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You've Got to be Efficient: Ambiguity, Misspecification and Variational Preferences
box1976science famously observed that all models are wrong, since they are necessarily approximations of reality. Any researcher or decision-maker who relies on a statistical model to learn about a parameter of interest must therefore contend with the possibility that the likelihood is misspecified. At the same time, researchers are often unable or unwilling to commit to a single prior over the parameter. In practice, then, decision-makers confront both prior ambiguity and likelihood misspecification.
This article introduces a framework for evaluating statistical decisions under both sources of concern. Following Bayesian practice, we define a statistical model as a joint distribution comprising a prior and a likelihood. We argue that both components are necessary because they capture fundamentally different types of uncertainty. The prior encodes epistemic uncertainty --- subjective uncertainty arising from incomplete knowledge about the parameter of interest --- while the likelihood captures aleatoric uncertainty --- the objective randomness inherent in any statistical experiment.
To account for prior ambiguity, we define an ambiguity set: a frequentist model that pairs a possibly misspecified likelihood with every possible prior. Following cerreia2026making, we then uniformly expand this set by a Kullback--Leibler radius to accommodate likelihood misspecification. The optimal decision rule is defined as the one that achieves the lowest expected loss under the worst-case model from this expanded set.
We show that optimal decisions under this formulation are equivalent to minimax decisions with an exponentially tilted loss function. Likelihood misspecification manifests as an exponential tilting of the loss, while prior ambiguity corresponds to a search for the least favorable prior. Our framework thus enables a clean separation between ambiguity and misspecification. Furthermore, when there is no fear misspecification, the optimal decisions reduce to the standard wald1950statistical formulation of minimax decisions under ambiguity alone --- the formulation underlying most of frequentist analysis.
This separation between ambiguity and misspecification also enables us to develop a local asymptotic theory under global misspecification, achieved by localizing the priors around a reference parameter. Under mild conditions, the finite-sample likelihoods, which may themselves be misspecified, can be replaced by a limit experiment involving the Gaussian family as the reference likelihood. While a substantial literature studies local asymptotics under local misspecification, where misspecification typically manifests as an added bias in the Gaussian limit, our framework permits the Gaussian family itself to be globally misspecified in the limit experiment, thereby accommodating much richer classes of misspecification.
Local asymptotics also simplifies the search for optimal decisions, as these are considerably easier to characterize in the limit experiment. Quite remarkably, we find that for estimation and treatment assignment problems, optimal decisions coincide with those under correct specification, regardless of the degree of global misspecification. For these problems, it is therefore always optimal for the decision-maker to proceed as if the likelihood were correctly specified and select the resulting optimal decision rule. Intuitively, these results arise because misspecification under our formulation is symmetric around the reference Gaussian likelihood. Since the estimation and treatment-assignment loss functions are also symmetric around the parameter of interest, any estimator that is not efficient under correct specification would break this symmetry. Because nature chooses the least favorable likelihood specification given the decision-maker's choice of estimator, departures from symmetry necessarily incur higher decision risk.
We extend our local asymptotic theory to semi-parametric models, and show that the above results on optimal decisions apply to that setting as well.
Our findings have a number of practical consequences. In applications, researchers often employ inefficient estimators over efficient ones, a practice frequently justified on the grounds that under misspecification, no estimand recovers the precise parameter of economic interest (andrews2025purpose). Our results, however, suggest that this reasoning is incomplete. While the parameter of interest cannot be recovered with certainty under misspecification, our decision-theoretic analysis shows that efficient estimators under correct specification also deliver the lowest decision risk under arbitrary misspecification. In the case of parametric models, these results suggest that practitioners should prefer maximum likelihood over the simulated method of moments, irrespective of the degree of misspecification. Similarly, in the context of GMM, practitioners should prefer efficient estimation methods, such as two-step GMM, over diagonally weighted or inefficient alternatives. Misspecification concerns alone cannot justify the use of the inefficient estimators over parametrically or semi-parametrically efficient alternatives.
This article relates to an extensive literature on ambiguity and misspecification spanning economics, statistics, and computer science. A detailed comparison of our approach with alternative decision-theoretic frameworks is deferred to Section (ref). Here, we restrict ourselves to a broad survey of the literature on ambiguity and misspecification.
The analysis of optimal decisions under prior ambiguity originates with wald1950statistical. A substantial body of work in statistics has extended this framework to local asymptotics; we refer to ibragimov1981problem,le1986asymptotic,van1996weak,van2000asymptotic for textbook treatments. A central result from this literature is that semi-parametrically efficient estimators are asymptotically minimax optimal under prior ambiguity.
The literature on misspecification is equally extensive. huber1964robust proposes a contamination model to address likelihood misspecification. hansen2011robustness develop an approach that involves selecting a worst-case likelihood from an ambiguity set defined by surrounding a reference or approximate likelihood with a Kullback--Leibler divergence ball of finite radius. The related field of Distributionally Robust Optimization (DRO) takes the reference distribution to be the empirical distribution $\hat{\mathbb{P}}$ of the data in the experiment, and employs more general measures of distance from $\hat{\mathbb{P}}$ to define ambiguity sets --- including $\phi$-divergence measures (e.g., reverse KL divergence and total-variation distance), Wasserstein distances and Levy-Prokhorov distances. We refer to Ben-Tal:RobustOptimization for a textbook treatment of DRO, and to rahimian2019distributionally, kuhn2025distributionally for recent surveys. These methods do not account for prior ambiguity and, consequently, do not reduce to the standard minimax formulation that underpins frequentist analysis in the absence of misspecification concerns. Our approach instead follows the recent work of cerreia2026making by first defining an ambiguity set to address prior ambiguity and then uniformly expanding this entire set by a KL divergence radius to accommodate likelihood misspecification. In contrast to the results from DRO, we find that optimal estimation and treatment assignment are invariant to the degree of misspecification.
This article adopts a decision-theoretic approach to ambiguity and misspecification, which is closely related to the literature in economic theory on variational preferences, as expounded in maccheroni2006ambiguity and cerreia2026making. The econometrics literature has also studied alternative, non-decision-theoretic approaches to misspecification, and we refer to armstrong2025misspecification for a recent survey. For instance, white1982maximum defines pseudo-parameters as the probability limits of estimators, and views them as suitably defined approximations to the underlying parameter of interest. The partial identification approach of manski2003partial proposes set-identifying the parameter under misspecification, while masten2021salvaging develop methods for sensitivity analysis. A limitation of these approaches relative to the decision-theoretic framework is that they do not directly identify the optimal statistical decision that a decision-maker should employ.
To illustrate our formalism, we introduce the following running example. A decision-maker, Alice, is tasked with determining whether a drug should be approved for use in the US population. She is therefore interested in learning about the parameter $\theta\in\Theta$, defined as the average population treatment effect. We assume binary outcomes, so that the population outcome distribution is $\text{Bernoulli}(\theta)$.
To assess the drug's efficacy, the pharmaceutical company has conducted a randomized controlled trial with $n$ observations. Given the observed data $\bm{x}\in\mathcal{X}$ from the trial, Alice seeks to choose a decision $\delta:\mathcal{X}\to\mathcal{A}$ so as to maximize her utility $u_{n}(\theta,\delta)$, or equivalently, minimize the loss function $l_{n}(\theta,\delta)=-u_{n}(\theta,\delta)$. Examples of loss functions include the estimation loss, \[ l_{n}(\theta,\delta)=\ell\bigl(\sqrt{n}(\theta-\delta)\bigr),\quad\delta\in\Theta, \] for some bowl-shaped function $\ell(\cdot)$, e.g., $\ell(z)=z^{2}$ for mean squared error, and the treatment-assignment loss, \[ l_{n}(\theta,\delta)=\sqrt{n}\bigl(\theta\,\mathbf{1}\{\theta\geq0\}-\theta\,\delta\bigr),\quad\delta\in\{0,1\}. \] Under the estimation loss, the goal is to learn directly about the parameter $\theta$, whereas under the treatment-assignment loss, the goal is to either approve ($\delta=1$) or reject ($\delta=0$) the drug for use in the entire population.
Unfortunately for Alice, the trial was conducted exclusively in the state of Pennsylvania. Because the drug is novel, she has no formal basis for judging whether, or to what degree, treatment effects observed in Pennsylvania are representative of those in the broader US population. This gives rise to model misspecification concerns. At the same time, Alice also faces ambiguity concerns, as she is unable to form an initial prior over $\theta$. We now describe a formalism that accommodates both.
We begin by formally defining the notion of a statistical model in the absence of ambiguity or misspecification concerns.
