Missing not at random and the nonparametric estimation of the spectral density

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. Letnandmbe 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 toln(n)n-1, while a differentiable likelihood may be estimated only with the MISE proportional tom(-2/3). On first glance, these familiar facts yield that the proposed approach is impractical becausemmust be in order larger thannto match the oracle. Surprisingly, the article presents the theory and a numerical study indicating thatmmay be in order smaller thannand 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.