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<p>In this paper, we examine a finite population distributed across a two-dimensional spatial domain, where a small subset of units exhibit substantially larger response values compared to the rest. Each unit is associated with a size variable, which is proportional to its response value and known prior to sampling. From this population, we develop an augmented sampling strategy involving two randomization stages: a spatially balanced sample (<i>SBS</i>) and a probability proportional-to-size (PPS) sample with unequal inclusion probabilities. First, we select an <i>SBS</i> of size <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, ensuring spatial balance across the population. Subsequently, a local pivotal method (<i>LPM</i>) sample of size <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> is drawn from the remaining unsampled units, using unequal inclusion probabilities proportional to the size variable. This two-stage sampling approach combines spatial balance with targeted selection of large-size units. Our results show that the population mean estimator derived from the proposed sampling design outperforms those based on individual <i>SBS</i> or <i>LPM</i> samples. Furthermore, we provide explicit expressions for the first-order inclusion probabilities associated with a given size variable. To illustrate the method’s effectiveness, we apply the proposed design to a population of poppy fields in Kandahar province of Afghanistan. Supplementary materials accompanying this paper appear online.</p>

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Augmented Spatially Balanced and Probability Proportional-to-Size Samples

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月18日

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  • Omer Ozturk,
  • Blair L. Robertson,
  • Olena Kravchuk,
  • Jennifer Brown

摘要

In this paper, we examine a finite population distributed across a two-dimensional spatial domain, where a small subset of units exhibit substantially larger response values compared to the rest. Each unit is associated with a size variable, which is proportional to its response value and known prior to sampling. From this population, we develop an augmented sampling strategy involving two randomization stages: a spatially balanced sample (SBS) and a probability proportional-to-size (PPS) sample with unequal inclusion probabilities. First, we select an SBS of size \(n_1\) n 1 , ensuring spatial balance across the population. Subsequently, a local pivotal method (LPM) sample of size \(n_2\) n 2 is drawn from the remaining unsampled units, using unequal inclusion probabilities proportional to the size variable. This two-stage sampling approach combines spatial balance with targeted selection of large-size units. Our results show that the population mean estimator derived from the proposed sampling design outperforms those based on individual SBS or LPM samples. Furthermore, we provide explicit expressions for the first-order inclusion probabilities associated with a given size variable. To illustrate the method’s effectiveness, we apply the proposed design to a population of poppy fields in Kandahar province of Afghanistan. Supplementary materials accompanying this paper appear online.