This paper proposes a positional poverty gap measure of multidimensional poverty within the Alkire-Foster counting framework. The measure captures the depth of deprivations even when indicators are ordinal, unlike the standard poverty gap, which requires cardinal variables. The proposed method draws on the fuzzy set literature and introduces a distribution-based measure of deprivation depth using the empirical cumulative distribution of each indicator, with the most deprived group as the benchmark. For each deprived individual, the method assigns a score based on the individual's relative position in the distribution. Depth is thus expressed as a difference in distributional positions, motivating the label positional poverty gap. The paper demonstrates that this measure preserves the identification and aggregation structure of the counting approach and satisfies its axiomatic properties when the reference distribution remains fixed over time. The framework remains flexible because it accommodates different identification rules, deprivation cutoffs, and variable types. Overall, it offers a simple, meaningful, and theoretically grounded way to incorporate depth into multidimensional poverty measurement with ordinal data.

翻译：本文在Alkire-Foster计数框架内提出了一种多维贫困的位置性贫困缺口测度方法。与需要基数变量的标准贫困缺口不同，该测度即使在指标为序数时仍能捕捉被剥夺的深度。所提出的方法借鉴模糊集理论文献，通过以最受剥夺群体为基准、利用各指标的经验累积分布，引入了一种基于分布的剥夺深度测度方法。对于每个被剥夺个体，该方法根据其在分布中的相对位置分配分数。因此，深度被表述为分布位置的差异，这构成了"位置性贫困缺口"这一命名的依据。本文证明，当参照分布随时间保持固定时，该测度保留了计数方法的识别与加总结构，并满足其公理性质。该框架具有灵活性，可容纳不同的识别规则、剥夺临界值和变量类型。总体而言，它为在序数数据条件下将深度纳入多维贫困测度提供了一种简明、有意义且理论依据充分的方法。