  We discuss the parameter estimation of the probability of default (PD), the
correlation between the obligors, and a phase transition. In our previous work,
we studied the problem using the beta-binomial distribution. A non-equilibrium
phase transition with an order parameter occurs when the temporal correlation
decays by power law. In this article, we adopt the Merton model, which uses an
asset correlation as the default correlation, and find that a phase transition
occurs when the temporal correlation decays by power law. When the power index
is less than one, the PD estimator converges slowly. Thus, it is difficult to
estimate PD with limited historical data. Conversely, when the power index is
greater than one, the convergence speed is inversely proportional to the number
of samples. We investigate the empirical default data history of several rating
agencies. The estimated power index is in the slow convergence range when we
use long history data. This suggests that PD could have a long memory and that
it is difficult to estimate parameters due to slow convergence.