Since the loss function takes the form $l_{n}(\theta,\delta(\bm{x}))$, the payoff-relevant state of the world is given by $\omega=(\theta,\bm{x})$: an oracle who knows $\omega$ would recover Alice's loss with certainty. Following the framework of Savage or Anscombe--Aumann (anscombe1963definition), we define a model $m\equiv m(\theta,\bm{x})$ as a probability distribution over the payoff-relevant state $\omega=(\theta,\bm{x})$. This distribution admits a natural decomposition into a prior and a likelihood:
Here, $\pi(\theta)$ denotes the posited prior, the marginal distribution over $\theta$, while $p_{\theta}(\bm{x})=p(\bm{x}\mid\theta)$ denotes the posited likelihood, the conditional distribution of $\bm{x}$ given $\theta$. Equation ((ref)) is nothing more than the definition of a Bayesian statistical model; see robert2007bayesian.
The decomposition of a model into a prior and a likelihood is a canonical feature of Bayesian decision-making. Given the importance of this decomposition for what follows, it is worth understanding why both components are necessary. As we argue below, they capture fundamentally different sources of uncertainty: epistemic and aleatoric. In the terminology of anscombe1963definition, these correspond to the uncertainties involved in horse gambles and roulette wheels.
Epistemic uncertainty refers to uncertainty arising from a lack of knowledge --- uncertainty that can, in principle, be reduced through the acquisition of additional data or evidence. In the Anscombe--Aumann framework, this is the uncertainty of a horse gamble. Because the parameter $\theta$ enters Alice's loss function directly, it is natural to regard it as a quantity that exists in principle but that Alice does not know. The prior $\pi(\theta)$ thus encodes Alice's epistemic uncertainty due to her imperfect knowledge of $\theta$. Crucially, Alice can conceptualize $\theta$ independently of the likelihood. She may, for instance, have access to prior information --- such as data from related studies --- that enables her to form a prior $\pi$ without reference to whatever experiment the pharmaceutical company may have conducted.
Aleatoric uncertainty, by contrast, refers to inherent randomness, which is implicit in the design of any statistical experiment. The likelihood $p_{\theta}(\bm{x})$ captures precisely this source of uncertainty. In conducting the trial, the pharmaceutical company presumably drew a random sample of $n$ observations from the population of Pennsylvania. This sampling procedure requires the use of an implicit or explicit random number generator and therefore introduces genuine randomness; in the Anscombe--Aumann framework, this is uncertainty generated by roulette wheels. The likelihood thus describes the distribution of the data $\bm{x}$ induced by this randomness, for any given value of $\theta$.
Importantly, in our framework, the likelihood does not rise to the status of a model. It provides only a mapping from the parameter $\theta$ to the distribution of $\bm{x}$. Because $\theta$ enters Alice's loss function directly, knowledge of the correct likelihood would not enable Alice to obtain a probabilistic forecast of her loss, as she would still face epistemic uncertainty over $\theta$.
We now incorporate prior ambiguity into our framework. Suppose that Alice is unable to form a single prior, perhaps because she is ambiguity-averse in the sense of maccheroni2006ambiguity. Instead, she posits a structured set of models, \[ \mathcal{Q}:=\bigl\{\pi(\theta)\otimes p_{\theta}(\bm{x}):\pi\in\Delta(\Theta)\bigr\}, \] where $\Delta(\Theta)$ denotes the set of all probability distributions over $\theta$, while continuing to treat the likelihood $p_{\theta}(\bm{x})$ as correctly specified. In the spirit of wald1950statistical, Alice could then choose the decision rule that performs best against the worst-case model in $\mathcal{Q}$ --- effectively, the one associated with the least favorable prior --- thereby guarding against prior ambiguity: \[ \delta_{n,f}^{*}:=\operatorname*{arg\,min}_{\delta}\left[\sup_{m\in\mathcal{Q}}\mathbb{E}_{m}\bigl[l_{n}(\theta,\delta)\bigr]\right]. \]
Because the Wald approach underpins much of frequentist analysis, we refer to $\mathcal{Q}$ as a frequentist model and to $\delta_{n,f}^{*}$ as a frequentist (or minimax) decision rule.
Now suppose that Alice entertains the possibility that the likelihood $p_{\theta}(x)$ employed in her frequentist model $\mathcal{Q}$ may not be correctly specified. Likelihood misspecification can arise in two distinct ways. The first is misspecification of functional form, e.g., specifying a Gaussian likelihood when the true data-generating process follows a $t$-distribution. In our running example, however, the outcomes are Bernoulli, so the likelihood is necessarily a product of Bernoulli distributions and functional-form misspecification is not a concern. In fact, as we show in Section 2.3, functional-form misspecification can always be ameliorated by making the likelihood sufficiently flexible. The misspecification that Alice confronts here is of the second kind: the link between the parameter of interest $\theta$ and the distribution over $x$ may be incorrectly specified. In the running example, the true outcome distribution is $Y\sim\text{Bernoulli}(\theta_{P})$, where $\theta_{P}$ is the average treatment effect in Pennsylvania. Likelihood misspecification arises because, in general, $\theta\neq\theta_{P}$.
To accommodate misspecification concerns, we follow cerreia2026making and place a protective belt around the frequentist model $\mathcal{Q}$. Formally, we posit a set of unstructured models of the form \[ \mathcal{M}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}}\,R_{q}(m)\leq K\Bigr\}, \] where $R_{q}(m)=\text{KL}(m\,\|\,q)$ denotes the Kullback--Leibler divergence and $K$ is a constant reflecting Alice's degree of ambiguity aversion. Alice could then choose the decision rule that performs best against the worst-case model in $\mathcal{M}$, thereby guarding against both prior ambiguity and misspecification: \[ \delta_{n}^{*}:=\operatorname*{arg\,min}_{\delta}\left[\sup_{m\in\mathcal{M}}\mathbb{E}_{m}\bigl[l_{n}(\theta,\delta)\bigr]\right]. \]
For the remainder of this article, we decompose any generic model $m$ as $m(\theta,\bm{x})=\pi(\theta)\otimes m_{\theta}(\bm{x})$, and reserve the notation $p_{\theta}(\bm{x})$ for a reference likelihood specification, which may itself be misspecified. Define $R_{\mathcal{Q}}(m):=\min_{q\in\mathcal{Q}}R_{q}(m)$. Straightforward algebra yields
so that the set of unstructured models can be equivalently written as \[ \mathcal{M}=\left\{ \pi(\theta)\otimes m_{\theta}(x)\;:\;\int\text{KL}\bigl(m_{\theta}(\cdot)\,\|\,p_{\theta}(\cdot)\bigr)\,d\pi(\theta)\leq K,\;\;\pi\in\Delta(\Theta)\right\} . \] The class of unstructured models thus comprises every possible prior $\pi$, paired with all likelihoods $m_{\theta}(\cdot)$ satisfying the integrated KL constraint.
Note that $\mathcal{M}$ expands $\mathcal{Q}$ by adding a protective radius of KL divergence of size $K$. Because $\mathcal{Q}$ already accommodates unrestricted prior ambiguity, this protective belt serves entirely to guard against likelihood misspecification. Within the class of alternative models, however, the prior and likelihood may interact in subtle ways: the constraint permits $m_{\theta}(\cdot)$ to deviate substantially from $p_{\theta}(\cdot)$ for certain values of $\theta$, provided the associated prior $\pi$ places low weight on those values.
To further understand this interaction between the prior and likelihood, it is instructive to see how likelihood misspecification would be modeled in the absence of prior ambiguity. If Alice were able to commit to a single prior $\pi$, then $\mathcal{Q}$ would be a singleton and $\mathcal{M}$ would consist of all models $\pi(\theta)\otimes m_{\theta}(\bm{x})$ such that $m_{\theta}(\cdot)\in\mathcal{M}_{\bm{x}|\theta}(\pi)$, where \[ \mathcal{M}_{\bm{x}|\theta}(\pi):=\left\{ m_{\theta}(\cdot):\int\text{KL}\bigl(m_{\theta}(\cdot)\,\|\,p_{\theta}(\cdot)\bigr)\,d\pi(\theta)\leq K\right\} . \] When the prior is fixed, the class of candidate likelihoods thus depends on $\pi$; the prior shapes which deviations from the reference likelihood are admissible. In the general case with unrestricted priors, $\mathcal{M}$ can be interpreted as the union of $\pi(\theta)\otimes\mathcal{M}_{\bm{x}|\theta}(\pi)$ over all $\pi\in\Delta(\Theta)$.
