Fine stratification survey is useful in many applications as its point estimator is unbiased, but the variance estimator under the design cannot be easily obtained, particularly when the sample size per stratum is as small as one unit. One common practice to overcome this difficulty is to collapse strata in pairs to create pseudo-strata and then estimate the variance. The estimator of variance achieved is not design-unbiased, and the positive bias increases as the population means of the paired pseudo-strata become more variant. The resulting confidence intervals can be unnecessarily large. In this paper, we propose a new Bayesian estimator for variance which does not rely on collapsing strata, unlike the previous methods given in the literature. We employ the penalized spline method for smoothing the mean and variance together in a nonparametric way. Furthermore, we make comparisons with the earlier work of Breidt et al. (2016). Throughout multiple simulation studies and an illustration using data from the National Survey of Family Growth (NSFG), we demonstrate the favorable performance of our methodology.

翻译：精细分层调查因其点估计量的无偏性而在众多应用中具有重要价值，但该设计下的方差估计量难以直接获得，尤其是当每层样本量小至一个单元时。为克服此困难，一种常见做法是将相邻层合并成对以构建伪层，进而估计方差。由此得到的方差估计量并非设计无偏，且其正偏差会随着配对伪层的总体均值差异增大而增加，最终导致置信区间可能不必要地扩大。本文提出一种新的方差贝叶斯估计方法，与文献中现有方法不同，该方法无需合并分层。我们采用惩罚样条方法，以非参数形式同时对均值与方差进行平滑处理。此外，本文还与Breidt等人（2016）的早期研究进行了比较。通过多项模拟研究以及基于全国家庭成长调查（NSFG）数据的实例分析，我们验证了所提方法的优越性能。