2 Variance estimation for the Horvitz-Thompson estimator

Peter M. Aronow and Cyrus Samii

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Consider a population $U$ indexed by $1,\dots ,k,\dots N$ and a sampling design such that the probability of inclusion in the sample for unit $k$ is ${\pi }_k,$ and the joint inclusion probability for units $k$ and $l$ is ${\pi }_{kl}.$ Under a measurable design, there are two conditions: (1) ${\pi }_k>0$ and ${\pi }_k$ is known for all $k\in U$ and (2) ${\pi }_{kl}>0$ and ${\pi }_{kl}$ is known for all $k,l\in U.$ Non-measurable designs include those for which either of the two conditions for a measurable design do not hold. Failure to meet the former condition precludes unbiased estimation of totals.

The Horvitz-Thompson estimator of a population total is $\widehat{t}={\displaystyle \sum _{k\in s}{y}_k/{\pi }_k}={\displaystyle \sum _{k\in U}{I}_k}{y}_k/{\pi }_k,$ where $I_k\in \{0,1\}$ is unit $k’\text{s}$ inclusion indicator, the only stochastic component of the expression, with $\text{E}(I_k)={\pi }_k,$ the inclusion probability, and $s$ and $U$ refer to the sample and the population, respectively. Define $\text{E}(I_k{I}_l)={\pi }_{kl},$ which is the probability that both units $k$ and $l$ from $U$ are included in the sample. Since $I_k{I}_k=I_k,\text{E}(I_k{I}_k)={\pi }_{kk}={\pi }_k$ by construction. When condition 1 holds, as we assume throughout, the Horvitz-Thompson estimator is unbiased.

2.1  Properties of the Horvitz-Thompson variance estimator under measurability

By Horvitz and Thompson (1952), the variance of the Horvitz-Thompson estimator for the total is

$$\text{Var}(\widehat{t})={\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in U}\text{Cov}(I_k,I_l)}\frac{y_k}{{\pi }_k}}\frac{y_l}{{\pi }_l}={\displaystyle \sum _{k\in U}\text{Var}}(I_k){\left(\frac{y_k}{{\pi }_k}\right)}^2+{\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in U\backslash k}\text{Cov}(I_k,I_l)\frac{y_k}{{\pi }_k}\frac{y_l}{{\pi }_l}}}.$$

We label a sample from a measurable design, $s^M,$ and an unbiased estimator for $\text{Var}(\widehat{t})$ on $s^M$ is

$$\widehat{\text{Var}}(\widehat{t})={\displaystyle \sum _{k\in s^M}{\displaystyle \sum _{l\in s^M}\frac{\text{Cov}(I_k,I_l)}{{\pi }_{kl}}}}\frac{y_k}{{\pi }_k}\frac{y_l}{{\pi }_l}={\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in U}{I}_k}}{I}_l\frac{\text{Cov}(I_k,I_l)}{{\pi }_{kl}}\frac{y_k}{{\pi }_k}\frac{y_l}{{\pi }_l},$$

where the only stochastic part of the expression is $I_k{I}_l,$ and unbiasedness is by $\text{E}(I_k{I}_l)={\pi }_{kl}.$

2.2  Properties of the Horvitz-Thompson variance estimator under non-measurability

We now examine the case where condition 2 does not hold: ${\pi }_{kl}=0$ for some units $k,l\in U.$ Because $I_k$ is a Bernoulli random variable with probability ${\pi }_k,\text{Cov}(I_k,I_l)={\pi }_{kl}-{\pi }_k{\pi }_l$ for $k\ne l,$ and $\text{Cov}(I_k,I_k)=\text{Var}(I_k)={\pi }_k(1-{\pi }_k).$ Then, we can re-express the variance above as,

$$\begin{array}{cc}\hfill \text{Var}(\widehat{t})& ={\displaystyle \sum _{k\in U}{\pi }_k}(1-{\pi }_k){\left(\frac{y_k}{{\pi }_k}\right)}^2+{\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in U\backslash k}({\pi }_{kl}-{\pi }_k{\pi }_l)\frac{y_k}{{\pi }_k}\frac{y_l}{{\pi }_l}}}\hfill \\ & ={\displaystyle \sum _{k\in U}{\pi }_k}(1-{\pi }_k){\left(\frac{y_k}{{\pi }_k}\right)}^2+{\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in \{U\backslash k:{\pi }_{kl}>0\}}({\pi }_{kl}-{\pi }_k{\pi }_l)\frac{y_k}{{\pi }_k}\frac{y_l}{{\pi }_l}}}-\underset{A}{\underbrace{{\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in \{U\backslash k:{\pi }_{kl}=0\}}{y}_k{y}_l}}}}.\hfill \end{array}$$

For $k$ and $l$ such that ${\pi }_{kl}=0,$ the sampling design will never permit unbiased estimation of the component of the variance labeled as $A$ above, since we will never observe $y_k$ and $y_l$ together. We label a sample from a design where condition 2 fails as $s^0.$ When $\widehat{\text{Var}}(\widehat{t})$ is applied to $s^0,$ the result is unbiased for $\text{Var}(\widehat{t})+A.$ We state this formally as follows:

Proposition 1. When $s^0$ refers to a sample from a design with some ${\pi }_{kl}=0,$

$\text{E}\left[\widehat{\text{Var}}(\widehat{t})\right]=\text{Var}(\widehat{t})+{\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in \{U\backslash k:{\pi }_{kl}=0\}}{y}_k{y}_l}}=\text{Var}(\widehat{t})+A.$

Proof. The result follows from,

$$\begin{array}{cc}\hfill \text{E}\left[{\displaystyle \sum _{k\in s^0}{\displaystyle \sum _{l\in s^0}\frac{\text{Cov}(I_k,I_l)}{{\pi }_{kl}}}}\frac{y_k}{{\pi }_k}\frac{y_l}{{\pi }_l}\right]& =\text{E}\left[{\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in \{U:{\pi }_{kl}>0\}}{I}_k}}{I}_l\frac{\text{Cov}(I_k,I_l)}{{\pi }_{kl}}\frac{y_k}{{\pi }_k}\frac{y_l}{{\pi }_l}\right]\hfill \\ & ={\displaystyle \sum _{k\in U}\text{Var}}(I_k){\left(\frac{y_k}{{\pi }_k}\right)}^2+{\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in \{U\backslash k:{\pi }_{kl}>0\}}\text{Cov}(I_k,I_l)\frac{y_k}{{\pi }_k}\frac{y_l}{{\pi }_l}}}\hfill \\ & =\text{Var}(\widehat{t})+{\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in \{U\backslash k:{\pi }_{kl}=0\}}{y}_k{y}_l}}=\text{Var}(\widehat{t})+A.\hfill \end{array}$$

The standard Horvitz-Thompson variance estimator, if applied to designs with zero pairwise inclusion probabilities, can therefore have a positive or a negative bias. If the $y_k,y_l$ values are always nonnegative (or always nonpositive), then the bias is always nonnegative. When values may be positive or negative, $A$ is the sum of cross-products of outcomes that never appear together under the design sample. If the jointly exclusive outcomes are centered over zero, then no correlation in these outcomes would tend to result in small bias, positive correlation in positive bias, and negative correlation in negative bias.

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