Improved Horvitz-Thompson estimator in survey sampling

Section 3. Improved HT estimator

• Previous
• Next

In this section, we improve the HT estimator in the sense of reducing its mean squared error (MSE). The resultant estimator is referenced as the IHT estimator. For doing this, we first propose the modified first-order inclusion probabilities, where the hard-threshold method is used to reduce the effect of those inclusion probabilities with relatively tiny values.

Definition 1. Let ${\pi }_{\left(1\right)}\le {\pi }_{\left(2\right)}\le \dots \le {\pi }_{\left(N\right)}$  be the ordered values of the first-oder inclusion probabilities $\left\{{\pi }_1\mathrm{,}{\pi }_2\mathrm{,}\dots ,{\pi }_N\right\}.$  Assume that there exists an integer $K\ge 2$  such that ${\pi }_{\left(K\right)}\le {\left(K+1\right)}^{-1}\mathrm{.}$  We define the modified first-order inclusion probabilities as follows

$${\pi }_k^{*}=(\begin{array}{cc}{\pi }_k& {\pi }_k>{\pi }_{\left(K\right)}\mathrm{,}\\ {\pi }_{\left(K\right)}& {\pi }_k\le {\pi }_{\left(K\right)}\mathrm{,}\end{array}1\le k\le N\mathrm{.}$$

From the definition, we partition the finite population into two parts: $U_1=\left\{k:{\pi }_k>{\pi }_{\left(K\right)}\right\}$ with size $N-K,$ and $U_2\mathrm{<mo>=</mo>}\left\{k\mathrm{<mo>:</mo>}{\pi }_k\le {\pi }_{\left(K\right)}\right\}$ with size $K.$ For $U_1,$ the first-order inclusion probabilities remain unchanged, while all of first-order inclusion probabilities for $U_2$ are replaced by ${\pi }_{\left(K\right)}.$ From this hard-threshold, we get our modified first-order inclusion probabilities ${\left\{{\pi }_k^{*}\right\}}_{k\mathrm{<mo>=</mo>\; 1}}^N.$ Obviously, the choice of $K$ is very important. In Section 3.2, we shall provide a simple way to choose $K.$

Remark on existence of $K.$ The assumption in Definition 1 is quite weak. If ${\pi }_{\left(2\right)}\mathrm{<mo>></mo>}1/\left(2+1\right),$ then the sampling fraction $f\mathrm{<mo>></mo>}\frac{1}{3}-\frac{1}{3N}.$ However that situation that $f\mathrm{<mo>></mo>}\frac{1}{3}$ rarely happens in practical surveys. Thus, the inequality that ${\pi }_{\left(2\right)}\le 1/\left(2+1\right)$ generally holds.

Instead of the original first-order inclusion probabilities ${\left\{{\pi }_k\right\}}_{k\mathrm{<mo>=</mo>\; 1}}^N,$ we use our defined modified first-order inclusion probabilities ${\left\{{\pi }_k^{*}\right\}}_{k\mathrm{<mo>=</mo>\; 1}}^N$ to construct an improved Horvitz-Thompson (IHT) estimator by inverse probability weighting.

Definition 2. The IHT estimator is defined as

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

Unlike the unbiased HT estimator, the IHT estimator is biased. However, this modification leads to much smaller MSE due to reducing the variance. It is worth pointing out that, although we focus on sampling without replacement in this paper, our modification idea is equally applicable to the Hansen-Hurwitz estimator (Hansen and Hurwitz, 1943) for sampling with replacement.

3.1  Properties of the IHT estimator

In this section, we derive the properties of the IHT estimator. We first provide the expressions of its bias, variance, MSE and an unbiased estimator of MSE in Theorem 1. Then we compare the IHT estimator with the HT estimator in Theorems 2 and 3.

