| Title: | Estimating the Degrees of Freedom of the Student's t-Distribution under a Bayesian Framework |
|---|---|
| Description: | A Bayesian framework to estimate the Student's t-distribution's degrees of freedom is developed. Markov Chain Monte Carlo sampling routines are developed as in <doi:10.3390/axioms11090462> to sample from the posterior distribution of the degrees of freedom. A random walk Metropolis algorithm is used for sampling when Jeffrey's and Gamma priors are endowed upon the degrees of freedom. In addition, the Metropolis-adjusted Langevin algorithm for sampling is used under the Jeffrey's prior specification. The Log-normal prior over the degrees of freedom is posed as a viable choice with comparable performance in simulations and real-data application, against other prior choices, where an Elliptical Slice Sampler is used to sample from the concerned posterior. |
| Authors: | Somjit Roy [aut, cre], Se Yoon Lee [aut, ctb] |
| Maintainer: | Somjit Roy <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 1.0.0 |
| Built: | 2025-01-10 05:47:43 UTC |
| Source: | https://github.com/roy-sr-007/bayesestdft |
BayesGA samples from the posterior distribution of the degrees of freedom (dof) with Gamma prior endowed upon the dof, using a random walk Metropolis (RMW) algorithm.
BayesGA(y, ini.nu = 1, S = 1000, delta = 0.001, a = 1, b = 0.1)BayesGA(y, ini.nu = 1, S = 1000, delta = 0.001, a = 1, b = 0.1)
y |
an N-dimensional vector of continuous observations supported on the real-line |
ini.nu |
the initial posterior sample value of the degrees of freedom (default is 1) |
S |
the number of posterior samples (default is 1000) |
delta |
the step size for the respective sampling engines (default is 0.001) |
a |
rate parameter of Gamma prior (default is 1, corresponds to an Exponential prior) |
b |
rate parameter of Gamma prior (default is 0.1) |
A vector of posterior sample estimates
res |
an S-dimensional vector with the posterior samples |
Lee, S. Y. (2022). "The Use of a Log-Normal Prior for the Student t-Distribution", Axioms, doi:10.3390/axioms11090462
Fernández, C., Steel, M. F. (1998). "On Bayesian modeling of fat tails and skewness", Journal of the American Statistical Association, doi:10.1080/01621459.1998.10474117
Juárez, M. A., Steel, M. F. (2010). "Model-Based Clustering of Non-Gaussian Panel Data Based on Skew-t Distributions", Journal of Business and Economic Statistics, doi:10.1198/jbes.2009.07145
# data from Student's t-distribution with dof = 0.1 y = rt(n = 100, df = 0.1) # running the random walk Metropolis algorithm with default settings nu = BayesGA(y) # reporting the posterior mean estimate of the dof mean(nu) # application to log-return (daily index values) of United States (S&P500) data(index_return) # log-returns of United States index_return_US <- dplyr::filter(index_return, Country == "United States") y = index_return_US$log_return_rate # running the random walk Metropolis algorithm with default settings nu = BayesGA(y) # reporting the posterior mean estimate of the dof from the log-return data of US mean(nu)# data from Student's t-distribution with dof = 0.1 y = rt(n = 100, df = 0.1) # running the random walk Metropolis algorithm with default settings nu = BayesGA(y) # reporting the posterior mean estimate of the dof mean(nu) # application to log-return (daily index values) of United States (S&P500) data(index_return) # log-returns of United States index_return_US <- dplyr::filter(index_return, Country == "United States") y = index_return_US$log_return_rate # running the random walk Metropolis algorithm with default settings nu = BayesGA(y) # reporting the posterior mean estimate of the dof from the log-return data of US mean(nu)
BayesJeffreys samples from the posterior distribution of the degrees of freedom (dof) with Jeffreys prior endowed upon the dof, using a random walk Metropolis (RMW) algorithm and Metropolis-adjusted Langevin algorithm (MALA).
