gm {ctmcd}

Generator Matrix Estimation

Description

Generic function to estimate the parameters of a continuous Markov chain

Usage

gm(tm, te, method, ...)

Arguments

tm

matrix of either absolute transition frequencies (if method is "EM" or "GS") or relative transition frequencies (if method is "DA", "WA" of "QO")

te

time elapsed in transition process

method

method to derive generator matrix: "DA" - Diagonal Adjustment, "WA" - Weighted Adjustment, "QO" - Quasi-Optimization, "EM" - Expectation-Maximization Algorithm, "GS" - Gibbs Sampler

...

Additional Arguments:

• gmguess: initial guess for generator matrix estimation procedure (if method is "EM")

• prior: prior parametrization (if method is "GS")

• burnin: burn-in period (if method is "GS")

• eps: convergence criterion (if method is "EM")

• conv_pvalue,conv_freq: convergence criterion (if method is "GS")

• niter: maximum number of iterations (if method is "EM" or "GS")

• sampl_func: optional self-written path sampling function for endpoint-conditioned Markov processes (if method is "GS")

• combmat: matrix stating combined use of modified rejection sampling / uniformization sampling algorithms (if method is "GS")

• sampl_method: sampling method for deriving endpoint-conditioned Markov process path: "Unif" - Uniformization Sampling, "ModRej" - Modified Rejection Sampling (if method is "GS")

• logmethod: method to compute matrix logarithm (if method is "DA", "WA" or "QO", see ?logm from expm package for more information)

• expmethod: method to compute matrix exponential (if method is "EM" or "GS", see ?expm from expm package for more information)

• verbose: verbose mode (if method is "EM" or "GS")

Details

The methods "DA", "WA" and "QO" provide adjustments of a matrix logarithm based candidate solution, "EM" gives the maximum likelihood estimate and "GS" a posterior mean estimate in a Bayesian setting with conjugate Gamma priors.

Value

generator matrix estimate

Author(s)

Marius Pfeuffer

References

G. dos Reis, M. Pfeuffer, G. Smith: Capturing Rating Momentum in the Estimation of Probabilities of Default, With Application to Credit Rating Migrations (In Preparation), 2018

M. Pfeuffer: Generator Matrix Approximation Based on Discrete Time Rating Migration Data. Master Thesis, University of Munich, 2016

Y. Inamura: Estimating Continuous Time Transition Matrices from Discretely Observed Data. Bank of Japan Working Paper Series, 2006

R. B. Israel et al.: Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings. Mathematical Finance 11(2):245-265, 2001

E. Kreinin and M. Sidelnikova: Regularization Algorithms for Transition Matrices. Algo Research Quarterly 4(1):23-40, 2001

M. Bladt and M. Soerensen: Statistical Inference for Discretely Observed Markov Jump Processes. Journal of the Royal Statistical Society B 67(3):395-410, 2005

See Also

gmDA, gmWA, gmQO, gmEM, gmGS

Examples

data(tm_abs)

## Maximum Likelihood Generator Matrix Estimate
gm0=matrix(1,8,8)
diag(gm0)=0
diag(gm0)=-rowSums(gm0)
gm0[8,]=0

gmem=gm(tm_abs,te=1,method="EM",gmguess=gm0)
gmem

## Quasi Optimization Estimate
tm_rel=rbind((tm_abs/rowSums(tm_abs))[1:7,],c(rep(0,7),1))

gmqo=gm(tm_rel,te=1,method="QO")
gmqo