Statistically rigorous inferences in the form of confidence intervals for map-based estimates require model-based inferential methods. Model-based mean square errors (MSE) incorporate estimates of both residual variability and sampling variability, of which the latter includes population unit variance estimates and pairwise population unit covariance estimates. Bootstrapping, which can be used with any prediction technique, provides a means of estimating the required variances and covariances.

The objectives of the study were to to demonstrate a method for estimating the sampling variability, ${\widehat{\mathrm{Var}}}_{\mathrm{sam}}\left(\widehat{\mathrm{\mu }}\right)$, that can be used with all prediction techniques, to develop an efficient method that map makers can use to disseminate metadata that facilitates calculation of ${\widehat{\mathrm{Var}}}_{\mathrm{sam}}\left(\widehat{\mathrm{\mu }}\right)$ for arbitrary subregions of maps, and to estimate the individual contributions of sampling variability and residual variability to the overall standard error of the prediction of the population mean.

The primary results were fourfold: (i) map makers must provide metadata that facilitate estimation of population unit variances and covariances for arbitrary map subregions, (ii) bootstrapping was demonstrated as an effective means of estimating the variances and covariances, (iii) the very large matrix of pairwise population unit covariances can be aggregated into a much smaller matrix that can be readily communicated by map makers to map users, and (iv) MSEs that include only estimates of residual variability and/or estimates of population unit variances, but not estimates of the pairwise population unit covariances, grossly under-estimate the actual MSEs.

基于地图的估计的置信区间形式的统计严谨推论需要基于模型的推论方法。基于模型的均方误差 (MSE) 包含残差变异性和抽样变异性的估计，后者包括总体单位方差估计和成对总体单位协方差估计。Bootstrapping 可以与任何预测技术一起使用，它提供了一种估计所需方差和协方差的方法。

该研究的目的是展示一种估计抽样变异性的方法，${\widehat{\mathrm{变量}}}_{\mathrm{山姆}}\left(\widehat{\mathrm{\mu }}\right)$，可以与所有预测技术一起使用，以开发一种有效的方法，地图制作者可以使用它来传播元数据，从而促进计算${\widehat{\mathrm{变量}}}_{\mathrm{山姆}}\left(\widehat{\mathrm{\mu }}\right)$对于地图的任意子区域，并估计抽样变异性和残差变异性对总体平均值预测的总体标准误差的个体贡献。

主要结果有四个：（i）地图制作者必须提供有助于估计任意地图子区域的人口单位方差和协方差的元数据，（ii）自举被证明是估计方差和协方差的有效手段，（iii）非常成对人口单位协方差的大矩阵可以聚合成一个更小的矩阵，地图制作者可以很容易地与地图用户沟通，以及 (iv) 仅包括残差变异性估计和/或人口单位方差估计的 MSE，但不包括成对人口单位协方差的估计，严重低估了实际的 MSE。