I'm trying to find an elementary derivation (proof was the wrong word) of the Horvitz-Thompson estimator: $$ \hat{Y}=\sum_{i\in s}\frac{y_{i}}{\pi_{i}} $$ where $i \in s$ if and only if unit $y_{i}$ a sampled unit is in a sample of interest, and $\pi_{i}$ is the probability that $y_{i}$ is in the sample of interest.
My apologies if I've messed up the definitions; I'm rather new to Sampling theory.
Can someone point me in the right direction? Many thanks.
Let $\pi_{i}$ be the probability that unit $U_{i}$ is included in sample of size $n$ by a without replacement sampling procedure. Now, define a random variable $t_{i}$, for $i=1,2,\cdots N$, by \begin{equation} t_{i}=\begin{cases} 1, & U_{i}\in s\\ 0, & \text{ otherwise. } \end{cases} \end{equation} Since a without replacement sampling procedure gives rise to $n$ distinct units, it is clear that \begin{equation} \sum_{i=1}^{N}t_{i}=n, \text{ and } \end{equation} \begin{equation} E(t_{i})=\pi_{i}. \end{equation}
A general linear function of the sample values can be written as \begin{eqnarray} T &=& \sum_{i=1}^{n}c_{i}y_{i},\qquad \text{ or }\\ T &=& \sum_{i=1}^{N}t_{i}c_{i}y_{i} \end{eqnarray} where $c_{i}$ is a constant attached to the unit $U_{i}$ whenever it is selected into the sample and Considering expectation of $T$, we get, \begin{equation} E(T)= \sum_{i=1}^{N}\pi_{i}c_{i}y_{i} \end{equation} For $T$ to be an unbiased estimator of the population total $\sum_{i=1}^{N}Y_{i}$, the constant $c_{i}$ should equal to $1/\pi_{i}$. Thus an unbiased estimator for the population total, as suggested by Horvitz-Thompson, is given by, \begin{eqnarray} \hat{Y}_{HT} &=& \sum_{i=1}^{N}t_{i}\left(\dfrac{Y_{i}}{\pi_{i}}\right)\qquad \text{ or }\\ \hat{Y}_{HT} &=& \sum_{i=1}^{n}\dfrac{y_{i}}{\pi_{i}} \end{eqnarray}
