# bmixgamma: Sampling algorithm for mixture of distributions In bmixture: Bayesian Estimation for Finite Mixture of Distributions

## Description

This function consists of several sampling algorithms for Bayesian estimation for finite mixture of Gamma distributions.

## Usage

 1 2 3 4 bmixgamma( data, k = "unknown", iter = 1000, burnin = iter / 2, lambda = 1, mu = NULL, nu = NULL, kesi = NULL, tau = NULL, k.start = NULL, alpha.start = NULL, beta.start = NULL, pi.start = NULL, k_max = 30, trace = TRUE ) 

## Arguments

 data  The vector of data with size n. k  The number of components of mixture distribution. Defult is "unknown". It can take an integer values. iter  The number of iteration for the sampling algorithm. burnin  The number of burn-in iteration for the sampling algorithm. lambda  For the case k = "unknown", it is the parameter of the prior distribution of number of components k. mu  The parameter of alpha in mixture distribution. nu  The parameter of alpha in mixture distribution. kesi  The parameter of beta in mixture distribution. tau  The parameter of beta in mixture distribution. k.start  For the case k = "unknown", initial value for number of components of mixture distribution. alpha.start Initial value for parameter of mixture distribution. beta.start  Initial value for parameter of mixture distribution. pi.start  Initial value for parameter of mixture distribution. k_max  For the case k = "unknown", maximum value for the number of components of mixture distribution. trace  Logical: if TRUE (default), tracing information is printed.

## Details

Sampling from finite mixture of Gamma distribution, with density:

Pr(x|k, \underline{π}, \underline{α}, \underline{β}) = ∑_{i=1}^{k} π_{i} Gamma(x|α_{i}, β_{i}),

where k is the number of components of mixture distribution (as a defult we assume is unknown) and

Gamma(x|α_{i}, β_{i})=\frac{(β_{i})^{α_{i}}}{Γ(α_{i})} x^{α_{i}-1} e^{-β_{i}x}.

The prior distributions are defined as below

P(K=k) \propto \frac{λ^k}{k!}, \ \ \ k=1,...,k_{max},

π_{i} | k \sim Dirichlet( 1,..., 1 ),

α_{i} | k \sim Gamma(ν, υ),

β_i | k \sim G(η, τ),

for more details see Mohammadi et al. (2013).

## Value

An object with S3 class "bmixgamma" is returned:

 all_k  A vector which includes the waiting times for all iterations. It is needed for monitoring the convergence of the BD-MCMC algorithm. all_weights  A vector which includes the waiting times for all iterations. It is needed for monitoring the convergence of the BD-MCMC algorithm. pi_sample  A vector which includes the MCMC samples after burn-in from parameter pi of mixture distribution. alpha_sample A vector which includes the MCMC samples after burn-in from parameter alpha of mixture distribution. beta_sample  A vector which includes the MCMC samples after burn-in from parameter beta of mixture distribution. data  original data.

## Author(s)

Reza Mohammadi [email protected]

## References

Mohammadi, A., Salehi-Rad, M. R., and Wit, E. C. (2013) Using mixture of Gamma distributions for Bayesian analysis in an M/G/1 queue with optional second service. Computational Statistics, 28(2):683-700

Mohammadi, A. and Salehi-Rad, M. R. (2012) Bayesian inference and prediction in an M/G/1 with optional second service. Communications in Statistics-Simulation and Computation, 41(3):419-435

Stephens, M. (2000) Bayesian analysis of mixture models with an unknown number of components-an alternative to reversible jump methods. Annals of statistics, 28(1):40-74

Richardson, S. and Green, P. J. (1997) On Bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society: series B, 59(4):731-792

Green, P. J. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4):711-732

Cappe, O., Christian P. R., and Tobias, R. (2003) Reversible jump, birth and death and more general continuous time Markov chain Monte Carlo samplers. Journal of the Royal Statistical Society: Series B, 65(3):679-700

Wade, S. and Ghahramani, Z. (2018) Bayesian Cluster Analysis: Point Estimation and Credible Balls (with Discussion). Bayesian Analysis, 13(2):559-626

bmixnorm, bmixt, bmixgamma
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ## Not run: # simulating data from mixture of gamma with two components n = 1000 # number of observations weight = c( 0.6, 0.4 ) alpha = c( 12 , 1 ) beta = c( 3 , 2 ) data <- rmixgamma( n = n, weight = weight, alpha = alpha, beta = beta ) # plot for simulation data hist( data, prob = TRUE, nclass = 50, col = "gray" ) x = seq( 0, 10, 0.05 ) truth = dmixgamma( x, weight, alpha, beta ) lines( x, truth, lwd = 2 ) # Runing bdmcmc algorithm for the above simulation data set bmixgamma.obj <- bmixgamma( data, iter = 1000 ) summary( bmixgamma.obj ) plot( bmixgamma.obj ) ## End(Not run)