doubly robust estimator

Suppose we are trying to estimate a parameter $\theta$ and we have estimates $Y_1,\dots ,Y_n$ of $\theta$ drawn from a meta-population $Y_1,\dots ,Y_N$. Suppose we also have a model $f$ which estimates $Y_i$ given some covariates $X_i$, and that the covariates $X_i$ have been sampled from some larger population with probability $\pi (X_i)$. Then the doubly robust estimator, also called the augmented IPW estimator, is

$$\hat{\theta }=\frac{1}{N}\sum _{i=1}^N\left(f(X_i)+(Y_i-f(X_i))\frac{{\xi }_i}{\pi (X_i)}\right),$$

where ${\xi }_i$ indicates whether $X_i$ has been sampled. Here $N$ is the size of the entire population, and we’ve sampled some number $n<N$ of labels $Y_i$. For the rest, we estimate the true value $Y_i$ with $f(X_i)$.

Since $\mathbb{E}[{\xi }_i]=\pi (X_i)$, this estimator is unbiased. It also tends to have lower variance. It’s used in survey sampling, reinforcement learning, and causal inference.