mcmc_mix3: Markov chain Monte Carlo for 3-component discrete extreme...

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Markov chain Monte Carlo for 3-component discrete extreme value mixture distribution

Description

mcmc_mix3 returns the posterior samples of the parameters, for fitting the 3-component discrete extreme value mixture distribution. The samples are obtained using Markov chain Monte Carlo (MCMC).

Usage

mcmc_mix3(
  x,
  count,
  v_set,
  u_set,
  v,
  u,
  alpha1,
  theta1,
  alpha2,
  theta2,
  shape,
  sigma,
  a_psi1,
  a_psi2,
  a_psiu,
  b_psiu,
  a_alpha1,
  b_alpha1,
  a_theta1,
  b_theta1,
  a_alpha2,
  b_alpha2,
  a_theta2,
  b_theta2,
  m_shape,
  s_shape,
  a_sigma,
  b_sigma,
  powerlaw1,
  positive1,
  positive2,
  a_pseudo,
  b_pseudo,
  pr_power2,
  iter,
  thin,
  burn,
  freq,
  invt,
  mc3_or_marg = TRUE
)

Arguments

x	Vector of the unique values (positive integers) of the data
count	Vector of the same length as x that contains the counts of each unique value in the full data, which is essentially rep(x, count)
v_set	Positive integer vector of the values v will be sampled from
u_set	Positive integer vector of the values u will be sampled from
v	Positive integer, initial value of the lower threshold
u	Positive integer, initial value of the upper threshold
alpha1	Real number greater than 1, initial value of the parameter
theta1	Real number in (0, 1], initial value of the parameter
alpha2	Real number greater than 1, initial value of the parameter
theta2	Real number in (0, 1], initial value of the parameter
shape	Real number, initial value of the parameter
sigma	Positive real number, initial value of the parameter
a_psi1, a_psi2, a_psiu, b_psiu, a_alpha1, b_alpha1, a_theta1, b_theta1, a_alpha2, b_alpha2, a_theta2, b_theta2, m_shape, s_shape, a_sigma, b_sigma	Scalars, real numbers representing the hyperparameters of the prior distributions for the respective parameters. See details for the specification of the priors.
powerlaw1	Boolean, is the discrete power law assumed for below v?
positive1	Boolean, is alpha1 positive (TRUE) or unbounded (FALSE)?
positive2	Boolean, is alpha2 positive (TRUE) or unbounded (FALSE)?
a_pseudo	Positive real number, first parameter of the pseudoprior beta distribution for theta2 in model selection; ignored if pr_power2 = 1.0
b_pseudo	Positive real number, second parameter of the pseudoprior beta distribution for theta2 in model selection; ignored if pr_power2 = 1.0
pr_power2	Real number in [0, 1], prior probability of the discrete power law (between v and u)
iter	Positive integer representing the length of the MCMC output
thin	Positive integer representing the thinning in the MCMC
burn	Non-negative integer representing the burn-in of the MCMC
freq	Positive integer representing the frequency of the sampled values being printed
invt	Vector of the inverse temperatures for Metropolis-coupled MCMC
mc3_or_marg	Boolean, is invt for parallel tempering / Metropolis-coupled MCMC (TRUE, default) or marginal likelihood via power posterior (FALSE)?

Details

In the MCMC, a componentwise Metropolis-Hastings algorithm is used. The thresholds v and u are treated as parameters and therefore sampled. The hyperparameters are used in the following priors: psi1 / (1.0 - psiu) ~ Beta(a_psi1, a_psi2); u is such that the implied unique exceedance probability psiu ~ Uniform(a_psi, b_psi); alpha1 ~ Normal(mean = a_alpha1, sd = b_alpha1); theta1 ~ Beta(a_theta1, b_theta1); alpha2 ~ Normal(mean = a_alpha2, sd = b_alpha2); theta2 ~ Beta(a_theta2, b_theta2); shape ~ Normal(mean = m_shape, sd = s_shape); sigma ~ Gamma(a_sigma, scale = b_sigma). If pr_power2 = 1.0, the discrete power law (between v and u) is assumed, and the samples of theta2 will be all 1.0. If pr_power2 is in (0.0, 1.0), model selection between the polylog distribution and the discrete power law will be performed within the MCMC.

Value

A list: $pars is a data frame of iter rows of the MCMC samples, $fitted is a data frame of length(x) rows with the fitted values, amongst other quantities related to the MCMC

See Also

mcmc_pol and mcmc_mix2 for MCMC for the Zipf-polylog and 2-component discrete extreme value mixture distributions, respectively.

crandep documentation built on Sept. 11, 2024, 8:01 p.m.