The aim of the article is twofold: (i) present a pivotal setting where using an extra experiment for restoring information lost due to missing not at random (MNAR) is practically feasible; (ii) attract attention to a wide spectrum of new research topics created by the proposed methodology of exploring the missing mechanism. It is well known that if the likelihood of missing an observation depends on its value, then the missing is MNAR, no consistent estimation is possible, and the only way to recover destroyed information is to study the likelihood of missing via an extra experiment. One of the main practical issues with an extra‐sample approach is as follows. Let n and m be the numbers of observations in a MNAR time series and in an extra sample exploring the likelihood of missing respectively. An oracle, that knows the likelihood of missing, can estimate the spectral density of an ARMA‐type spectral density with the MISE proportional to $\ln (n)n^{-1}$, while a differentiable likelihood may be estimated only with the MISE proportional to m^−2/3. On first glance, these familiar facts yield that the proposed approach is impractical because m must be in order larger than n to match the oracle. Surprisingly, the article presents the theory and a numerical study indicating that m may be in order smaller than n and still the statistician can match performance of the oracle. The proposed methodology is used for the analysis of MNAR time series of systolic blood pressure of a person with immunoglobulin D multiple myeloma. A number of possible extensions and future research topics are outlined.

中文翻译：

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随机丢失和频谱密度的非参数估计

本文的目的是双重的：（i）提供一个关键的设置，在该设置中，使用额外的实验来恢复由于随机丢失（MNAR）而丢失的信息实际上是可行的；（ii）引起人们对由探索缺失机制的方法论提出的新研究主题的关注。众所周知，如果遗漏某个观测值的可能性取决于其值，那么该遗漏就是MNAR，则不可能进行一致的估计，并且恢复被破坏信息的唯一方法是通过额外的实验来研究遗漏的可能性。额外样本方法的主要实际问题之一如下。令n和m分别是MNAR时间序列和其他样本中分别观察缺失可能性的观察次数。知道丢失可能性的预言家可以估计ARMA型光谱密度的光谱密度，而MISE与$\ln （ñ）{ñ}^{-\mathrm{1个}}$，而仅在MISE与m ^-2/3成正比的情况下，才可以估计出可能的可能性。乍一看，这些熟悉的事实导致所提出的方法不切实际，因为m必须大于n才能与oracle匹配。出人意料的是，本文介绍了该理论和数值研究，表明m的顺序可能小于n，而统计学家仍然可以与Oracle的性能相匹配。所提出的方法用于分析免疫球蛋白D多发性骨髓瘤患者的MNAR收缩压的时间序列。概述了许多可能的扩展和将来的研究主题。