bmixture (version 1.7)

mixt: Mixture of t-distribution

Description

Random generation and density function for a finite mixture of univariate t-distribution.

Usage

rmixt( n = 10, weight = 1, df = 1, mean = 0, sd = 1 ) 
dmixt( x, weight = 1, df = 1, mean = 0, sd = 1 )

Arguments

n

number of observations.

x

vector of quantiles.

weight

vector of probability weights, with length equal to number of components (\(k\)). This is assumed to sum to 1; if not, it is normalized.

df

vector of degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.

mean

vector of means.

sd

vector of standard deviations.

Value

Generated data as an vector with size \(n\).

Details

Sampling from finite mixture of t-distribution, with density:

$$Pr(x|\underline{w}, \underline{df}, \underline{\mu}, \underline{\sigma}) = \sum_{i=1}^{k} w_{i} t_{df}(x| \mu_{i}, \sigma_{i}),$$ where $$ t_{df}(x| \mu, \sigma) = \frac{ \Gamma( \frac{df+1}{2} ) }{ \Gamma( \frac{df}{2} ) \sqrt{\pi df} \sigma } \left( 1 + \frac{1}{df} \left( \frac{x-\mu}{\sigma} \right) ^2 \right) ^{- \frac{df+1}{2} }.$$

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, 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, 10.1080/03610918.2011.588358

See Also

rt, rmixnorm, rmixgamma

Examples

# NOT RUN {
n      = 10000   
weight = c( 0.3, 0.5, 0.2 )
df     = c( 4  , 4  , 4   )
mean   = c( 0  , 10 , 3   )
sd     = c( 1  , 1  , 1   )
    
data = rmixt( n = n, weight = weight, df = df, mean = mean, sd = sd )
  
hist( data, prob = TRUE, nclass = 30, col = "gray" )
  
x           = seq( -20, 20, 0.05 )
densmixt = dmixt( x, weight, df, mean, sd )
      
lines( x, densmixt, lwd = 2 )
# }