  We propose a new method for changepoint estimation in partially-observed,
high-dimensional time series that undergo a simultaneous change in mean in a
sparse subset of coordinates. Our first methodological contribution is to
introduce a 'MissCUSUM' transformation (a generalisation of the popular
Cumulative Sum statistics), that captures the interaction between the signal
strength and the level of missingness in each coordinate. In order to borrow
strength across the coordinates, we propose to project these MissCUSUM
statistics along a direction found as the solution to a penalised optimisation
problem tailored to the specific sparsity structure. The changepoint can then
be estimated as the location of the peak of the absolute value of the projected
univariate series. In a model that allows different missingness probabilities
in different component series, we identify that the key interaction between the
missingness and the signal is a weighted sum of squares of the signal change in
each coordinate, with weights given by the observation probabilities. More
specifically, we prove that the angle between the estimated and oracle
projection directions, as well as the changepoint location error, are
controlled with high probability by the sum of two terms, both involving this
weighted sum of squares, and representing the error incurred due to noise and
the error due to missingness respectively. A lower bound confirms that our
changepoint estimator, which we call 'MissInspect', is optimal up to a
logarithmic factor. The striking effectiveness of the MissInspect methodology
is further demonstrated both on simulated data, and on an oceanographic data
set covering the Neogene period.