A nuisance parameter is an unknown quantity that enters the likelihood but does not affect the loss function. In our running example, if the outcomes are distributed as $\mathcal{N}(\mu,\sigma^{2})$ but Alice is solely interested in learning about the mean treatment effect $\mu$, then $\sigma^{2}$ is a nuisance parameter. Nuisance parameters make the likelihood more flexible and can be used to ameliorate functional-form misspecification; indeed, one can allow for nonparametric specifications by making the nuisance parameters infinite-dimensional. As noted earlier, however, nuisance parameters cannot address the second type of misspecification concerning the link between the parameter of interest and the data. Even if Alice were to adopt a fully nonparametric specification of the outcome distribution, she would still face misspecification concerns, as treatment effects in Pennsylvania may differ fundamentally from those in the broader US population.
In contrast, we refer to the parameters that enter the utility function directly as structural parameters. In what follows, $\theta$ denotes the full collection of unknown parameters, which may include both structural and nuisance components. The structural parameters are modeled as known functions $\mu(\theta)$ of $\theta$. With this notation, the loss functions introduced earlier take the more general form \[ l_{n}(\theta,\delta)=\ell\bigl(\sqrt{n}\,(\mu(\theta)-\delta)\bigr),\quad\delta\in\mu(\Theta), \] for estimation loss, and \[ l_{n}(\theta,\delta)=\sqrt{n}\bigl(\mu(\theta)\,\mathbf{1}\{\mu(\theta)\geq0\}-\mu(\theta)\,\delta\bigr),\quad\delta\in\{0,1\}, \] for treatment-assignment loss.
The definitions of Bayesian, frequentist, and misspecified models remain unchanged; the introduction of nuisance parameters affects only the form of the loss functions.
andrews2025purpose define an econometric model $(\theta,p_{\theta}(\cdot))$ as a combination of the parameter and the likelihood.\footnote{In the terminology of andrews2025purpose, the likelihood is referred to as a data-generating process.} Apart from the prior over $\theta$, this coincides with the definition of a Bayesian statistical model. Introducing the prior allows us to account for prior ambiguity, which, as we have seen, plays a key role even in the standard frequentist approach.
A growing recent literature has considered accounting for misspecification through partial identification; see, e.g., ishihara2021evidence,yata2021optimal,christensen2022optimal,montiel2026decision. This literature supposes that the parameter of interest $\theta$ lies within a bounded distance, $d(\theta,\theta_{P})\leq L$, of an identifiable parameter $\theta_{P}$. In our running example, this would require Alice to assume that the population treatment effect $\theta$ differs from the treatment effect in Pennsylvania $\theta_{P}$ by at most $L$. While bounds of this form can arise naturally in a number of applications, the approach falls short as a general framework for misspecification for several reasons.
First, unlike our formalism, bounding the parameters directly lacks an axiomatic justification. Second, the approach is sensitive to the choice of the distance measure $d(\cdot)$, which in turn makes $L$ difficult to calibrate. In the Bernoulli setting, for instance, it would not be reasonable to use Euclidean distance $d(\theta,\theta_{P})=|\theta-\theta_{P}|$, as it does not respect the constraint $\theta\in[0,1]$. Third, and most importantly, imposing a uniform bound $d(\theta,\theta_{P})\leq L$ for all $\theta_{P}$ implies comparisons across different values of $\theta_{P}$ that may be at odds with the decision-maker's actual preferences over ambiguity and misspecification. To see why, note that there is always epistemic uncertainty over the value of $\theta_{P}$. There is no a priori reason to believe that the bound $d(\theta,\theta_{P})\leq L$ provides the same degree of protection against misspecification when $\theta_{P}=0.9$ as when $\theta_{P}=0.1$. Depending on Alice's preferences, e.g., her loss function, she may be less concerned about misspecification at high values of $\theta_{P}$ (which suggests the treatment is highly effective) than at low values. The constraint $d(\theta,\theta_{P})\leq L$ is not directly linked to her attitudes toward misspecification; it is a constraint on parameters, not on payoff-relevant quantities.
In more closely related work, andrews2020informativeness, bonhomme2022minimizing and christensen2023counterfactual characterize misspecification through a statistical distance $d(m_{\theta}(\cdot),p_{\theta}(\cdot))$, e.g., KL divergence, over likelihoods. The specific setups and goals of these works differ substantially from our own.\footnote{andrews2020informativeness study the relationship between descriptive statistics and structural parameters. bonhomme2022minimizing analyze estimation under local misspecification, i.e., when $d(m_{\theta}(\cdot),p_{\theta}(\cdot))\to0$. christensen2023counterfactual study partial identification of $\theta$. These goals are all distinct from ours: devising optimal decisions under global ambiguity and misspecification.} Quite apart from this, however, a uniform bound on the KL divergence over likelihoods, of the form $\sup_{\theta}d(m_{\theta}(\cdot),p_{\theta}(\cdot))\leq L$, is subject to the same criticism as a bound $d(\theta,\theta_{P})\leq L$ over parameters: there is no a priori reason to believe that it provides the same degree of protection against misspecification across different values of $\theta$. As before, Alice may be less concerned about misspecification at high values of $\theta$ than at low values. Furthermore, quantities such as KL divergence are sensitive to the choice of the reference measure $p_{\theta}(\cdot)$, and KL divergences evaluated at different parameter values such as $R_{p_{\theta_{1}}}(m_{\theta_{1}})$ and $R_{p_{\theta_{2}}}(m_{\theta_{2}})$ are not directly comparable.
Our formulation avoids this problem because we first postulate an infinite-dimensional ambiguity set $\mathcal{Q}$ and then uniformly expand it by a KL radius $K$. As in cerreia2026making, the value of $K$ can be tied to the decision-maker's underlying preferences over ambiguity and misspecification. But rather remarkably, as it turns out, our optimal decisions are asymptotically independent of the choice of $K$.
We now characterize optimal decisions under ambiguity and misspecification. For reasons that will become apparent shortly, it is convenient to start with utility maximization rather than loss minimization. Following the framework described in Section (ref), the optimal decision rule takes the form: \[ \delta_{n}^{*}:=\operatorname*{arg\,max}_{\delta}\left[\inf_{m\in\mathcal{M}}\,\mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]\right]=\operatorname*{arg\,max}_{\delta}\,\inf_{m}\,\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]:R_{\mathcal{Q}}(m)\le K\right\} . \]
The function $R_{\mathcal{Q}}(\cdot):\Delta(\Theta\times\mathcal{X})\to\mathbb{R}$ is strictly convex, so we may apply a minimax theorem to show that for each $K$ there exists a unique multiplier $\lambda$ such that \[ V_{n}(\delta)=\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\,R_{\mathcal{Q}}(m)\right\} . \] Following cerreia2026making, we refer to $V_{n}(\cdot)$ as the variational decision criterion.
Recalling the definition $R_{\mathcal{Q}}(m):=\min_{q\in\mathcal{Q}}R_{q}(m)$ and interchanging the the order of the $\min_{q\in\mathcal{Q}}$ and $\inf_{m}$ operations, we can write \[ V_{n}(\delta)=\min_{q\in\mathcal{Q}}\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\,R_{q}(m)\right\} . \] The Donsker-Varadhan variational formula yields
Converting back to the loss function via $l_{n}(\theta,\delta)=-u_{n}(\theta,\delta)$, we obtain
and consequently, the optimal decision can be characterized as
So far, the calculations above follow cerreia2026making. However, due to the special structure of $\mathcal{Q}$ in our setting---comprising all possible priors paired with the reference likelihood $p_{\theta}(x)$---we can simplify ((ref)) further:
In this expression, the quantity $R_{n}(\theta,\delta):=\mathbb{E}_{p(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]$ admits a natural interpretation as the frequentist risk of the decision rule $\delta$ under the reference likelihood $p(x|\theta)$, evaluated with respect to the exponentiated loss $e^{l_{n}(\theta,\delta)/\lambda}$.
Notice that ((ref)) corresponds to a standard minimax decision framework under exponentiated loss: we can interpret the optimal decision as the result of a two-player game in which nature chooses the least favorable prior while the decision-maker chooses the optimal rule. Equation ((ref)) is therefore a key result of this article. It establishes that optimal decisions under both ambiguity and misspecification are equivalent to optimal decisions under ambiguity alone, but with an exponentiated loss function. The result also reveals how the two sources of concern separate naturally. The effect of misspecification is to transform the loss function into an exponentiated version; intuitively, the decision-maker magnifies the impact of large losses while attenuating the impact of small ones. The effect of ambiguity, as in the setting without misspecification, manifests in the search for the least favorable prior.
It is rarely feasible to solve the minimax problem ((ref)) exactly in finite samples. Instead, as is standard even in classical frequentist (i.e., minimax) settings, we turn to local asymptotic approximations.
