Quasi-random ranked set sampling

Abstract

This article presents a quasi-random ranked set sampling approach based on the Halton sequence for natural resource surveys. A design-based variance estimator for the sample mean is also presented.

Introduction

To estimate a population characteristic of a natural resource, a sample consisting of some part of the population is used. The main goal of a sampling design is to select a representative sample from which the population characteristic can be estimated with low bias and high precision. This article considers drawing sample locations from two-dimensional continuous study regions to estimate a population mean $\mu$. One of the most commonly used designs is simple random sampling (SRS), where the sample locations are independently drawn from a uniform distribution over the resource. Although SRS yields unbiased estimates of $\mu$, there is no guarantee that a specific sample is representative. Many sampling designs have been proposed in the statistical literature that improve on SRS in a variety of ways (Hankin et al., 2019). One effective strategy is to spread the sample locations evenly over the resource, called spatially balanced sampling (Stevens and Olsen, 2004). These designs are known to be efficient when sampling natural resources because nearby locations tend to have more similar response values than distant locations (Stevens and Olsen, 2004, Robertson et al., 2013, Grafström and Schelin, 2014). In this article, we consider incorporating spatial spread into a two-phase sampling design, ranked set sampling.

Ranked set sampling (RSS) was first proposed by McIntyre (1952) for estimating the average yield of an arable crop in an agricultural field trial. Measuring yield for each field plot was time consuming because it involved removing, and then weighing, the crop. With some experience, it was relatively quick to estimate, by eye, the yield to the extent that a group of plots could be ranked in estimated yield order. McIntyre proposed the design, later named RSS (Halls and Dell, 1966), where a subset of plots is first ranked. The ranking variable can be a quick estimate of the response variable, a visual comparison, an expert opinion, or some other variable known to be correlated with the response variable, but it need not involve actual measurements of the response variable. Then, based on these rankings, a sample is drawn from the ranked plots. The goal of RSS is to collect observations from the resource that are likely to span the full range of response values in the population. This approach to data collection has spawned an active field of research and many RSS approaches have been proposed (Wolfe, 2010, Wolfe, 2012).

In practice, randomness is introduced into sampling designs using pseudo-random sequences which are irregular, non-repetitive and designed to mimic true random sequences. This article considers replacing the pseudo-random sequence in RSS with a quasi-random sequence to increase the spatial spread of the ranking variable. These sequences have been used as a substitute for random numbers in many fields, including numerical integration (Niederreiter, 1978, Niederreiter, 2003), optimization (Sobol, 1979, Robertson et al., 2014) and sampling (Robertson et al., 2013, Robertson et al., 2017).

A $d$-dimensional quasi-random sequence $H={\left\{{\mathit{x}}_j\right\}}_{j=1}^n\subset {[0,1)}^d$ is a low-discrepancy sequence with the property that for all values of $n$, the sequence has low discrepancy. The discrepancy of $H$ is (Niederreiter, 1978)

$$D_n\left(H\right)=\underset{B\in \mathcal{J}}{sup}|\frac{A_H\left(B\right)}{n}-\lambda \left(B\right)|,$$

where $\lambda$ is the Lebesgue measure, $A_H\left(B\right)$ is the number of points from $H$ in $B$ and $\mathcal{J}$ is the set of boxes of the form $\{\mathit{x}\in {[0,1)}^d:a_i\le x_i<b_i\}$ with $0\le a_i<b_i<1$. Loosely speaking, a sequence is considered low-discrepancy if the fraction of points in $B\in \mathcal{J}$ is proportional to $\lambda \left(B\right)$.

A number of quasi-random sequences have been proposed (Halton, 1960, Sobol, 1976, Faure, 1982), but this article focuses on the random-start Halton sequence (Wang and Hickernell, 2000) because of its simplicity. The $i$th coordinate of the $j$th point in a random-start Halton sequence $\mathcal{H}={\left\{{\mathit{x}}_j\right\}}_{j=1}^{\infty }\subset {[0,1)}^d$ is (Price and Price, 2012, Robertson et al., 2017)

$$x_j^{\left(i\right)}=\sum _{p=0}^{\infty }\left\{⌊\frac{u_i+j-1}{b_i^p}⌋\text{ mod }{b}_i\right\}\frac{1}{b_i^{p+1}},$$

where $b_1,\dots ,b_d$ are pair-wise co-prime bases, $\lfloor \cdot \rfloor$ is the floor function and $u_1,\dots ,u_d$ are independently generated integers. To find $u_i$, pick $v_i\in [0,1]$ randomly and round $v_i$ to $r$ digits in base $b_i$, giving $v_i=0.d_1\dots d_r$ (base $b_i$), where $r=max\{q:{\sum }_{t=1}^q{d}_t{b}_i^{t-1}\le 10^{15}\}$. Radix inversion gives $u_i=d_r\dots d_1$ (base $b_i$). For example, the first two points in $\mathcal{H}\subset {[0,1)}^2$ with $b_1=2$, $b_2=3,u_1=2$ and $u_2=5$ are $\{{\mathit{x}}_1,{\mathit{x}}_2\}=\{(1∕4,7∕9),(3∕4,2∕9)\}$ (see supplementary material for calculations and an R function). This corresponds to the Halton sequence skipping two points in the first dimension and five points in the second dimension. Each ${\mathit{x}}_j\in \mathcal{H}$ is a random point with uniform distribution on ${[0,1)}^d$ (Wang and Hickernell, 2000).

Taking the first $n$ points from $\mathcal{H}$ that fall within a study region $\Omega \subset {[0,1)}^2$ is called balanced acceptance sampling (BAS) (Robertson et al., 2013), and its modification (Robertson et al., 2017) requires ${\mathit{x}}_1\in \Omega$. BAS is a spatially balanced sampling design that spreads sample locations evenly over $\Omega$. BAS is efficient when sampling natural resources because well-spread sample locations are likely to span the full range of response values due to the locally similar property of natural resources (Stevens and Olsen, 2004). Rather than having spatially balanced response values, this article considers a spatially balanced ranking variable in RSS, called quasi-random RSS.

The rest of this article is organized as follows. In Section 2, RSS is explained and a quasi-random approach is presented in Section 3. Both approaches are numerically tested in Section 4 and concluding remarks are given in Section 5.