  We propose tensor time series imputation when the missing pattern in the
tensor data can be general, as long as any two data positions along a tensor
fibre are both observed for enough time points. The method is based on a tensor
time series factor model with Tucker decomposition of the common component. One
distinguished feature of the tensor time series factor model used is that there
can be weak factors in the factor loadings matrix for each mode. This reflects
reality better when real data can have weak factors which drive only groups of
observed variables, for instance, a sector factor in financial market driving
only stocks in a particular sector. Using the data with missing entries,
asymptotic normality is derived for rows of estimated factor loadings, while
consistent covariance matrix estimation enables us to carry out inferences. As
a first in the literature, we also propose a ratio-based estimator for the rank
of the core tensor under general missing patterns. Rates of convergence are
spelt out for the imputations from the estimated tensor factor models. We
introduce a new measure for gauging imputation performances, and simulation
results show that our imputation procedure works well, with asymptotic
normality and corresponding inferences also demonstrated. Re-imputation
performances are also gauged when we demonstrate that using slightly larger
rank then estimated gives superior re-imputation performances. An NYC taxi
traffic data set is also analyzed by imposing general missing patterns and
gauging the imputation performances.
