  While the test-negative design (TND), which is routinely used for monitoring
seasonal flu vaccine effectiveness (VE), has recently become integral to
COVID-19 vaccine surveillance, it is susceptible to selection bias due to
outcome-dependent sampling. Some studies have addressed the identifiability and
estimation of causal parameters under the TND, but efficiency bounds for
nonparametric estimators of the target parameter under the unconfoundedness
assumption have not yet been investigated. We propose a one-step doubly robust
and locally efficient estimator called TNDDR (TND doubly robust), which
utilizes sample splitting and can incorporate machine learning techniques to
estimate the nuisance functions. We derive the efficient influence function
(EIF) for the marginal expectation of the outcome under a vaccination
intervention, explore the von Mises expansion, and establish the conditions for
$\sqrt{n}-$consistency, asymptotic normality and double robustness of TNDDR.
The proposed TNDDR is supported by both theoretical and empirical
justifications, and we apply it to estimate COVID-19 VE in an administrative
dataset of community-dwelling older people (aged $\geq 60$y) in the province of
Qu\'ebec, Canada.
