Auxiliary information \({\varvec{x}}\) is commonly used in survey sampling at the estimation stage. We propose an estimator of the finite population distribution function \(F_{y}(t)\) when \({\varvec{x}}\) is available for all units in the population and related to the study variable y by a superpopulation model. The new estimator integrates ideas from model calibration and penalized calibration. Calibration estimates of \(F_{y}(t)\) with the weights satisfying benchmark constraints on the fitted values distribution function \(\hat{F}_{\hat{y}}=F_{\hat{y}}\) on a set of fixed values of t can be found in the literature. Alternatively, our proposal \(\hat{F}_{y\omega }\) seeks an estimator taking into account a global distance \(D(\hat{F}_{\hat{y}\omega },F_{\hat{y}})\) between \(\hat{F}_{\hat{y}\omega }\) and \({F}_{\hat{y}},\) and a penalty parameter \(\alpha \) that assesses the importance of this term in the objective function. The weights are explicitly obtained for the \(L^2\) distance and conditions are given so that \(\hat{F}_{y\omega }\) to be a distribution function. In this case \(\hat{F}_{y\omega }\) can also be used to estimate the population quantiles. Moreover, results on the asymptotic unbiasedness and the asymptotic variance of \(\hat{F}_{y\omega }\), for a fixed \(\alpha \), are obtained. The results of a simulation study, designed to compare the proposed estimator to other existing ones, reveal that its performance is quite competitive.

中文翻译：

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使用全局惩罚校准方法估算有限人口分布函数

辅助信息\（{\ varvec {x}} \）通常在估计阶段的调查抽样中使用。我们提出了有限的人口分布函数的估计\（F_ {Y}（T）\）时\（{\ varvec {X}} \）可用于人口各单位及相关的研究变量ÿ由人口模型。新的估算器整合了模型校准和惩罚性校准的思想。权重满足拟合值分布函数\（\ hat {F} _ {\ hat {y}} = F _ {\ hat {y}}的权重\（F_ {y}（t）\）的校准估计\）关于t的一组固定值可以在文献中找到。另外，我们的建议\（\ hat {F} _ {y \ omega} \）在考虑全局距离的情况下寻找一个估计量\（D（\ hat {F} _ {\ hat {y} \ omega}，F _ {\ hat {y }}）\）在\（\ hat {F} _ {\ hat {y} \ omega} \）和\（{F} _ {\ hat {y}}，\）和惩罚参数\（\ alpha \）评估此术语在目标函数中的重要性。明确地获得了\（L ^ 2 \）距离的权重，并给出了条件，以使\（\ hat {F} _ {y \ omega} \）成为分布函数。在这种情况下，\（\ hat {F} _ {y \ omega} \）也可以用于估计总体分位数。此外，关于\（\ hat {F} _ {y \ omega} \）的渐近无偏性和渐近​​方差的结果对于固定的\（\ alpha \），获得。仿真研究的结果旨在将建议的估算器与其他现有估算器进行比较，结果表明其性能颇具竞争力。