  In statistical problems, a set of parameterized probability distributions is
used to estimate the true probability distribution. If Fisher information
matrix at the true distribution is singular, then it has been left unknown what
we can estimate about the true distribution from random samples. In this paper,
we study a singular regression problem and prove a limit theorem which shows
the relation between the singular regression problem and two birational
invariants, a real log canonical threshold and a singular fluctuation. The
obtained theorem has an important application to statistics, because it enables
us to estimate the generalization error from the training error without any
knowledge of the true probability distribution.
