In the geosciences, a recurring problem is one of estimating spatial means of a physical field using weighted averages of point observations. An important variant is when individual observations are counted with some probability less than one. This can occur in different contexts: from missing data to estimating the statistics across subsamples. In such situations, the spatial mean is a ratio of random variables, whose statistics involve approximate estimators derived through series expansion. The present paper considers truncated estimators of variance of the spatial mean and their general structure in the presence of missing data. To all orders, the variance estimator depends only on the first and second moments of the underlying field, and convergence requires these moments to be finite. Furthermore, convergence occurs if either the probability of counting individual observations is larger than 1/2 or the number of point observations is large. In case the point observations are weighted uniformly, the estimators are easily found using combinatorics and involve Stirling numbers of the second kind.

翻译：在地球科学中，一个常见问题是通过点观测值的加权平均来估算物理场的空间均值。一个重要变体是当单个观测值以小于1的某种概率被计数时。这可能出现于不同情境下：从缺失数据到跨子样本的统计量估计。在此类情况下，空间均值成为随机变量的比值，其统计量涉及通过级数展开导出的近似估计量。本文研究了缺失数据存在时空间均值方差的截断估计量及其一般结构。对所有阶次而言，方差估计量仅依赖于基础场的一阶和二阶矩，且收敛要求这些矩为有限值。此外，当单个观测值的计数概率大于1/2，或点观测值数量较大时，收敛得以实现。若点观测值采用均匀加权，则可通过组合数学方法简便求得估计量，并涉及第二类斯特林数。