Improved Horvitz-Thompson estimator in survey sampling

Section 2. HT estimator and its drawback

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Consider a finite population $U\mathrm{<mo>=</mo>}\left\{U_1\mathrm{,}\dots ,U_N\right\}$ of size $N,$ where $U_k$ denotes the $k^{\text{th}}$ unit. For simplicity, we write $U\mathrm{<mo>=</mo>}\left\{\mathrm{1,}\dots \mathrm{,}k\mathrm{,}\dots \mathrm{,}N\right\}.$ For each unit $k,$ suppose that the value $y_k$ of the target characteristic $Y$ is measured. Our aim is to estimate the total, $t_y={\displaystyle {\sum }_U{y}_k},$ using a sample $s$ of size $n$ which is randomly drawn from the population $U.$ We implement unequal probability sampling without replacement. Denote ${\left\{{\pi }_k\right\}}_{k\mathrm{<mo>=</mo>\; 1}}^N$ as the first-order inclusion probabilities and ${\left\{{\pi }_{kl}\right\}}_{k\ne l}$ as the second-order inclusion probabilities.

Horvitz and Thompson (1952) proposed the HT estimator as follows

$${\widehat{t}}_{\text{HT}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s}}\frac{y_k}{{\pi }_k}\mathrm{.}$$

The HT estimator ${\widehat{t}}_{\text{HT}}$ is an unbiased estimator of $t_y$ and its variance is

$$V\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_{\text{HT}}\mathrm{<mo\; stretchy="false">)</mo>\; <mo>=</mo>}{{\displaystyle \sum }}_U\frac{{\Delta }_{kk}}{{\pi }_k^2}{y}_k^2+\underset{k\ne l}{{\displaystyle \sum }{{\displaystyle \sum }}_U}\frac{{\Delta }_{kl}}{{\pi }_k{\pi }_l}{y}_l{y}_k\mathrm{,}(2.1)$$

where ${\Delta }_{kk}\mathrm{<mo>=</mo>}{\pi }_k-{\pi }_k^2$ for all $k$ and ${\Delta }_{kl}\mathrm{<mo>=</mo>}{\pi }_{kl}-{\pi }_k{\pi }_l$ for all $k\ne l.$ When the inclusion probabilities are highly imbalanced, i.e., some ${\pi }_k’\text{s}$ are very small, the variance of the HT estimator may be very large.

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