Following the usual approach, we fix a reference parameter $\theta_{0}$ and consider local perturbations of the form $\theta_{0}+h/\sqrt{n}$. Priors $\pi(\theta)$ over $\theta$ are then mapped to local priors $\pi(h)$ over $h$.
In the case of treatment-assignment loss, $l_{n}(\theta,\delta)=n\bigl(\mu(\theta)\,\mathbf{1}\{\mu(\theta)\geq0\}-\mu(\theta)\,\delta\bigr)$, local asymptotics arise naturally from the global minimax problem ((ref)). The key observation is that the least favorable prior concentrates its mass on regions where the treatment effect $\mu(\theta)$ is of order $1/\sqrt{n}$. When $\mu(\theta)$ is of a higher order of magnitude than $1/\sqrt{n}$, determining the optimal assignment is asymptotically trivial. Conversely, when $\mu(\theta)$ is of a lower order of magnitude than $1/\sqrt{n}$, the difference between treatment and status quo is negligible, so the loss is close to zero regardless of the choice of $\delta$. It is therefore natural to choose a reference parameter $\theta_{0}$ satisfying $\mu(\theta_{0})=0$, since the least favorable prior would concentrate around this value in any case. As hirano2009asymptotics showed, these same considerations apply to treatment-assignment problems in the absence of misspecification as well.
But how should one choose a reference $\theta_{0}$ for estimation loss, and what is the meaning of local asymptotics in this setting? We offer two interpretations.
The first is that local asymptotics amounts to localizing prior ambiguity around a reference parameter $\theta_{0}$ that the decision-maker believes is close to the true value. In our running example, Alice may have a priori reason to believe that the true treatment effect lies near $\theta_{0}$, even if she is uncertain about its exact value. It would then be natural to restrict her ambiguity set to a $1/\sqrt{n}$ neighborhood of $\theta_{0}$. Because this prior information is obtained independently of the data, likelihood misspecification does not affect the choice of $\theta_{0}$. Consequently, we can localize the priors, even as we can --- and do --- allow for global misspecification of the likelihood.
Under the second interpretation, the choice of the reference $\theta_{0}$ is itself subject to adversarial optimization. The basic idea, following ibragimov1981problem, is to decompose the global minimax problem into two stages: first, fix a reference $\theta_{0}$ and evaluate the local minimax performance of a decision rule $\delta$ against local alternatives of the form $\theta_{0}+h/\sqrt{n}$; then, in an outer step, select the least favorable reference $\theta_{0}$.
Here, we focus primarily on a theoretical development of the first interpretation. The second interpretation is detailed in Section (ref), while the theory is developed in Appendix (ref).
We assume that the data consists of an i.i.d collection of outcomes $\ensuremath{\bm{x}:=\{Y_{i}^ {}\}_{i=1}^{n}}$. Under the reference likelihood, $Y_{i}$ is distributed as $P_{\theta}$. Let $\nu$ denote a dominating measure for $\{P_{\theta}:\theta\in\mathbb{R}^{d}\}$, and set $p_{\theta}^ {}:=dP_{\theta}^ {}/d\nu$. We require the reference class of likelihoods, $\{P_{\theta}^ {}\}_{\theta}$, to be quadratic mean differentiable (qmd):
\begin{asm1} The class $\{P_{\theta}^ {}:\theta\in\mathbb{R}^{d}\}$ is qmd around $\theta_{0}^ {}$, i.e., there exists a score function $\psi_{}(\cdot)$ such that for each $h^ {}\in\mathbb{R}^{d},$ \[ \int\left[\sqrt{p_{\theta_{0}^ {}+h^ {}}}-\sqrt{p_{\theta_{0}}}-\frac{1}{2}h^{\intercal}\psi_{}\sqrt{p_{\theta_{0}^ {}}}\right]^{2}d\nu=o(\vert h^ {}\vert^{2}). \] Furthermore, the information matrix $I_{0}:=\mathbb{E}_{\theta_{0}}[\psi\psi_{}^{\intercal}]$ is invertible. \end{asm1}
In the illustrative example, the outcomes are modeled as Bernoulli, so Assumption 1 holds with $\psi_{}(y)=\left(\theta_{0}^ {}(1-\theta_{0}^ {})\right)^{-1}(y-\theta_{0}^ {})$. More broadly, this assumption is rather mild and satisfied for almost all commonly used likelihood models, including the Normal, Cauchy, Exponential, and Poisson distributions. It is important to bear in mind that Assumption 1 constrains the reference class of likelihoods, $p_{\theta}$, not the actual likelihoods, which are unknown.
We also assume that the function $\mu(\theta)$, which maps $\theta$ to structural parameters, satisfies a mild differentiability condition:
\begin{asm2} There exists $\dot{\mu}_{0}\in\mathbb{R}^{d}$ and $\epsilon_{n}\to0$ independent of $h$ such that $\sqrt{n}\left(\mu(\theta_{0}+h/\sqrt{n})-\mu(\theta_{0})\right)=\dot{\mu}_{0}^{\intercal}h+\epsilon_{n}\vert h\vert^{2}$ for all bounded $h$. \end{asm2}
Let $P_{n,h}$ denote the joint probability measure over the iid $Y_{1},\dots,Y_{n}$ when each $Y_{i}\sim P_{\theta_{0}+h/\sqrt{n}}$, and let $\mathbb{E}_{n,h}[\cdot]$ denote the corresponding expectation. Under local asymptotics, the minimal risk attained under the minimax problem ((ref)) can be written as:
Define the standardized score statistic as \[ x_{n}=\frac{I_{0}^{-1/2}}{\sqrt{n}}\sum_{i=1}^{n}\psi(Y_{i}). \] It is well known, see e.g., van2000asymptotic, that quadratic mean differentiability (Assumption 1) implies $\mathbb{E}_{n,0}[\psi(Y_{i})]=0$. Then, by the central limit theorem,
Assumption 1 also implies the important property of Local Asymptotic Normality (LAN; van2000asymptotic):
Consider now a limit experiment in which the decision-maker observes a $d$-dimensional signal $x$, posited to be drawn from a reference Gaussian likelihood, $P_{h}(x)\sim\mathcal{N}(I_{0}^{-1/2}h,I)$. By the properties of the Gaussian distribution, \[ \ln\frac{dP_{h}}{dP_{0}}=h^{\intercal}I_{0}^{1/2}x_{}-\frac{1}{2}h^{\intercal}I_{0}h. \] It follows from ((ref)) and ((ref)) that the reference likelihood ratios in the finite-sample experiment converge to their counterparts in the limit experiment: \[ \ln\frac{dP_{n,\theta_{0}+h/\sqrt{n}}}{dP_{n,\theta_{0}}}\xrightarrow[P_{n,0}]{d}\ln\frac{dP_{h}}{dP_{0}},\ \textrm{for each }h. \] Furthermore, Assumption 2 implies that the loss functions admit asymptotic approximations. For estimation-loss, defining $\tilde{\delta}_{n}=\sqrt{n}\dot{\mu}_{0}^{\intercal}(\delta_{n}-\theta_{0})$ and assuming $\tilde{\delta}_{n}$ has a weak limit $\tilde{\delta}$, we have
where $`$$\rightsquigarrow$' represents weak convergence. For treatment-assignment loss, since the reference parameter satisfies $\mu(\theta_{0})=0$, Assumption 2 implies
Convergence of likelihood ratios implies asymptotic equivalence between the actual and limit experiments in the sense of le1986asymptotic. Combined with the loss function approximations above, this suggests that the minimax value $V_{n}^{*}$ in ((ref)) should converge to the minimax value $V^{*}$ in the limit experiment, where
Formal statements to this effect are provided in Section (ref).
It is instructive to compare our asymptotic approach with the more traditional analysis of local asymptotics under local misspecification. In locally misspecified models, the KL divergence between the true and reference likelihoods is assumed to decline at a $1/n$ rate. Consequently, as highlighted in andrews2020informativeness, bonhomme2022minimizing and muller2024locally, local misspecification manifests as asymptotic bias in the Gaussian limit experiment. Our framework differs fundamentally since it permits the Gaussian likelihood approximation itself to be globally misspecified. This is possible because our asymptotic theory approximates only the reference finite-sample likelihood ratios with Gaussian likelihoods; it makes no claim about the convergence of the true likelihood ratios. Global misspecification consequently manifests not as bias in the Gaussian limit, but as an exponential tilting of the loss function.
Equation ((ref)) suggests that asymptotically optimal decision rules can be derived by solving the minimax problem in the limit experiment and mapping the solutions back to the finite-sample setting. Since optimal decisions are considerably easier to characterize under Gaussian likelihoods, this reduction illustrates the key benefit of the local asymptotic approach.
