Response-probability weighting for a sample mean

Let $(\mathcal{X},\mathcal{A})$ be a measurable covariate space and let $n\ge 1$. In the one-outcome full/observed-data convention of Full and observed data under a missingness pattern, $X_i\in \mathcal{X}$ is observed for every unit, $R_i\in \{0,1\}$ indicates whether the real outcome $Y_i$ is observed, and the encoded input stores $Z_i=Y_i$ when $R_i=1$ and the placeholder $Z_i=0$ when $R_i=0$. The mask $R_i$ distinguishes the placeholder from an observed zero; it is not an imputation or a replacement of the missing full-data value. Let $\pi :\mathcal{X}\to (0,1]$ be a supplied, known, measurable response-probability function, with $\pi (x_i)>0$ for every record in the input.

Input: $n$, records $(x_i,r_i,z_i{)}_{i=1}^n$ satisfying the stated conventions, and the function $\pi$.

Procedure: For each record use weight $1\mathrm{/}\pi (x_i)$ when $r_i=1$ and weight zero when $r_i=0$. Return the unnormalized Horvitz–Thompson weighted mean ${\widehat{\mu }}_{\mathrm{H}\mathrm{T}}=\frac{1}{n}{\sum }_{i=1}^n\frac{z_i}{\pi (x_i)}\mathrm{.}$

The procedure takes the true known response probabilities as input. Substituting fitted probabilities is a different procedure; this definition makes no finite-sample stability or estimated-weight inference claim. The weighting convention is the one used for a single response indicator; no joint treatment/response weight is defined here.