Simulation-based Quantiles of the Multivariate-t Distribution

DESCRIPTION:

Approximates a quantile of a pivotal quantity for use in multiple comparisons in nonstandard cases.

USAGE:

qmvt.sim(vmat, lmat, error.df=Inf, simsize=12999, alpha=.05, 
         two.sided=T) 

REQUIRED ARGUMENTS:

vmat
the variance-covariance matrix of an underlying vector of normal variables.
lmat
a matrix whose columns specify linear combinations of the underlying normal vector.
error.df
an integer, the degrees of freedom associated with vmat (if any). Inf is accepted.
simsize
the desired simulation size. The default 12999 is a standard choice that will provide an actual alpha within .005 of a nominal .05 in 99% of simulations. For other choices, see Edwards and Berry (1987). To assure stability of the critical point, simsize should be chosen to be on the order of at least the default size.
alpha
the function will generate an approximate 1-alpha quantile for the distribution of the maximum of multivariate-t random variables.
two.sided
a logical value, if TRUE, the default, the quantile is for the distribution of the maximum absolute value of the multivariate-t random variables. Otherwise, the absolute value is not taken.

VALUE:

the simulation-based ( 1-alpha) quantile of the distribution of the pivotal quantity specified below.

DETAILS:

Suppose Z = (Z1, Z2, ... Zk) are multivariate normal variables with means 0 and covariance matrix vmat. Let U be a random variable independent of Z, such that df*U^2 is Chisquare with df degrees of freedom. Suppose lmat has p columns. If two.sided is TRUE, the pivotal quantity for qmvt.sim is defined to be D = max{1<=j<=p: | lmat[,j]*Z | / U}. If two.sided is FALSE, the pivotal quantity is the above without absolute values. The function approximates the 1-alpha quantile by simulating the pivotal quantity the specified number of times and selecting the (1-alpha)*(simsize+1) smallest value from the simulated versions. Of course, the actual value of the critical point generated will vary between simulations. Repeated generations of critical points in an attempt to obtain smaller values renders this method invalid.

REFERENCES:

Edwards, D. and Berry, J. J. (1987). The efficiency of simulation-based multiple comparisons. Biometrics 43, 913-928.

SEE ALSO:

, , , .

EXAMPLES:

# compare to qmvt(.90, 20, 60, .10) 
qmvt.sim(diag(20), diag(20), 60, alpha=.10)