mcmc_mix2: Markov chain Monte Carlo for 2-component discrete extreme...

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Markov chain Monte Carlo for 2-component discrete extreme value mixture distribution

Description

mcmc_mix2 returns the posterior samples of the parameters, for fitting the 2-component discrete extreme value mixture distribution. The samples are obtained using Markov chain Monte Carlo (MCMC).

Usage

mcmc_mix2(
  x,
  count,
  u_set,
  u,
  alpha,
  theta,
  shape,
  sigma,
  a_psiu,
  b_psiu,
  a_alpha,
  b_alpha,
  a_theta,
  b_theta,
  m_shape,
  s_shape,
  a_sigma,
  b_sigma,
  positive,
  a_pseudo,
  b_pseudo,
  pr_power,
  iter,
  thin,
  burn,
  freq,
  invt,
  mc3_or_marg = TRUE,
  constrained = FALSE
)

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)
u_set	Positive integer vector of the values u will be sampled from
u	Positive integer, initial value of the threshold
alpha	Real number greater than 1, initial value of the parameter
theta	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_psiu, b_psiu, a_alpha, b_alpha, a_theta, b_theta, 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.
positive	Boolean, is alpha positive (TRUE) or unbounded (FALSE)? Ignored if constrained is TRUE
a_pseudo	Positive real number, first parameter of the pseudoprior beta distribution for theta in model selection; ignored if pr_power = 1.0
b_pseudo	Positive real number, second parameter of the pseudoprior beta distribution for theta in model selection; ignored if pr_power = 1.0
pr_power	Real number in [0, 1], prior probability of the discrete power law (below u). Overridden if constrained is TRUE
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)?
constrained	Boolean, are alpha & shape constrained such that 1/shape+1 > alpha > 1 with the powerlaw assumed in the body & "continuity" at the threshold u (TRUE), or is there no constraint between alpha & shape, with the former governed by positive, and no powerlaw and continuity enforced (FALSE, default)?

Details

In the MCMC, a componentwise Metropolis-Hastings algorithm is used. The threshold u is treated as a parameter and therefore sampled. The hyperparameters are used in the following priors: u is such that the implied unique exceedance probability psiu ~ Uniform(a_psi, b_psi); alpha ~ Normal(mean = a_alpha, sd = b_alpha); theta ~ Beta(a_theta, b_theta); shape ~ Normal(mean = m_shape, sd = s_shape); sigma ~ Gamma(a_sigma, scale = b_sigma). If pr_power = 1.0, the discrete power law (below u) is assumed, and the samples of theta will be all 1.0. If pr_power 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_mix3 for MCMC for the Zipf-polylog and 3-component discrete extreme value mixture distributions, respectively.

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