We begin with estimation-loss. Since $\ell(\cdot)$ is bowl-shaped, so is $e^{\ell(\cdot)/\lambda}$. It then follows from Anderson's lemma, see e.g., van2000asymptotic, that the minimax-optimal estimator in the limit experiment --- the solution to ((ref)) --- is simply \[ \tilde{\delta}^{*}=\dot{\mu}_{0}^{\intercal}I_{0}^{-1/2}x. \] Remarkably, $\tilde{\delta}^{*}$ is independent of $\lambda$, which governs the degree of misspecification. In fact, $\tilde{\delta}^{*}$ coincides with most efficient estimator, the maximum likelihood estimator, under correct misspecification, which corresponds to $\lambda=\infty$. In other words, the optimal estimator under ambiguity and misspecification is identical to the optimal estimator under ambiguity alone.
For treatment-assignment loss, Anderson's lemma does not apply. Nevertheless, as the following proposition shows, the optimal decision rule again takes a simple form: it recommends treatment whenever the MLE of the treatment effect under correct specification is positive.
As with the optimal estimator, the optimal treatment-assignment rule is independent of the degree of misspecification.
Intuitively, these results arise because misspecification under our formulation is unstructured and therefore symmetric around the reference Gaussian likelihood. Since the loss functions are also symmetric around the reference $\theta_{0}$, any estimator that is not efficient under correct specification would break this symmetry. Because nature chooses the least favorable likelihood specification given the decision-maker's choice of estimator, departures from symmetry would necessarily incur higher decision risk. It is therefore always optimal for the decision-maker to proceed as if the likelihood were correctly specified and select the resulting optimal decision rule.
We now formally establish the asymptotic equivalence of experiments through two results. First, we show that the minimax value $V^{*}$ in the limit experiment forms an asymptotic lower bound on the sequence of optimal decision risks in the finite-sample experiments. Second, we show that plug-in versions of the optimal limit-experiment decision rules $\tilde{\delta}^{*}$ -- obtained by replacing $I_{0}^{-1/2}x$ with the finite sample MLE $\hat{\theta}_{\textrm{mle}}$ -- are asymptotically optimal, in the sense that their decision-risks converge to $V^{*}$. Specifically, we argue that the asymptotically decisions are given by
As discussed at the beginning of this section our formal results require localization of ambiguity. This involves restricting attention to the set of compactly supported priors $\Delta_{M}(\mathcal{H})\equiv\{\pi(h):\textrm{supp}(\pi)\in[-M,M]\}$ for some $M<\infty$. This restriction is not needed for the lower bound, but plays a role in establishing that the bound is attained by the plug-in rules.
We place a mild regularity condition on the MLE:
\begin{asm3} The maximum-likelihood estimator $\hat{\theta}_{\textrm{mle}}$ admits a locally linear score-function approximation: \[ \hat{\theta}_{\textrm{mle}}-\theta_{0}=I_{0}^{-1/2}x_{n}+o_{P_{n,0}}(1). \] \end{asm3}
To avoid technical issues relating to the existence of moments for the estimation problem, our theory also requires that $\ell(\cdot)$ be bounded. We state this as an additional assumption.
\begin{asm4} The function $\ell(\cdot)$ is bounded.\end{asm4}
Assumption 4 implies the estimation loss $l_{n}(\theta,\delta)$ is bounded. As for the treatment-assignment loss, since we work with compact priors, it is automatically bounded.
In fact, the MLE can be replaced by any asymptotically efficient estimator satisfying Assumption 3. All such estimators attain the same minimax lower bound $V^{*}$; our theory therefore does not distinguish among them.
The statements of Theorems (ref) and (ref) are new even in the absence of misspecification. Standard local asymptotic minimax theorems are typically stated in terms of a maximum over discrete sets of $h$ values, rather than the $\max_{\pi(h)\in\Delta_{M}(\mathcal{H})}$ operation employed here, see, e.g., van2000asymptotic. This distinction is consequential for our setting, as the discrete formulation does not lend itself to a natural interpretation of prior ambiguity. Our results avoid this limitation through a different method of proof that directly accommodates optimization over a compact space of local priors.
The formal results above were derived under priors localized around a reference parameter $\theta_{0}$. As noted earlier, with a slight strengthening of the assumptions, we can allow for global priors over a compact set $\Theta$; essentially, we require the assumptions to be valid uniformly over $\theta_{0}\in\Theta$. In this case, the lower bounds on decision risk correspond to the least favorable choice of the reference.
Formally, we show that under both loss functions,
where $V_{\theta_{0}}^{*}$ represents the minimal decision-risk in the limit experiment (as defined in Section (ref)) evaluated at a given reference parameter $\theta_{0}$. The formal statement, together with the required assumptions, is provided in Appendix (ref).
In the same appendix, we show that the decisions in ((ref)) are asymptotically optimal under global priors as well, in that they attain this bound:
where $l_{n,K}(\cdot):=K\wedge l_{n}(\cdot)$ denotes the loss function truncated at level $K$.
To the best of our knowledge, these results appear new even in the context of no misspecification concerns. A number of authors, e.g., hirano2025wald, have raised concerns about the interpretation of asymptotic analysis under local reparameterizations. Our results address these concerns by showing that global minimax analysis is equivalent to local asymptotic analysis with a least favorable choice of reference.
As noted by andrews2025purpose, researchers in applied work often prefer the simulated method of moments --- which involves subjective selection of moment functions --- over maximum likelihood, despite the latter's superior efficiency. This preference is frequently justified by the argument that “with misspecification concerns, moment estimators are often more reliable\textquotedbl (bordalo2020overreaction).
Our results suggest, however, that this reasoning is incomplete. Under our framework, there is no tradeoff between misspecification robustness and efficiency. Efficient estimators are misspecification-robust to an arbitrary degree, and inefficient estimators remain suboptimal even in the presence of potential misspecification. Misspecification concerns alone therefore cannot justify the use of the simulated method of moments over maximum likelihood.
This is not to say that the simulated method of moments should never be used. The selection of moment functions can often be understood as a form of model selection. For instance, if the parameter of interest is the treatment effect on a subgroup of the population, it may be reasonable to selectively overweight the relevant portion of the sample. This, however, is a problem of model selection rather than model misspecification per se.
The preceding sections focused on parametric models for the likelihood. In many applications, however, the exact distributional form is left unspecified, motivating the use of semi-parametric models. As discussed in (ref), our methodology extends naturally to the semi-parametric and nonparametric settings by allowing for infinite-dimensional nuisance parameters.
In semi-parametric models, the structural parameter is typically a regular functional $\mu:=\mu(P)$, of the unknown population distribution $P$. Common examples of regular functionals include the mean, median, and quantiles. For simplicity, we assume that $\mu$ is scalar-valued. The loss functions are then \[ l_{n}(P,\delta)=\ell\bigl(\sqrt{n}\,(\mu(P)-\delta)\bigr), \] for estimation loss, and \[ l_{n}(P,\delta)=\sqrt{n}\bigl(\mu(P)\,\mathbf{1}\{\mu(P)\geq0\}-\mu(P)\,\delta\bigr),\quad\delta\in\{0,1\}, \] for treatment-assignment loss. The population distribution $P$ plays the same role as $\theta$ in the parametric analysis. While $P$ is unknown, we are only interested in its scalar functional $\mu(P)$; the remaining features of $P$ are treated as an infinite dimensional nuisance parameter.
As in Section (ref), we consider a framework in which the decision-maker confronts both prior ambiguity --- an inability to form a single prior over the parameter $P$ --- and model misspecification --- the concern that the population distribution $P$ may not coincide with the outcome distribution $\hat{P}$ in the experimental sample. In our running example, Alice may adopt a fully nonparametric specification of the outcome distribution yet still face misspecification concerns, as treatment effects in Pennsylvania could differ fundamentally from those in the broader US population. More generally, misspecification also arises when the semi-parametric model imposes restrictions that are not satisfied by the combination of the true $\mu$ and the sample outcome distribution $\hat{P}$. For instance, in the GMM framework, the researcher specifies a moment condition $\mathbb{E}_{P}[m(Y_{i},\mu)]=0$, where $m(\cdot)\in\mathbb{R}^{p}$ is a known vector of moment functions, $\mu\in\mathbb{R}^{d}$ is the structural parameter of interest, and $P$ is the unknown population distribution. Misspecification arises if this restriction does not hold in the sample, i.e., $\mathbb{E}_{\hat{P}}[m(Y_{i},\mu)]\neq0$ for the true $\mu$. In our running example, which fits within the GMM framework with $m(Y_{i},\mu)=Y_{i}-\mu$, Alice may be concerned that $\mathbb{E}_{\hat{P}}[Y_{i}-\mu]\neq0$.
