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 a beta-binomial distribution. A non-equilibrium phase transition with an order parameter occurs when the temporal correlation decays by a power law. In this study, 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 the power law. When the power index is less than one, the PD estimator converges slowly. Thus, it is difficult to estimate the 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 the PD can have a long memory and that it is difficult to estimate parameters due to slow convergence.

我们讨论了违约概率（PD），债务人之间的相关性以及相变的参数估计。在我们以前的工作中，我们使用β-二项分布来研究该问题。当时间相关性因幂律衰减时，会发生具有阶数参数的非平衡相变。在这项研究中，我们采用默顿模型，该模型使用资产相关性作为默认相关性，并发现当时间相关性因幂定律衰减时发生相变。当功率指数小于1时，PD估计器收敛缓慢。因此，难以利用有限的历史数据来估计PD。相反，当功率指数大于1时，收敛速度与样本数成反比。我们调查了几个评级机构的经验默认数据历史记录。当我们使用长历史数据时，估计的功率指数在慢收敛范围内。这表明PD可能具有较长的存储空间，并且由于收敛速度较慢而难以估计参数。