Theorem 1. The bias and variance of the IHT estimator ${\widehat{t}}_{IHT}$  are expressed as

$$Bias\left({\widehat{t}}_{IHT}\right)\mathrm{<mo>=</mo>}{{\displaystyle \sum }}_{U_2}\left(\frac{{\pi }_k}{{\pi }_{\left(K\right)}}-1\right)y_k\mathrm{,}$$

and

$$Var\left({\widehat{t}}_{IHT}\right)\mathrm{<mo>=</mo>}{{\displaystyle \sum }}_U\frac{{\Delta }_{kk}}{{\pi }_k^{*2}}{y}_k^2+\underset{k\ne l}{{\displaystyle \sum }{{\displaystyle \sum }}_U}\frac{{\Delta }_{kl}}{{\pi }_k^{*}{\pi }_l^{*}}{y}_k{y}_l\mathrm{,}$$

respectively, where ${\Delta }_{kk}\mathrm{<mo>=</mo>}{\pi }_k\left(1-{\pi }_k\right)\mathrm{,}{\Delta }_{kl}\mathrm{<mo>=</mo>}{\pi }_{kl}-{\pi }_k{\pi }_l$   $\left(k\ne l\right)$  as defined before. Therefore, its MSE is given by

$$MSE\left({\widehat{t}}_{IHT}\right)\mathrm{<mo>=</mo>}{\left[{{\displaystyle \sum }}_{U_2}\left(\frac{{\pi }_k}{{\pi }_{\left(K\right)}}-1\right)y_k\right]}^2+{{\displaystyle \sum }}_U\frac{{\Delta }_{kk}}{{\pi }_k^{*2}}{y}_k^2+\underset{k\ne l}{{\displaystyle \sum }{{\displaystyle \sum }}_U}\frac{{\Delta }_{kl}}{{\pi }_k^{*}{\pi }_l^{*}}{y}_k{y}_l\mathrm{.}(3.1)$$

An unbiased estimator of the MSE is

$$\begin{array}{ll}\widehat{MSE}\left({\widehat{t}}_{IHT}\right)\hfill & \mathrm{<mo>=</mo>}{{\displaystyle \sum }}_{s_2}\frac{{\left({\pi }_k-{\pi }_{\left(K\right)}\right)}^2}{{\pi }_{\left(K\right)}^2{\pi }_k}{y}_k^2+\underset{k\ne l}{{\displaystyle \sum }{{\displaystyle \sum }}_{s_2}}\frac{\left({\pi }_k-{\pi }_{\left(K\right)}\right)\left({\pi }_l-{\pi }_{\left(K\right)}\right)}{{\pi }_{\left(K\right)}^2{\pi }_{kl}}{y}_k{y}_l\hfill \\ \hfill & +{{\displaystyle \sum }}_s\frac{{\stackrel{⌣}{\Delta }}_{kk}}{{\pi }_k^{*2}}{y}_k^2+\underset{k\ne l}{{\displaystyle \sum }{{\displaystyle \sum }}_s}\frac{{\stackrel{⌣}{\Delta }}_{kl}}{{\pi }_k^{*}{\pi }_l^{*}}{y}_k{y}_l\mathrm{,}\hfill \end{array}$$

where ${\stackrel{⌣}{\Delta }}_{kk}\mathrm{<mo>=</mo>}\frac{{\Delta }_{kk}}{{\pi }_k}\mathrm{,}{\stackrel{⌣}{\Delta }}_{kl}\mathrm{<mo>=</mo>}\frac{{\Delta }_{kl}}{{\pi }_{kl}},$   $s$  is the sample set, and $s_2\mathrm{<mo>=</mo>}s\cap U_2.$

Proof. See Appendix A.1.

To derive the properties of the IHT estimator, we need the following regularity conditions:

Condition C.1. ${{\displaystyle \min }}_{i\in U}{\pi }_i\ge \lambda \mathrm{<mo>></mo>\; 0,}{{\displaystyle \min }}_{i\mathrm{,}j\in U}{\pi }_{ij}\ge {\lambda }^{*}\mathrm{<mo>></mo>\; 0},$ and

$$\underset{Narrow\infty }{{\displaystyle \lim sup}}n\underset{i\ne j\in U}{{\displaystyle \max }}\left|{\pi }_{ij}-{\pi }_i{\pi }_j\right|\mathrm{<mo><</mo>}\infty \mathrm{.}$$

Condition C.2. ${{\displaystyle \max }}_{i\in U}\left|y_i\right|\le C$ with $C$  a positive constant not depending on $N.$

Condition C.1 is a common condition imposed on the first-order and second-order inclusion probabilities. The same conditions are used in Breidt and Opsomer (2000), where further comments on C.1 are provided. Condition C.2 is also a common condition.

