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This paper studies the maximum likelihood estimation of nonlinear state space models. Particle Markov chain Monte Carlo method is introduced to implement the Monte Carlo expectation maximization algorithm for more accurate and robust estimation. Under this framework, an automated sample size selection criterion is constructed via renewal theory. This criterion would increase the sample size when the relative likelihood indicates that the parameters are close to each other. The proposed methodology is applied to the stochastic volatility model and another nonlinear state space model for illustration, where the results show better estimation performance.

Credit migrations constitute the building blocks of modern risk management. A firm-specific structural model of credit migration that incorporates the firm's capital structure and the risk perception of rating agencies is proposed. The proposed model employs the notion of distance-to-default, which quantifies default probability. The properties of Brownian excursions play an essential role in the analysis. The proposed model not only allows the derivation of closed-form credit transition probability, but also provides plausible explanations for certain empirical evidence, such as the default probability overlaps in ratings and the slow-to-respond feature of rating agencies. The proposed model is calibrated through simulations and applied to empirical data, which show rating agencies' risk perceptions to be significant. The calibrated model allows calculation of the firm-specific transition probabilities of rated companies.

Bandwidth selection has been an important topic in nonparametric density estimation. In this paper an effective method for local bandwidth selection is proposed. For local bandwidth selection, due to data sparsity and other reasons, extremely small bandwidths are sometimes selected, which lead to severe undersmoothing. To circumvent this difficulty, the main idea behind the proposed method is to choose the largest bandwidth that still achieves the optimal rate. When coupled with practical bias reduction techniques, the bandwidth selected from this method can be applied simultaneously to conduct both local point and interval estimation. Simulation studies demonstrate the effectiveness of the proposed approach, which compares favorably with other existing approaches.