Estimate Parameters of a Multivariate t Distribution Using the ECME Algorithm

Description

Use the Expectation/Conditional Maximization Either (ECME) algorithm to obtain estimate of parameters of a multivariate t distribution.

Usage

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  mvt.ecme(X, lower.v, upper.v, err=1e-4)

Arguments

X

a matrix of observations with one subject per row.

lower.v

lower bound of degrees of freedom (df).

upper.v

upper bound of df.

err

the iteration stops when consecutive difference in percentage of df reaches this bound. The default value is 1e-4.

Details

They are number of forms of the generalization of the univariate student-t distribution to multivariate cases. This function adopts the widely used representation as a scale mixture of normal distributions.

To obtain the estimate, the algorithm adopted is the Expectation/Conditional Maximization Either (ECME), which extends the Expectation/Conditional Maximization (ECM) algorithm by allowing CM-steps to maximize either the constrained expected complete-data log-likelihood, as with ECM, or the correspondingly constrained actual log-likelihood function.

Value

Mu

estimate of location.

Sigma

estimate of scale matrix.

v

estimate of df.

References

Chuanhai Liu (1994) Statistical Analysis Using the Multivariate t Distribution Ph. D. Dissertation, Harvard University

Examples

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  mu1 <- mu2 <- sigma12 <- sigma22 <- 100
  rho12 <- 0.7
  Sigma <- matrix(c(sigma12, rho12*sqrt(sigma12*sigma22),
                    rho12*sqrt(sigma12*sigma22), sigma22),
                  nrow=2)
  k <- 5
  N <- 100
  require(mvtnorm)
  X <- rmvt(N, sigma=Sigma, df=k, delta=c(mu1, mu2))

  result <- mvt.ecme(X, 3, 300)
  result$Mu
  result$Sigma
  result$v