## Abstract

We address the characterization of problems in which a consistent estimator exists in a union of two models, also termed as a doubly robust estimator. Such estimators are important in missing information, including causal inference problems. Existing characterizations, based on the semiparametric theory of projections, have seen sufficient progress, but can still leave one’s understanding less than satisfied as to when and especially why such estimation works. We explore here a different, explanatory characterization – an exegesis based on logical operators. We show that double robustness exists if and only if we can produce consistent estimators for each contributing model based on an “AND” estimator, i. e., an estimator whose consistency generally needs both models to be correct. We show how this characterization explains double robustness through falsifiability.

## 1 Motivation

Consider the following problem that often motivates doubly robust estimation [1]. For patients with a particular disease, one of two treatments, *i*, physicians first measure a set of covariates

The average

One usual assumption is that the regression *y* is the general form of the probability limit of *e* is the general form of the probability limit of

An interesting case is if one assumes that, for the true distribution of the data,

is consistent for

Although the verification of this result is easy, standard *derivations* of the above estimator in the first place are relatively lengthy and involve semiparametric projection theory. One such derivation includes finding the form of all regular and asymptotically linear estimators of

If the estimator solves the nonparametric influence function evaluated at some working models, then an alternative way to find the robustness conditions of such estimator is to find the zeros of the second order remainder in the von Mises expansion. Specifically if the remainder has factors that are differences between “posited minus true” value of certain component, then correct specification of that component is a sufficient condition for robustness (Mark van der Laan, personal communication).

More generally, such characterization based on semiparametric theory, have seen sufficient progress [7], [3], but still leave one’s understanding less than satisfied as to when and especially why such estimation works. Our purpose here is to add to this understanding by presenting an exegesis – an alternative characterization based on logical operators that focuses on falsifiability.

## 2 Characterization emphasizing falsifiability

We argue that double robustness is critically related to how to produce an OR operator using an AND operator. By an operator here we mean the equivalence class that contains a possibly random function with a probability limit, and all other functions with the same limit. Denote “share same equivalence class” by

For the example with the ignorable assignment of the previous section, consider an operator

Then, the operator defined as

This operator provides explanatory and predictive power on a number of important points regarding the structure of doubly robust estimators.

**(a) The logical operator reproduces easily well known results.** To see how the logical operation argument can easily reproduce this result in the ignorable assignment example, we can choose

satisfying (2). Then, we can obtain

So, estimating the components

which, by (3), has the property that

For the above example, there exist a number of other estimators that are doubly robust [7]. However, all such estimators are in the same equivalence class as above, and they share the same influence function. Also, although the above example is relatively simple, many enriched variations, including parametric vs single value for

**(b) The three-term structure arises from the general logical relation between an OR and an AND operator.** We can generalize the above arguments as follows.

## Result.

If a consistently estimable operator

The first claim is easy to check from above. For the second claim, note that if there exists an

This result predicts that the three-term structure of (1) is not specific to the above example, but rather obeys the logical structure that any OR operator has as constructed from an AND operator (3).

**(c) The role of the AND operator in falsifiability.** The AND operator can be seen more practically as essential for falsifiability. To see this, focus in the case where indeed either *incorrect* value of *τ*, and it is in this way that we can conclude that its alternative is the true value.

**(d) Connection with variation dependent components.** Double robustness is usually examined only when the components

Consider an extreme example where the estimand now is the true regression *larger* (worse) of the two estimated mean squared errors,

still produces the true regression

**(e) Further comments.** The interpretation of why and how one of the two component models is misspecified is important in the discussion between the physician and the statistician. As [3] (p. 923) also note, the construction of a doubly robust estimator allows us to point to which of the two component estimators it disagrees with the most, and interpret this as evidence of which component is misspecified. Although useful, this observation is different from that discussed in **(c)** above, and is formed *after* construction and *in terms of* the doubly robust estimator. Therefore, such observation by itself does not quite answer why and how this evidence arises, as with the more elemental use of the components in the three terms of the OR operator.

The present characterization does not antagonize use of semiparametric methods, which can be very useful for deriving, when possible, AND operators with regularly estimable marginal components. On the other hand, there exist union models with consistent estimators that are irregular [8], but which still obey the three-term structure discussed above.

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**Received:**2018-06-04

**Revised:**2018-10-25

**Accepted:**2018-11-09

**Published Online:**2019-03-14

**Published in Print:**2019-04-26

© 2019 Walter de Gruyter GmbH, Berlin/Boston

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.