mvtcdf

Multivariate t cumulative distribution function

Syntax

y = mvtcdf(X,C,DF)
y = mvtcdf(xl,xu,C,DF)
[y,err] = mvtcdf(...)
[...] = mvntdf(...,options)

Description

y = mvtcdf(X,C,DF) returns the cumulative probability of the multivariate t distribution with correlation parameters C and degrees of freedom DF, evaluated at each row of X. Rows of the n-by-d matrix X correspond to observations or points, and columns correspond to variables or coordinates. y is an n-by-1 vector.

C is a symmetric, positive definite, d-by-d matrix, typically a correlation matrix. If its diagonal elements are not 1, mvtcdf scales C to correlation form. mvtcdf does not rescale X. DF is a scalar, or a vector with n elements.

The multivariate t cumulative probability at X is defined as the probability that a random vector T, distributed as multivariate t, will fall within the semi-infinite rectangle with upper limits defined by X, i.e., Pr{T(1)X(1),T(2)X(2),...T(d)X(d)}.

y = mvtcdf(xl,xu,C,DF) returns the multivariate t cumulative probability evaluated over the rectangle with lower and upper limits defined by xl and xu, respectively.

[y,err] = mvtcdf(...) returns an estimate of the error in y. For bivariate and trivariate distributions, mvtcdf uses adaptive quadrature on a transformation of the t density, based on methods developed by Genz, as described in the references. The default absolute error tolerance for these cases is 1e-8. For four or more dimensions, mvtcdf uses a quasi-Monte Carlo integration algorithm based on methods developed by Genz and Bretz, as described in the references. The default absolute error tolerance for these cases is 1e-4.

[...] = mvntdf(...,options) specifies control parameters for the numerical integration used to compute y. This argument can be created by a call to statset. Choices of statset parameters are:

  • 'TolFun' — Maximum absolute error tolerance. Default is 1e-8 when d < 4, or 1e-4 when d ≥ 4.

  • 'MaxFunEvals' — Maximum number of integrand evaluations allowed when d ≥ 4. Default is 1e7. 'MaxFunEvals' is ignored when d < 4.

  • 'Display' — Level of display output. Choices are 'off' (the default), 'iter', and 'final'. 'Display' is ignored when d < 4.

Examples

expand all

Compute the Multivariate t Distribution cdf

Compute the cdf of a multivariate t distribution with correlation parameters C = [1 .4; .4 1] and 2 degrees of freedom.

C = [1 .4; .4 1];
df = 2;
[X1,X2] = meshgrid(linspace(-2,2,25)',linspace(-2,2,25)');
X = [X1(:) X2(:)];
p = mvtcdf(X,C,df);

Plot the cdf.

figure;
surf(X1,X2,reshape(p,25,25));

References

[1] Genz, A. "Numerical Computation of Rectangular Bivariate and Trivariate Normal and t Probabilities." Statistics and Computing. Vol. 14, No. 3, 2004, pp. 251–260.

[2] Genz, A., and F. Bretz. "Numerical Computation of Multivariate t Probabilities with Application to Power Calculation of Multiple Contrasts." Journal of Statistical Computation and Simulation. Vol. 63, 1999, pp. 361–378.

[3] Genz, A., and F. Bretz. "Comparison of Methods for the Computation of Multivariate t Probabilities." Journal of Computational and Graphical Statistics. Vol. 11, No. 4, 2002, pp. 950–971.

See Also

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