bmixgamma {bmixture} | R Documentation |

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

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 )

`data ` |
vector of data with size |

`k ` |
number of components of mixture distribution. It can take an integer values. |

`iter ` |
number of iteration for the sampling algorithm. |

`burnin ` |
number of burn-in iteration for the sampling algorithm. |

`lambda ` |
For the case |

`mu ` |
parameter of alpha in mixture distribution. |

`nu ` |
parameter of alpha in mixture distribution. |

`kesi ` |
parameter of beta in mixture distribution. |

`tau ` |
parameter of beta in mixture distribution. |

`k.start ` |
For the case |

`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 |

`trace ` |
Logical: if TRUE (default), tracing information is printed. |

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 Gamma(η, τ),*

for more details see Mohammadi et al. (2013), doi: 10.1007/s00180-012-0323-3.

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 |

`alpha_sample` |
a vector which includes the MCMC samples after burn-in from parameter |

`beta_sample ` |
a vector which includes the MCMC samples after burn-in from parameter |

`data ` |
original data. |

Reza Mohammadi a.mohammadi@uva.nl

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, doi: 10.1007/s00180-012-0323-3

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, doi: 10.1080/03610918.2011.588358

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, doi: 10.1214/aos/1016120364

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, doi: 10.1111/1467-9868.00095

Green, P. J. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. *Biometrika*, 82(4):711-732, doi: 10.1093/biomet/82.4.711

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, doi: 10.1214/17-BA1073

## Not run: set.seed( 70 ) # 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)

[Package *bmixture* version 1.7 Index]