The central message of our semi-parametric results is that the sample analogs of the optimal decisions characterized in Section (ref) remain optimal in the semi-parametric setting. In essence, one replaces the score statistic from the parametric setting with an efficient influence function process associated with the functional of interest. For example, if the goal is to conduct inference on the mean, one replaces $x_{n}$ with the cumulative sum of outcomes $n^{-1/2}\sum_{i=1}^{n}(Y_{i}/\sigma)$, where $\sigma^{2}:=\textrm{Var}[Y_{i}]$.
It is easiest to discuss ambiguity and misspecification in semi-parametric settings using local asymptotics. Our local asymptotic analysis employs the formalism of van2000asymptotic. Let $\mathcal{P}$ denote the class of candidate population distributions with bounded variance, dominated by some measure $\nu$. We fix a reference distribution $P_{0}\in\mathcal{P}$, and surround it with various smooth one-dimensional parametric sub-models, $\{P_{s,h}:s\le\zeta\}$ for some $\zeta>0$, whose score function is $h(\cdot)$, and that pass through $P_{0}$ at $s=0$ (i.e., $P_{0,h}=P_{0}$). Formally, these sub-models satisfy
As in the parametric setting, for the treatment-assignment problem, we choose $P_{0}$ such that $\mu(P_{0})=0$.
By van2000asymptotic, condition ((ref)) implies $\int hdP_{0}=0$ and $\int h^{2}dP_{0}<\infty$. The set of all such functions $h$ is termed the tangent space $T(P_{0})$, which is a subset of the Hilbert space $L^{2}(P_{0})$ endowed with the inner product $\left\langle f,g\right\rangle =\mathbb{E}_{P_{0}}[fg]$ and norm $\left\Vert f\right\Vert =\mathbb{E}_{P_{0}}[f^{2}]^{1/2}$. For any $h\in T(P_{0})$, let $P_{n,h}$ denote the joint probability measure over $Y_{1},\dots,Y_{n}$, when each $Y_{i}$ is an iid draw from $P_{1/\sqrt{n},h}$, and let $\mathbb{E}_{n,h}[\cdot]$ denote the corresponding expectation. An important consequence of ((ref)) is the LAN property:
Let $\psi\in T(P_{0})$ denote the efficient influence function corresponding to $\mu$, defined by the property that for any $h\in T(P_{0})$,
Set $\sigma^{2}=\mathbb{E}_{P_{0}}[\psi^{2}]$. The semi-parametric analogue of the score statistic in the semi-parametric setting is the standardized efficient influence function process \[ x_{n}:=\frac{\sigma^{-1}}{\sqrt{n}}\sum_{i=1}^{n}\psi(Y_{i}). \]
Any element $h\in T(P_{0})$ admits the orthogonal decomposition $h=\left\langle \psi/\sigma,h\right\rangle \psi/\sigma+\tilde{h}$, where $\tilde{h}$ is orthogonal to $\psi$ (i.e., $\left\langle \psi,\tilde{h}\right\rangle =0$). The component $\mu=\left\langle \psi,h\right\rangle $ represents the structural parameter, while $\tilde{h}$ represents an infinite dimensional nuisance parameter. Although the full perturbation direction $h$ is unknown, only the projection onto the efficient influence function is relevant for learning about $\mu$.
For each $h\in T(P_{0})$, define $\mu_{n}(h):=\mu(P_{1/\sqrt{n},h})$, and let $\bm{x}:=(Y_{1},\dots,Y_{n})$ denote the collection of outcomes. We can rewrite the loss functions in terms of $h$ as \[ l_{n}(h,\delta)=
\]
In the semi-parametric setting, a Bayesian statistical model, $m(h,\bm{x})$, is defined as a joint probability distribution over both $h,\bm{x}$. As in Section (ref), it admits the decomposition \[ m(h,\bm{x})=\pi(h)\otimes p_{n,h}(\bm{x}), \] where $\pi\in\Delta(T(P_{0}))$ denotes a prior over the tangent space $T(P_{0})$, and $p_{n,h}(\bm{x}):=\prod_{i}dP_{1/\sqrt{n},h}(Y_{i})/d\nu$ represents the likelihood, a parametric sub-model. Prior ambiguity is incorporated by defining a frequentist model, a structured set of models, as \[ \mathcal{Q}:=\bigl\{\pi(h)\otimes p_{n,h}(\bm{x}):\pi\in\Delta(T(P_{0}))\bigr\}. \] Finally, misspecification concerns are addressed by placing a protective belt around $\mathcal{Q}$, yielding the set of unstructured models \[ \mathcal{M}=\Bigl\{ m\in\Delta(T(P_{0}),\mathcal{X}):\min_{q\in\mathcal{Q}}\,R_{q}(m)\leq K\Bigr\}. \] Expanding the set of structured models allows us to account for the possibility that the true distribution of the experimental data $\bm{x}$ is not captured by $p_{n,h}(\bm{x})$ for any $h$.
The decision-maker chooses the decision rule that performs best against the worst-case model in $\mathcal{M}$, thereby guarding against both prior ambiguity and misspecification: \[ \delta_{n}^{*}:=\operatorname*{arg\,min}_{\delta}\left[\sup_{m\in\mathcal{M}}\mathbb{E}_{m}\bigl[l_{n}(h,\delta)\bigr]\right]. \] As in Section (ref), applying a Lagrangian formulation and the same sequence of calculations yields the following characterization of minimal decision risk:
We impose the following regularity conditions throughout this section:
\begin{asm5}The sub-models $\{P_{s,h};h\in T(P_{0})\}$ satisfy ((ref)). Furthermore, they admit an efficient influence function, $\psi(\cdot)$, for $\mu(P)$ such that \[ \sqrt{n}\left(\mu(P_{1/\sqrt{n},h})-\mu_{0}\right)=\left\langle \psi,h\right\rangle +\epsilon_{n}\left\Vert h\right\Vert ^{2}, \] where $\mu_{0}:=\mu(P_{0})$, and $\epsilon_{n}$ is independent of $h$ for bounded $\left\Vert h\right\Vert $. \end{asm5}
The first part of Assumption 5 simply states the definition of parametric sub-models. The second part of Assumption 5 slightly strengthens ((ref)).
Since $L^{2}(P_{0})$ is a Hilbert space, it is possible to select $\{\phi_{1},\phi_{2},\dots\}\in L^{2}(P_{0}^ {})$ in such a manner that $\{\psi/\sigma,\phi_{1},\phi_{2},\dots\}$ is a set of orthonormal basis functions for the closure of $T(P_{0}^ {})$; the division by $\sigma_{}$ in the first component ensures $\left\Vert \psi/\sigma_{}\right\Vert ^{2}=1$. We can also choose these bases so they lie in $T(P_{0}^ {})$, i.e., $\mathbb{E}_{P_{0}}[\phi_{j}]=0$ for all $j$. By the Hilbert space isometry, each $h_{}\in T(P_{0}^ {})$ is then associated with an element from the $l_{2}$ space of square integrable sequences, $(\mu/\sigma,\gamma_{1},\gamma_{2},\dots)$, where $\mu=\left\langle \psi,h\right\rangle $ and $\gamma_{k}=\left\langle \phi_{k},h\right\rangle _{}$ for all $k\neq0$. Consequently, any prior $\pi(h)$ over $T(P_{0})$ can be represented as a prior over $l_{2}$.
As in Section (ref), our formal results require localization of ambiguity. This involves restricting attention to priors that are supported on a compact subset, $K_{M}$, of $l_{2}$, defined as \[ K_{M}\equiv\left\{ h=\left(\mu/\sigma,\gamma_{1},\dots\right):\left\Vert h\right\Vert \le M,\ \lim_{J\to\infty}\sup_{(\gamma_{1},\gamma_{2}\dots)}\sum_{j=J}^{\infty}\vert\gamma_{j}\vert^{2}=0\right\} . \] The compactness condition essentially requires the set of candidate $h$ to be sufficiently smooth.
As in the parametric setting, we show that the minimax value $V^{*}$ in the limit experiment also forms an asymptotic lower bound on the sequence of optimal decision risks, $V_{n}^{*}$, in the finite sample semi-parametric experiments. However, since the previous definition of the limit experiment used a different interpretation of $h$, we will need to modify the construction slightly.
