Semiparametric efficiency and nearly-true models

Suppose you have $N$ people with some variables measured, and you choose a subset of $n$ to measure additional variables. I’m going to assume the probability ${\pi }_i$ that you measure the additional variables on person $i$ is known, so it has to be a setting where non-response isn’t an issue – eg, choosing which frozen blood samples to analyse, or which free-text questionnaire responses to code, or which medical records to pull for abstraction. As an example, if you have a binary outcome $Y$ you might take a case–control sample and measure $X$ on everyone with $Y=1$ and the same number of people with $Y=0$.

Suppose in addition that you want to fit a particular parametric or semiparametric model $P_{\theta ,\eta }$ to the data, where $\theta$ are parameters of interest and $\eta$ are nuisance parameters. For example, you might want to fit a logistic regression model where the coefficients are $\theta$ and the density of $X$ is $\eta$.

There are now two possible semiparametric models for the observed data. Let $R_i$ be the indicator that person $i$ is sampled. We could have

• Model D: ${\pi }_i=E\left[R_i|\text{variables available on everyone}\right]$
• Model M: the submodel of $D$ that satisfies $P_{\theta ,\eta }$

Typically, estimation under model M will be more efficient. For example, in the case-control setting with a logistic regression model for $Y|X$ we know that the efficient estimator under model M is unweighted logistic regression (per Prentice & Pyke 1979), and that the efficient estimator under model D is weighted logistic regression with weights $w_i=1/{\pi }_i$.

I want to consider slight misspecifications, where model M is ‘nearly true’. Gross misspecifications aren’t interesting: if the data don’t look anything like a sample from $P_{\theta ,\eta }$, a careful data analyst will notice and pick a different model. However, the difference between the efficient estimators under M and under D is $O_p\left(n^{-1/2}\right)$, so a bias of the same order is enough to outweigh the precision gain. It’s not obvious that we should expect to detect a misspecification of this size, so more precise investigation is needed.

The efficient estimator under $D$ is an Augmented Inverse Probability Weighted (AIPW) estimator (if you’re a biostatistician) or a calibration estimator (if you’re a survey statistician), and we can get reasonably close to it (Breslow et al, 2009). Write ${\hat{\theta }}_{\text{wtd}}$ for this estimator, and ${\hat{\theta }}_{\text{eff}}$ for the efficient estimator under $M$.

Models M and D agree when there is complete data, so I will define the true value ${\theta }_0$ of $\theta$ as the common limit of ${\hat{\theta }}_{eff}$ and ${\hat{\theta }}_{wtd}$ with complete data. Survey statisticians call this the ‘census estimator.’ Biostatisticians call it ‘our next grant proposal’.

We now need a mathematical characterisation of ‘nearly true’. I will use contiguity. A sequence of distributions $Q_n$ is contiguous to a sequence $P_n$ if for every event $A$, $P_{n}A\to 0$ implies $Q_{n}A\to 0$. They are mutually contiguous if the implication goes both ways. Let $A$ be the event that a model diagnostic accepts model $M$, and let $P_n$ be a sequence of distributions in model M. If this is a useful diagnostic, $P_{n}A\nrightarrow 0$, so for a mutually contiguous sequence of distributions $Q_n$ in model D but not in model M, $Q_{n}A\nrightarrow 0$.

Now, under M $$\sqrt{n}({\hat{\theta }}_{\text{eff}}-{\theta }_0)\stackrel{d}{\to }N(0,{\sigma }^2)$$ and $$\sqrt{n}({\hat{\theta }}_{\text{wtd}}-{\theta }_0)\stackrel{d}{\to }N(0,{\sigma }^2+{\omega }^2)$$

By the Convolution Theorem, the extra variance for ${\hat{\theta }}_{\text{wtd}}$ under model M is pure noise, so $$\sqrt{n}({\hat{\theta }}_{\text{eff}}-{\hat{\theta }}_{\text{wtd}})\stackrel{d}{\to }N(0,{\omega }^2)$$

Now, by LeCam’s Third Lemma, if we switch from $P_n$ to $Q_n$ as the data distribution there is no change in variance, but there is bias $$\sqrt{n}({\hat{\theta }}_{\text{eff}}-{\hat{\theta }}_{\text{wtd}})\stackrel{d}{\to }N(-\kappa \rho \omega ,{\omega }^2)$$ where $\kappa$ is the limit of the log likelihood ratio $\log d{Q}_n/d{P}_n$, which governs the power of the Neyman–Pearson Lemma test, and $\rho$ measures whether the misspecification is in a direction that matters for $\theta$ or not.

Substituting back, under the contiguous misspecified model sequence $Q_n$, $$\sqrt{n}({\hat{\theta }}_{\text{eff}}-{\theta }_0)\stackrel{d}{\to }N(-\kappa \rho \omega ,{\sigma }^2)$$ and $$\sqrt{n}({\hat{\theta }}_{\text{wtd}}-{\theta }_0)\stackrel{d}{\to }N(0,{\sigma }^2+{\omega }^2)$$ So, the mean squared error of ${\hat{\theta }}_{\text{wtd}}$ is lower if ${\kappa }^2{\rho }^2>1$. If $\rho \approx 1$, this happens when $\kappa \approx 1$, at which point the most powerful test for $Q_n$ vs $P_n$ has power about 24%.

That is, the least-favourable misspecification of model M leads to worse mean squared error for ${\hat{\theta }}_{\text{eff}}$ than ${\hat{\theta }}_{\text{wtd}}$ before the most powerful test of misspecification is even moderately reliable, even if we (unrealistically) knew exactly the form of the misspecification.

Since the sense in which ${\hat{\theta }}_{\text{eff}}$ is optimal is precisely this local asymptotic minimax sense within $P_{\theta ,\eta }$, it seems reasonable to use the same description of optimality outside the model. Under this description of optimality, the ‘efficient’ estimator’s optimality is not robust to undetectable model misspecification.