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 \({\cal P}_{\theta,\eta}\) 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: \(\pi_i=E[R_i|\textrm{variables available on everyone}]\)
- Model M: the submodel of \(D\) that satisfies \({\cal P}_{\theta,\eta}\)

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 \({\cal P}_{\theta,\eta}\), a careful data analyst will notice and pick a different model. However, the difference between the efficient estimators under M and under D is \(O_p(n^{-1/2})\), 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 \(\hat\theta_{\textrm{wtd}}\) for this estimator, and \(\hat\theta_{\textrm{eff}}\) 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 \(A\), \(P_nA\to0\) implies \(Q_nA\to 0\). 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, \(P_nA\not\to 0\), so for a mutually contiguous sequence of distributions \(Q_n\) in model D but not in model M, \(Q_nA\not\to 0\).

Now, under M \[\sqrt{n}(\hat\theta_{\textrm{eff}}-\theta_0) \stackrel{d}{\to}N(0,\sigma^2)\] and \[\sqrt{n}(\hat\theta_{\textrm{wtd}}-\theta_0) \stackrel{d}{\to}N(0,\sigma^2+\omega^2)\]

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

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 \[\sqrt{n}(\hat\theta_{\textrm{eff}}-\hat\theta_{\textrm{wtd}}) \stackrel{d}{\to} N(-\kappa\rho\omega,\omega^2)\] where \(\kappa\) is the limit of the log likelihood ratio \(\log dQ_n/dP_n\), 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 \(Q_n\), \[\sqrt{n}(\hat\theta_{\textrm{eff}}-\theta_0) \stackrel{d}{\to}N(-\kappa\rho\omega,\sigma^2)\] and \[\sqrt{n}(\hat\theta_{\textrm{wtd}}-\theta_0) \stackrel{d}{\to}N(0,\sigma^2+\omega^2)\] So, the mean squared error of \(\hat\theta_{\textrm{wtd}}\) 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 \(\hat\theta_{\textrm{eff}}\) than \(\hat\theta_{\textrm{wtd}}\) 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 \(\hat\theta_{\textrm{eff}}\) is optimal is precisely this local asymptotic minimax sense within \({\cal P}_{\theta,\eta}\), 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.