mixt function

Mixture of t-distribution

Mixture of t-distribution

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

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 (kk). 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.

Details

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

Pr(x∣w‾,df‾,μ‾,σ‾)=∑i=1kwitdf(x∣μi,σi), Pr(x|\underline{w}, \underline{df}, \underline{\mu}, \underline{\sigma}) = \sum_{i=1}^{k} w_{i} t_{df}(x| \mu_{i}, \sigma_{i}),

where

tdf(x∣μ,σ)=Γ(df+12)Γ(df2)πdfσ(1+1df(x−μσ)2)−df+12. 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} }.

Returns

Generated data as an vector with size nn.

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, tools:::Rd_expr_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, tools:::Rd_expr_doi("10.1080/03610918.2011.588358")

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

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 )
## End(Not run)
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