1. Introduction

Stephen Ash

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Fay and Train (1995) present a method called successive difference replication (SDR) that can be used to estimate the variance of an estimated total from a systematic random sample from an ordered list. The estimator uses the general form of a replication variance estimator where the replicate factors are constructed such that it mimics the successive difference (SD) estimator.

The paper establishes and uses new concepts to gain more understanding into the methodology originally proposed by Fay and Train (1995), hereafter referred to as F&T. The new concepts help to explain the impact of the row assignments on the variance estimator, show how a reduced set of replicates leads to a reasonable estimator, and establish conditions for successive difference replication to be equivalent to the successive difference estimator. It is our hope that this additional understanding of SDR will make it less mysterious and thereby more accessible to anyone estimating variances for a systematic random sample.

The paper begins by reviewing the SD estimator and how it is suited for variance estimation of systematic random samples. The main section of the paper presents two theorems that provide conditions for the SDR estimator to be equivalent to the SD estimator. The paper concludes with empirical examples that examine alternative row assignments and the suitability of using a reduced set of replicates.

For the remainder of the paper, $sys$ will be used as shorthand for systematic random sampling from an ordered list. We abbreviate $sys$ this way because systematic sampling from an unordered or randomly ordered list can be shown to be equivalent to simple random sampling (Madow and Madow 1944). For our discussion, we focus solely on equal probability selection and methods for selecting a sample in only one dimension. Excellent summaries of $sys$ and estimating variances from $sys$ can be found in Iachan (1982), Wolter (1985, chapter 7), Murthy and Rao (1988), and Bellhouse (1988).

1.1  Review of successive differences

Wolter (1984; estimator 2) provides a form of the successive difference estimator of the variance of an estimated mean $\left(\overline{y}\right)$ for a $sys$ sample design as

$${\widehat{v}}_{\text{SD}1}\left(\widehat{\overline{y}}\right)=\left(1-f\right)\frac{1}{2n\left(n-1\right)}{\displaystyle \sum _{k=2}^n{\left(y_k-y_{k-1}\right)}^2},$$

where $y_k$ is the variable of interest, $k$ indexes the units of the ordered sample, and $f=n/N$ is the sampling fraction. The statistic of interest is $Y$ or the total of $y_k$ over the universe of interest and $\widehat{Y}$ is an estimator of $Y.$ Let $N$ and $n$ be the size of the universe and sample, respectively. The mean of $y_k$ and its estimator are defined as $\overline{y}=Y/N$ and $\widehat{\overline{y}},$ respectively. We also define the estimator of the total $Y$ as $\widehat{Y}={\displaystyle {\sum }_{k=1}^n{\stackrel{⌣}{y}}_k,}$ where the weighted variable of interest with equal weights is ${\stackrel{⌣}{y}}_k=\left(N/n\right)y_k;$ with unequal sample weights $w_k,$ it is defined as ${\stackrel{⌣}{y}}_k=w_k{y}_k.$ The estimator ${\widehat{v}}_{\text{SD}1}\left(\widehat{\overline{y}}\right)$ has been described by Yates (1953; pages 229-231) and recommended by Wolter (1984). Murthy and Rao (1988, equation 32) provide a sketch of why the estimator works. The short version is that since $sys$ only selects one unit within each implicit stratum, SD's solution is to collapse adjacent implicit strata. With two units, we can estimate the variance of an implicit stratum. Implicit strata are collapsed and averaged over all possible pairs and then multiplied by $n,$ the number of implicit strata, to give the variance of all the implicit strata.

One SD variance estimator of a total from a $sys$ sample is given by F&T as

$${\widehat{v}}_{\text{SD}1}\left(\widehat{Y}\right)=\left(1-f\right)\frac{n}{2\left(n-1\right)}{\displaystyle \sum _{k=2}^n{\left({\stackrel{⌣}{y}}_k-{\stackrel{⌣}{y}}_{k-1}\right)}^2}.$$

Wolter (1985, equation 7.7.4) defined the same estimator where $w_k={\left(n{p}_k\right)}^{-1}$ and $p_k$ is the with replacement probability of selection for unit $k.$ F&T defined a second SD estimator

$${\widehat{v}}_{\text{SD}2}\left(\widehat{Y}\right)=\frac{1}{2}\left(1-f\right)\left[{\displaystyle \sum _{k=2}^n{\left({\stackrel{⌣}{y}}_k-{\stackrel{⌣}{y}}_{k-1}\right)}^2}+{\left({\stackrel{⌣}{y}}_n-{\stackrel{⌣}{y}}_1\right)}^2\right],$$

which is "circular� in that it includes an extra squared difference that links the first and last unit from the sorted list.

We express the SD2 estimator more generally as a quadratic form as ${\stackrel{⌣}{y}}^{\prime }C\stackrel{⌣}{y},$ where ${\stackrel{⌣}{y}}^{\prime }=\left[{\stackrel{⌣}{y}}_1{\stackrel{⌣}{y}}_2\dots {\stackrel{⌣}{y}}_n\right]$ is defined as the $n\times 1$ weighted observation vector and $C$ is a square matrix with 2 for each value of the diagonal, -1 for every value of the superdiagonal and subdiagonal, and -1 for the bottom left and top right value. Here the superdiagonals are defined as the diagonals adjacent to the main diagonal. The exception is a $2\times 2$ matrix.

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