BayesJeffreys( y, ini.nu = 1, S = 1000, delta = 0.001, sampling.alg = c("MH", "MALA") )BayesJeffreys( y, ini.nu = 1, S = 1000, delta = 0.001, sampling.alg = c("MH", "MALA") )
y |
an N-dimensional vector of continuous observations supported on the real-line |
ini.nu |
the initial posterior sample value of the degrees of freedom (default is 1) |
S |
the number of posterior samples (default is 1000) |
delta |
the step size for the respective sampling engines (default is 0.001) |
sampling.alg |
takes the choice of the sampling algorithm to be performed, either 'MH' or 'MALA' |
A vector of posterior sample estimates
res |
an S-dimensional vector with the posterior samples |
Lee, S. Y. (2022). "The Use of a Log-Normal Prior for the Student t-Distribution", Axioms, doi:10.3390/axioms11090462
Gustafson, P. (1998). "A guided walk Metropolis algorithm", Statistics and Computing, doi:10.1023/A:1008880707168
# data from Student's t-distribution with dof = 0.1 y = rt(n = 100, df = 0.1) # running the random walk Metropolis algorithm with default settings nu1 = BayesJeffreys(y, sampling.alg = "MH") # reporting the posterior mean estimate of the dof mean(nu1) # running MALA with default settings nu2 = BayesJeffreys(y, sampling.alg = "MALA") # reporting the posterior mean estimate of the dof mean(nu2) # application to log-return (daily index values) of United States (S&P500) data(index_return) # log-returns of United States index_return_US <- dplyr::filter(index_return, Country == "United States") y = index_return_US$log_return_rate # running the random walk Metropolis algorithm with default settings nu1 = BayesJeffreys(y, sampling.alg = "MH") # reporting the posterior mean estimate of the dof from the log-return data of US mean(nu1) # running MALA with default settings nu2 = BayesJeffreys(y, sampling.alg = "MALA") # reporting the posterior mean estimate of the dof from the log-return data of US mean(nu2)# data from Student's t-distribution with dof = 0.1 y = rt(n = 100, df = 0.1) # running the random walk Metropolis algorithm with default settings nu1 = BayesJeffreys(y, sampling.alg = "MH") # reporting the posterior mean estimate of the dof mean(nu1) # running MALA with default settings nu2 = BayesJeffreys(y, sampling.alg = "MALA") # reporting the posterior mean estimate of the dof mean(nu2) # application to log-return (daily index values) of United States (S&P500) data(index_return) # log-returns of United States index_return_US <- dplyr::filter(index_return, Country == "United States") y = index_return_US$log_return_rate # running the random walk Metropolis algorithm with default settings nu1 = BayesJeffreys(y, sampling.alg = "MH") # reporting the posterior mean estimate of the dof from the log-return data of US mean(nu1) # running MALA with default settings nu2 = BayesJeffreys(y, sampling.alg = "MALA") # reporting the posterior mean estimate of the dof from the log-return data of US mean(nu2)
BayesLNP samples from the posterior distribution of the degrees of freedom (dof) with Log-normal prior endowed upon the dof, using an Elliptical Slice Sampler (ESS).
BayesLNP(y, ini.nu = 1, S = 1000, mu = 1, sigma.sq = 1)BayesLNP(y, ini.nu = 1, S = 1000, mu = 1, sigma.sq = 1)
y |
an N-dimensional vector of continuous observations supported on the real-line |
ini.nu |
the initial posterior sample value of the degrees of freedom (default is 1) |
S |
the number of posterior samples (default is 1000) |
mu |
mean of the Log-normal prior density (default is 1) |
sigma.sq |
variance of the Log-normal prior density (default is 1) |
A vector of posterior sample estimates
res |
an S-dimensional vector with the posterior samples |
Lee, S. Y. (2022). "The Use of a Log-Normal Prior for the Student t-Distribution", Axioms, doi:10.3390/axioms11090462
Murray, I., Prescott Adams, R., MacKay, D. J. (2010). "Elliptical slice sampling", Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics
# data from Student's t-distribution with dof = 0.1 y = rt(n = 100, df = 0.1) # running the Elliptical Slice Sampler (ESS) with default settings nu = BayesLNP(y) # reporting the posterior mean estimate of the dof mean(nu) # application to log-return (daily index values) of United States (S&P500) data(index_return) # log-returns of United States index_return_US <- dplyr::filter(index_return, Country == "United States") y = index_return_US$log_return_rate # running the Elliptical Slice Sampler (ESS) with default settings nu = BayesLNP(y) # reporting the posterior mean estimate of the dof from the log-return data of US mean(nu)# data from Student's t-distribution with dof = 0.1 y = rt(n = 100, df = 0.1) # running the Elliptical Slice Sampler (ESS) with default settings nu = BayesLNP(y) # reporting the posterior mean estimate of the dof mean(nu) # application to log-return (daily index values) of United States (S&P500) data(index_return) # log-returns of United States index_return_US <- dplyr::filter(index_return, Country == "United States") y = index_return_US$log_return_rate # running the Elliptical Slice Sampler (ESS) with default settings nu = BayesLNP(y) # reporting the posterior mean estimate of the dof from the log-return data of US mean(nu)
The stock market returns are recorded for four countries viz., United States (S&P500), Japan (NIKKEI225), Germany (DAX Index), and South Korea (KOSPI). Specifically log return rates (as computed in Section 5 of doi:10.3390/axioms11090462) are recorded for 5 months in the year 2009 for all the four countries, where these rates are considered to be Student's t-distributed and used for the purpose of estimating the corresponding degrees of freedom using a Bayesian model-based framework, developed in doi:10.3390/axioms11090462.
index_returnindex_return
A data frame with 4 columns:
name of the country to which the log return rate corresponds to: 'United States', 'Japan', 'Germany', and 'South Korea'
value of the log return rate
an index for the log return rate observations
the date on which the log return rate was recorded
(Lee, 2022), doi:10.3390/axioms11090462.