Specifically, we now consider a limit experiment where we observe a one dimensional signal $x$, posited to be drawn from a reference Gaussian likelihood, $P_{\mu}(x)\sim\mathcal{N}(\mu/\sigma,1)$. Let $\mathbb{E}_{\mu}[\cdot]$ denote the expectation corresponding to $P_{\mu}$. The minimax value $V^{*}$ in this limit experiment is then defined as
It is straightforward to verify that the value of $V^{*}$ in ((ref)) is, in fact, the same as that in ((ref)) when $\theta=\mu$ and $I_{0}=1/\sigma^{2}$.
As in the parametric setting, asymptotically optimal decisions under ambiguity and misspecification are the same as those under ambiguity along. Let $\hat{\mu}_{n}$ denote any semi-parametrically efficient estimator for $\mu$, understood as satisfying the following assumption:
\begin{asm6} The estimator $\hat{\mu}$ attains the semi-parametric efficiency bound, in that it admits a locally linear influence-function approximation: \[ \hat{\mu}_{n}-\mu_{0}=\sigma x_{n}+o_{P_{n,0}}(1). \] \end{asm6}
As we show below, asymptotically optimal decisions are then given by \[ \hat{\delta}_{n}^{*}=
\]
The Generalized Method of Moments (GMM) is an example of a semi-parametric model that is widely used in economic applications. Recall that in the GMM framework, the researcher specifies a moment condition $\mathbb{E}_{P}[m(Y_{i},\mu)]=0$, where $m(\cdot)\in\mathbb{R}^{p}$ is a known vector of moments, $\mu\in\mathbb{R}^{d}$ is the structural parameter, and $P$ is the population distribution. When $p>d$, the GMM model is said to be over-identified. In this setting, the efficient influence function is given by \[ \psi(Y_{i})=G_{0}^{\intercal}\Omega_{0}^{-1}m(Y_{i},\mu_{0}), \] where $\mu_{0}:=\mu(P_{0})$ is the unique solution to $\mathbb{E}_{P_{0}}[m(Y_{i},\mu)]=0$ under the reference distribution $P_{0}$, $G_{0}:=\mathbb{E}_{P_{0}}[\nabla_{\mu}m(Y_{i},\mu)]$ and $\Omega_{0}:=\mathbb{E}_{P_{0}}[m(Y_{i},\mu_{0})m(Y_{i},\mu_{0})^{\intercal}]$.
In the absence of misspecification concerns, it is well known that several estimators that are asymptotically efficient, including 2-step GMM and continuously updated GMM (CUGMM), among others. In practice, however, researchers are often concerned that the model may be misspecified, i.e., $\mathbb{E}_{\hat{P}}[m(Y_{i},\mu)]\neq0$ under the true structural parameter $\mu^ {}$ and the sample outcome distribution $\hat{P}$. Under misspecification, different estimators converge to different limits under the distribution of sample outcomes $\hat{P}$, and researchers often resort to inefficient estimators employing a weighing matrix other than the optimal $W^{*}=G_{0}^{\intercal}\Omega_{0}^{-1}$. This practice is frequently justified on the grounds that “...under misspecification the two-step GMM estimator is no more efficient than any other estimator... each weighting leads us to recover a different parameter” (andrews2025purpose).
Our results suggest, however, that this reasoning is incomplete. When the decision-maker confronts both prior ambiguity and model misspecification as in our framework, the optimal estimator coincides with that under prior ambiguity alone. Consequently, 2-step GMM remains superior to diagonally-weighted GMM, even under an arbitrary degree of misspecification. Researchers who wish to employ inefficient weighting should therefore provide explicit justification for doing so. If, for instance, the researcher believes that misspecification is not unstructured but that certain forms or directions of misspecification are more likely than others, this could in principle lead to different estimation strategies. Even so, it would be difficult to justify use of identify or diagonal weighting on the basis of directional misspecification alone, since such weighting typically preserves symmetry across directions.
Apart from requiring efficiency, our results do not distinguish among different efficient estimators. Under local asymptotics, all efficient estimators attain the same decision risk $V^{*}$, so additional criteria must be employed to select among them. For instance, imposing invariance would lead one to prefer CUGMM over two-step GMM.
In this section, we discuss several variations and extensions of our framework.
So far, we have used relative entropy $R_{q}(m)$ to measure the discrepancy between statistical models. hansen2011robustness provide two important reasons for using relative entropy. First, it leads to a tractable characterization of minimal decision risk and optimal decisions, as demonstrated in Section (ref). Second, as discussed in hansen2011robustness, it can be linked to risk-sensitivity adjustment through the theory of large deviations.
It is, however, possible to employ alternative discrepancy measures. Relative entropy, is but a special case of the $\phi$-divergence class of discrepancies, which take the general form \[ D_{\phi}(m\|q)=\int\phi\left(\frac{dm}{dq}\right)dq, \] where $\phi:[0,\infty)\to(-\infty,\infty]$ is a convex function satisfying $\phi(x)<\infty$ for all $x>0$, $\phi(1)=0$ and $\phi(0)=\lim_{x\to0}\phi(x)$. Setting $\phi(x)=x\ln x$ recovers KL divergence.
Under this more general class of discrepancies, the set of unstructured models takes the form \[ \mathcal{M}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}}\,D_{\phi}(m\|q)\leq K\Bigr\}, \] and following arguments analogous to those in Section (ref), the decision-risk of a rule $\delta$ can be characterized as \[ V_{n,\phi}(\delta)=\min_{q\in\mathcal{Q}}\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\,D_{\phi}(m\|q)\right\} . \] Let $\phi^{*}(\cdot)$ denote the convex conjugate of $\phi(\cdot)$. The decision risk $V_{n,\phi}(\delta)$ then admits the variational representation
where the first equality is due to cerreia2026making and the second equality exploits the specific structure of $\mathcal{Q}$ in our setting.
By standard properties of convex conjugates, $\phi^{*}(x)<\infty$ for all bounded $x$ if and only if $\lim_{x\to\infty}\phi(x)/x=\infty$. Since our loss functions $l_{n}(\theta,\delta)$ are generally unbounded, $V_{n,\phi}(\delta)$ is therefore $\infty$ for any $\phi$-divergence measure satisfying $\lim_{x\to\infty}\phi(x)/x<\infty$, unless sharp support restrictions are imposed on the class of priors. In other words, the decision risk is trivially infinite for these divergence measures whenever the class of priors is sufficiently rich. We therefore argue that such discrepancies are not well suited for a framework that accommodates both ambiguity and misspecification. Notable examples in this category include total variation $(\phi(x)=\vert x-1\vert/2$), squared Hellinger distance ($\phi(x)=(\sqrt{x}-1)^{2}/2)$ and Pearson $\chi^{2}$ divergence ($\phi(x)=(x-1)^{2}/x$).
The underlying issue with these divergence measures is that the misspecification they permit is too broad. Consider, for instance, the total-variation metric: it is possible to have $dm/dq=\infty$, meaning $q\in\mathcal{Q}$ may not share the same support as the true model, even as total variation remains finite. This would imply that $\mathcal{Q}$ is blatantly misspecified, in the sense that any specification test would surely reject it almost surely. In contrast, as hansen2011robustness argue, when $R_{\mathcal{Q}}(m)<\infty$, the approximating model can still be regarded as plausibly correct, since it would not be rejected with probability one.
Among the $\phi$-divergence measures satisfying $\lim_{x\to\infty}\phi(x)/x=\infty$, the only commonly used divergence apart from KL divergence is the Neyman $\chi^{2}$ divergence ($\phi(x)=(x-1)^{2}$). The Neyman $\chi^{2}$ divergence is a stronger discrepancy measure than KL divergence: it is possible to have $\chi^{2}(m\|q)<\infty$ while $R_{q}(m)=\infty$. Consequently, a KL-based a misspecification set $\mathcal{M}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}}\,D_{q}(m)\leq K\Bigr\}$ is strictly larger than the corresponding $\chi^{2}$-based set $\mathcal{M}_{\chi^{2}}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}}\,D_{\chi^{2}}(m\|q)\leq K\Bigr\}$. Based on this insight, we conjecture that our results on the optimality of efficient decisions continue to hold for the Neyman $\chi^{2}$ divergence as well, though we leave the formal analysis to future work.
The estimation and treatment assignment loss functions considered thus far share a crucial property: they are symmetric in the sense that overestimating $\theta$ by a given amount incurs the same loss as underestimating it by that amount. This symmetry is crucial to the result that optimal decisions do not depend on the degree of misspecification.
