Extrapolated Spectral Density Estimation Of Missing Not At Random Time Series

Missing data exist in all fields of study,and resea.rchers want the data.to be ascomplete and accurate as possible,so the study of how to deal with missing data has become an important topic.Missing time series is an important branch of missing data,which is widely used in the fields of medicine,finance,and sociology.However,there are dependencies and correlations among time series data,which leads to higher complexity compared with linear models and other general data models.Therefore,the study of missing time series is practical,but difficult.According to the missing mechanism,the missing data can be classified into three categories,namely,missing completely at random(MCAR),missing at random(MAR)and missing not at random(MNAR).However,in some cases,it is reasonable and practical to assume that there is an association between the missing mechanism and the real data.For example,in the ca.se of height and weight records,people outside the normal range of data may prefer not to disclose specific data;in the case of reports,people may also prefer to exclude poor data,which can lead to missing not at random.However,there is a paucity of research literature on missing not at random(MNAR)time series.Sam Eformovich proposed a spectral density estimation method for the case where the random variables at any time in the time series have a range of values within a deterministic interval.However,most of the common time series are normal series in which the range of values of each random variable is not limited to a finite interval,but to the whole real number R.Therefore,the results have some limitations.To address the above limitations,we use tail probabilities to generalize Sam Eformovich’s results under reasonable assumptions and relax the range of values of the random variables in the time series to real number field R.First,we obtain estimates of the missing mechanism based on an set of independent exploratory experimental data using nonparametric Bernoulli regression.First,we obtain an estimate of the missing mechanism based on an set of independent exploratory experimental data using nonparametric Bernoulli regression,and simply treat the estimate to prevent any subsequent inestimation of the spectral density.Subsequently,the spectral density is estimated using the Fourier series.Under the assumptions of this paper,we theoretically prove that the mean integrated squared error(MISE)of the spectral density estimation maintains the convergence property under the condition that the exploratory experimental data volume m and the actual time series data volume n satisfy m ≤Cn/ln(n).Finally,by changing the time series model and the missing mechanism,two sets of data simulations are made and the applicability of the method in different,contexts is verified in terms of images and mean integrated squared error(MISE).In addition,this paper also conducts an empirical analysis with the real data of the aggregate financing to the real economy(AFRE)to further verify the validity and practical value of this method.