around a fixed transformed parameter. To claim a standard-normal limit in that setting, one must account for the joint estimation effect, or impose additional conditions that make it negligible. The local variance function is still stabilized by the plug-in derivative, but that is not a substitute for analyzing the full estimator.

Common Mistake

• Wrong reasoning: “The nuisance estimate is consistent, so replacing the true nuisance parameter by its estimate cannot affect the first-order distribution of any transformed estimate.” Consistency only says \(\hat{\lambda}-\lambda=o_p(1)\); it does not say this difference is \(o_p(n^{-1/2})\).
• Correction: Use consistency to justify convergence of the plug-in variance function and derivative. Then state the centering carefully. For the local difference \(g_{\hat{\lambda}}(\hat{\theta})-g_{\hat{\lambda}}(\theta)\), the plug-in theorem applies. For the fixed-target difference \(g_{\hat{\lambda}}(\hat{\theta})-g_\lambda(\theta)\), include the nuisance contribution whenever it is first-order.