  Existing high-dimensional statistical methods are largely established for
analyzing individual-level data. In this work, we study estimation and
inference for high-dimensional linear models where we only observe "proxy
data", which include the marginal statistics and sample covariance matrix that
are computed based on different sets of individuals. We develop a rate optimal
method for estimation and inference for the regression coefficient vector and
its linear functionals based on the proxy data. Moreover, we show the intrinsic
limitations in the proxy-data based inference: the minimax optimal rate for
estimation is slower than that in the conventional case where individual data
are observed; the power for testing and multiple testing does not go to one as
the signal strength goes to infinity. These interesting findings are
illustrated through simulation studies and an analysis of a dataset concerning
the genetic associations of hindlimb muscle weight in a mouse population.
