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：地球科学中，一个经常出现的问题是使用点观测的加权平均来估计物理场的空间平均值。一个重要的变体是当对个别观测进行计数时，每个观测的计数概率都小于1。这可以发生在不同的情况下：从缺失数据到估计子样本间的统计数据。在这种情况下，空间平均值是随机变量的比率，其统计量涉及通过级数展开导出的近似估计量。本文考虑了在存在缺失数据时空间均值方差的截断估计量及其一般结构。到所有阶段，方差估计量仅取决于基础场的一阶和二阶矩，收敛需要这些矩是有限的。此外，如果个别观测的计数概率大于1/2或点观测数量很大，将会发生收敛。在点观测加权均匀的情况下，估计量易于使用组合数学找到，并涉及第二类斯特林数。