R中多元t分布的估计

Question

I would like to konw if there is any function in R that allows to estimate the df of a multivariate t distribution.

The problem is easy: I have a matrix of 5 variables (columns) with 75 observations (rows). I would like to estimate the df of a multivariate t on that sample.

Thanks,

Juan.

*** Edition: after fabians suggestions I implemented the dmvt() formula * ** *

# "residuals" is a matrix with residuals from a model. I want to estimate the df of  
# that sample assuming multivariate-t

sigma<-cor(residuals, use="pairwise.complete.obs", method="pearson")
my_means<-vector(length = 8)

for (i in 1:8){
  my_means[i]<-mean(my_matrix[,i]) 
}

residuals.scaled<-scale(residuals)
df.1 <-dmvt(residuals.scaled, my_means, sigma, log= FALSE, type = "shifted", df = 1)

I have some doubts regarding: 1) Scaling: I'm also centering the data. Don't know if this is correct. 2) Using log = FALSE as I don't know why densities should be given as log(d) in my case 3) From here I should estimate the likehood of the sample data for each df. Thus, more code lines like df.2, df.3, etc should be added and then calculate the likelihood of each. Then, choose the highest. Is that correct?

Answer 1

Package mvtnorm supplies the density of a (shifted) multivariate t-distribution in function dmvt . You could enter your (scaled) data and its sample correlation and compute the likelihood of your data for different values of df . Pick the value of df that maximizes the likelihood of your data.

EDIT:

library(mvtnorm)
set.seed(12121212)
################################################################################
## simulate n vectors of p-dim. t-distributed data in matrix X:
n <- 300
p <- 8

# draw random column means
means <- 10 * rnorm(p)

# correlation is AR(1) with correlation rho=.8
rho <- 0.8
sigma <- rho ^ abs(outer(1:p, 1:p, "-"))

# column s.d.s are sqrt(1:8)
df <- 3
X <- t(t(rmvt(n, sigma=sigma, delta=means, df=df)) * sqrt(1:8))


################################################################################
# evaluate t-likelihood for scaled X:

X_scale <- scale(X)
sigma_est <- cor(X_scale)

df_candidates <- seq(1, 20, by=2)
loglik <- numeric(length(df_candidates))
names(loglik) <- df_candidates
for(df in df_candidates){
    # no need for delta since we're working on scaled & centered data.
    # use sum(log(likelihood)), not prod(likelihood) to avoid numeric over/underflow 
    loglik[as.character(df)] <- sum(dmvt(x=X_scale, sigma=sigma_est, 
                                         df=df, log=TRUE))
}
loglik
#        1         3         5         7         9        11        13 
#-1788.219 -1756.301 -1768.885 -1783.724 -1797.386 -1809.556 -1820.382 
#       15        17        19 
#-1830.066 -1838.788 -1846.698 
## --> maximal for df=3, as used for the simulation.

## verify that mean shift can be incorporated into pre-processing as above:
dmvt(X[1,], delta=means) == dmvt(X[1,] - means)
#[1] TRUE