Comparison between weighted adjustment, diagonal adjustment, quasi optimization and Gibbs sampler methods in deriving the generator matrix of an $8\times 8$ credit transition matrix

Abstract

A transition matrix P is said to be embeddable if it has a generator matrix Q such that P =exp (Q) If the approximated transition matrix P is embeddable, then the estimator Q can be got for the generator matrix Q. What of cases when P is not embeddable? This paper will show how to evaluate P and find estimatorˆin cases where Pˆ is embeddable (Kingman, 1962). This problem can be solved by using four methods namely.1. Diagonal and Weighted adjustments Method 2. The Generator Quasi-Optimization method 3. The EM logarithm Method 4. The Gibbs sampler (Markov Chain Monte Carlo Method) Later after sampling each of the above methods, an L norm will be performed on the results so as to deduce which method is the best for estimating the Generator Matrix for an Credit transition Matrix.