Incremental effects for continuous exposures

Causal inference problems often involve continuous treatments, such as dose, duration, or frequency. However, continuous exposures bring many challenges, both with identification and estimation. For example, identifying standard dose-response estimands requires that everyone has some chance of receiving any particular level of the exposure (i.e., positivity). In this work, we explore an alternative approach: rather than estimating dose-response curves, we consider stochastic interventions based on exponentially tilting the treatment distribution by some parameter $\delta$, which we term an incremental effect. This increases or decreases the likelihood a unit receives a given treatment level, and crucially, does not require positivity for identification. We begin by deriving the efficient influence function and semiparametric efficiency bound for these incremental effects under continuous exposures. We then show that estimation of the incremental effect is dependent on the size of the exponential tilt, as measured by $\delta$. In particular, we derive new minimax lower bounds illustrating how the best possible root mean squared error scales with an effective sample size of $n/\delta$, instead of usual sample size $n$. Further, we establish new convergence rates and bounds on the bias of double machine learning-style estimators. Our novel analysis gives a better dependence on $\delta$ compared to standard analyses, by using mixed supremum and $L_2$ norms, instead of just $L_2$ norms from Cauchy-Schwarz bounds. Finally, we show that taking $\delta \to \infty$ gives a new estimator of the dose-response curve at the edge of the support, and we give a detailed study of convergence rates in this regime.