Theorem 2. For the HT estimator ${\widehat{t}}_{HT}$  and the IHT estimator ${\widehat{t}}_{IHT},$  under the Conditions C.1-C.2, we have

$$Bias\left(N^{-1}{\widehat{t}}_{HT}\right)\mathrm{<mo>=</mo>\; 0,}Bias\left(N^{-1}{\widehat{t}}_{IHT}\right)\mathrm{<mo>=</mo>}O\left(n^{-1}\right)\mathrm{<mo>;</mo>}$$

and

$$MSE\left(N^{-1}{\widehat{t}}_{HT}\right)\mathrm{<mo>=</mo>}O\left(n^{-1}\right)\mathrm{,}MSE\left(N^{-1}{\widehat{t}}_{IHT}\right)\mathrm{<mo>=</mo>}O\left(n^{-1}\right)\mathrm{.}$$

Proof. See Appendix A.2.

From Theorem 2, the squared-bias of our IHT estimator is very small compared to its MSE. Although our IHT estimator brings a bias to reduce the variance, the price for this is relatively small. The following theorem theoretically compares the efficiency of the two estimators.

Theorem 3. Under the Conditions C.1-C.2, we have

$$MSE\left(N^{-1}{\widehat{t}}_{IHT}\right)\le MSE\left(N^{-1}{\widehat{t}}_{HT}\right)+o\left(n^{-1}\right)\mathrm{.}(3.2)$$

Especially, for Poisson sampling, we obtain

$$MSE\left(N^{-1}{\widehat{t}}_{IHT}\right)\le MSE\left(N^{-1}{\widehat{t}}_{HT}\right)\mathrm{,}$$

where the strict inequality is true if there exist $k\ne l\in U_2$  such that $\left({\pi }_k-{\pi }_{\left(K\right)}\right)y_k\ne \left({\pi }_l-{\pi }_{\left(K\right)}\right)y_l.$

Proof. See Appendix A.3.

Theorem 3 shows that, under some mild conditions, the proposed IHT estimator is asymptotically more efficient than the HT estimator. From the proof in Appendix A.3, the term $o\left(n^{-1}\right)$ in equation (3.2) is due to the interaction term from the second-order inclusion probabilities. We theoretically bound the term as $o\left(n^{-1}\right).$ For Poisson sampling, the term does not exist, so the MSE of the IHT estimator is uniformly not larger than that of the HT estimator. Empirically, we compare the IHT estimator with the HT estimator in Section 5.

3.2  The choice of $K$

The efficiency of the IHT estimator relies on the choice of $K,$ which provides a control of the variance-and-bias tradeoff. The choice of $K$ needs to satisfy the condition that ${\pi }_{\left(K\right)}\mathrm{<mo><</mo>}1/\left(K+1\right)$ of Definition 1, since the modified inclusion probabilities would cause large bias when $K$ becomes large. On the other hand, the improvement of the IHT estimator would not be significant if $K$ is small. In the proofs of Theorem 3, equation (A.5) provides a lower bound of the main term of $\text{MSE}\left(N^{-1}{\widehat{t}}_{\text{HT}}\right)-\text{MSE}\left(N^{-1}{\widehat{t}}_{\text{IHT}}\right).$ The lower bound increases as ${\pi }_{\left(K\right)}$ increases. Therefore, denoting the maximum value $K^{*}\mathrm{<mo>=</mo>}\max \left\{i\mathrm{<mo>:</mo>}{\pi }_{\left(i\right)}\le 1/\left(i+1\right)\right\},$ we choose $K^{*}$ as the threshold. In practice, we propose the following algorithm to find the maximum value $K^{*}.$

Note that the choice of $K^{*}$ based on Algorithm 1 is not optimal in terms of MSE. However, we simulate an example in Section 5 where the performance of Algorithm 1 is very close to that of the theoretically ideal choice.

• Previous
• Article main page
• Next