There are, however, many loss functions that lack this symmetry. A prominent example is the linex loss $l(\theta,\delta)=e^{(\mu(\theta)-\delta)}-(\mu(\theta)-\delta)-1$ which penalizes positive errors much more than negative errors. Under such losses, the optimal estimator is biased even in the absence of misspecification. Incorporating misspecification concerns introduces additional bias whose magnitude depends on the misspecification parameter $\lambda$; see Appendix (ref) for details in the context of linex loss.\footnote{Under misspecification, the linex loss function must be truncated to keep the minimax risk finite. The optimal estimator therefore also depends on the level of truncation. However, for any level of truncation, the bias of the optimal estimator decreases in $\lambda$ (note that $\lambda=0$ corresponds to no misspecification risk). } Intuitively, misspecification entails an exponential tilting of the loss function, which further exacerbates any asymmetry already present in the loss. Consequently, for asymmetric losses, optimal decisions under ambiguity and misspecification may not coincide with those under ambiguity alone.
Our framework employs the Waldian approach to ambiguity by selecting the worst-case model within the ambiguity set -- the structured class of models $\mathcal{Q}$. cerreia2026making term the preference axiom underlying this approach as `caution'. An alternative is to adopt smooth ambiguity aversion, as in klibanoff2005smooth, by introducing a probability distribution $\varrho_{Q}(\cdot)$ over $\mathcal{Q}$. Intuitively, smooth ambiguity aversion corresponds to making the decision-maker less cautious: rather than guarding against the worst case, she averages over models according to $\varrho_{Q}(\cdot)$. Since $\mathcal{Q}$ pairs a candidate likelihood with every possible prior, a distribution over $\mathcal{Q}$ is equivalent to a distribution $\varrho(\pi)$ over the space of priors $\Delta(\Theta)$: the latter can simply be interpreted as a hyperprior in a Bayesian hierarchical model.
cerreia2026making show that under smooth ambiguity aversion, the decision risk takes the form
where $\phi_{Q}(\cdot)$ is a monotone function and the second equality uses ((ref)). Setting $\phi_{Q}(t)=-e^{-t/\lambda}$ and converting $\varrho_{Q}(q)$ to a prior $\varrho(\pi)$ over $\Delta(\Theta)$ yields
where $\bar{\pi}(\theta)$ is the effective prior induced by the hyperprior $\varrho(\cdot)$ over $\Delta(\Theta)$. Hence, under this specification of $\phi_{Q}(\cdot)$, misspecification combined with smooth ambiguity aversion with $\phi_{Q}(t)=-e^{-t/\lambda}$ is equivalent to misspecification with a single hierarchical prior.
The choice of $\phi_{Q}(t)=-e^{-t/\lambda}$, however, uses the same parameter $\lambda$ to govern both aversion to prior ambiguity and sensitivity to model misspecification. It may therefore be more natural to set $\phi_{Q}(t)=-e^{-t/\xi}$, where $\xi$ captures aversion to prior uncertainty separately from the misspecification parameter $\lambda$. With this choice, the decision risk becomes
As $\xi\to0$, aversion to prior ambiguity grows without bound and the objective $\tilde{V}_{n}(\delta)$ reduces to the decision risk $V_{n}(\delta)$ from ((ref)) --- the Waldian formulation involving the least favorable prior, and the primary focus of this article. For $\xi\in(0,\infty)$, the decision-maker exhibits less aversion to prior ambiguity, but this comes at the cost of a nonlinear objective when $\lambda\neq\xi$, which complicates the analysis. A formal treatment of the more general criterion ((ref)) when $\xi\notin\{\lambda,\infty\}$ is therefore left for future research.
While we have so far focused on misspecification of a single likelihood, practitioners are often interested in selecting among multiple competing likelihood specifications, each potentially subject to varying degrees of misspecification concern.
Let $p_{1,\theta}(\bm{x})$ and $p_{2,\theta}(\bm{x})$ denote two candidate likelihoods, and let $\mathcal{Q}_{1},\mathcal{Q}_{2}$ denote the corresponding frequentist models, where, as in Section (ref), \[ \mathcal{Q}_{a}:=\bigl\{\pi(\theta)\otimes p_{a,\theta}(\bm{x}):\pi\in\Delta(\Theta)\bigr\},\quad a\in\{1,2\}. \] Suppose that, treating each likelihood in isolation, Alice contemplates a misspecification set for each of the form \[ \mathcal{M}_{a}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}_{a}}R_{q_{a}}(m)\leq K_{a}\Bigr\},\quad a\in\{1,2\}. \] Here, $K_{a}$ quantifies the decision-maker's misspecification concern for each model: $K_{1}<K_{2}$ implies that Alice has greater concern about likelihood 2 being misspecified than about likelihood 1.
Rather than treating the models in isolation, however, Alice may wish to combine them. The overall set of misspecified models under consideration can then be taken to be the intersection $\mathcal{M}=\mathcal{M}_{1}\cap\mathcal{M}_{2}.$ With this choice, the decision risk becomes
As $R_{\mathcal{Q}_{a}}(\cdot)$ is strictly convex, standard duality arguments yield the Lagrangian form \[ V_{n}(\delta)=\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\left(\alpha\,R_{\mathcal{Q}_{1}}(m)+(1-\alpha)\,R_{\mathcal{Q}_{2}}(m)\right)\right\} , \] for some $\alpha\in[0,1]$ and $\lambda\geq0$ that depend on $K_{1},K_{2}$. In particular, $\alpha>1/2$ whenever $K_{1}<K_{2}$: the decision risk places greater weight on the likelihood that is less likely to be misspecified.
Recalling the decomposition $m=\pi(\theta)\otimes m_{\theta}(\bm{x})$ and applying ((ref)) yields
where the last step makes use of the fact that the weighted sum of KL divergences can be expressed as a single KL divergence against a geometric mixture. Applying the Donsker--Varadhan variational formula again then gives
where $p_{\alpha}(\bm{x}|\theta):=p_{1}^{\alpha}(\bm{x}|\theta)\cdot p_{2}^{1-\alpha}(\bm{x}|\theta)$ is the geometric mixture likelihood that combines $p_{1}(\bm{x}|\theta)$ and $p_{2}(\bm{x}|\theta)$ with mixing weights $\alpha$ and $1-\alpha$, respectively.
The optimal decision therefore solves \[ \delta_{n}^{*}=\operatorname*{arg\,min}_{\delta}\,\max_{\pi\in\Delta(\Theta)}\int\mathbb{E}_{p_{\alpha}(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]d\pi(\theta). \] Optimal decisions under multiple possibly misspecified candidate likelihoods are thus equivalent to minimax decisions with an exponentiated loss function and a single mixture likelihood $p_{\alpha}(\bm{x}|\theta)$. All of our theoretical results therefore continue to apply upon reinterpreting $p_{\alpha}(\bm{x}|\theta)$ as the relevant reference likelihood.
Recall that the mixture likelihood places greater weight on likelihood 1 when $K_{1}<K_{2}$. In the extreme case where$K_{1}\to0$ --- that is, Alice has no misspecification concerns about likelihood 1 --- we obtain $\alpha\to1$ and $\lambda\to\infty$, so the optimal decision reduces to the minimax-optimal decision under likelihood 1 alone, irrespective of the degree of misspecification concern about likelihood 2. This holds even if likelihood 2 is more efficient than likelihood 1 under correct specification. Intuitively, when likelihood 2 is globally misspecified but likelihood 1 is not, the likelihoods are far apart in terms of misspecification risk, and it is always optimal to place all weight on the correctly specified model. To generate more meaningful tradeoffs between efficiency and misspecification, one would need to bring the misspecification risks closer together by taking $|K_{1}-K_{2}|=O(1/n)$. We leave the analysis of such a regime of closely competing models to future research.
In this article, we have introduced a framework for evaluating statistical decisions under both prior ambiguity and likelihood misspecification. Misspecification manifests as an exponential tilting of the loss function, while ambiguity corresponds to a search for the least favorable prior. We also develop a theory of local asymptotics under global misspecification, achieved by localizing the priors around a reference parameter, and use this theory to characterize optimal estimation and treatment-assignment decisions. Remarkably, in both cases, optimal decisions coincide with those under correct likelihood specification.
The proposed framework opens several avenues for further research. While we discuss some examples of asymmetric loss functions in Appendix C, a general theory for characterizing optimal decisions under such losses remains to be developed. As noted there, optimal decisions under asymmetric loss may depend on the degree of misspecification. On model selection, while Section (ref) provides an initial treatment, a richer characterization would require taking the misspecification risks of competing likelihoods to converge to each other, so that meaningful tradeoffs between efficiency and misspecification robustness may emerge. A further extension would be to separate ambiguity concerns over the prior from those over candidate likelihoods, employing smooth ambiguity aversion for the latter, as discussed in Section (ref). This would bring the framework closer to the literature on Bayesian model averaging and model selection.