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MultivariateTDistribution

MultivariateStatistics`

MultivariateStatistics`

MultivariateTDistribution

As of Version 8, MultivariateTDistribution is part of the built-in Wolfram Language kernel.

MultivariateTDistribution[Σ,m]

represents the multivariate Student distribution with scale matrix Σ and degrees of freedom parameter m.

MultivariateTDistribution[μ,Σ,m]

represents the multivariate Student distribution with location μ, scale matrix Σ, and m degrees of freedom.

Details and Options

To use MultivariateTDistribution, you first need to load the Multivariate Statistics Package using Needs["MultivariateStatistics`"].
The probability density for vector x in a multivariate t distribution is proportional to (1+(x-μ).Σ^-1.(x-μ)/m)^{-(m+Length[Σ])/2}.
The scale matrix Σ can be any real‐valued symmetric positive definite matrix.
With specified location μ, μ can be any vector of real numbers, and Σ can be any symmetric positive definite p×p matrix with p=Length[μ].
The multivariate Student distribution characterizes the ratio of a multinormal to the covariance between the variates.
MultivariateTDistribution can be used with such functions as Mean, CDF, and RandomReal.

Examples

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Basic Examples (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

The mean of a bivariate distribution with 10 degrees of freedom:

Wolfram Language code: Mean[MultivariateTDistribution[{{1, ρ}, {ρ, 1}}, 10]]

Wolfram Language code: Needs["MultivariateStatistics`"]

The variances of each dimension:

Wolfram Language code: Variance[MultivariateTDistribution[{{1, ρ}, {ρ, 1}}, 10]]

Wolfram Language code: Needs["MultivariateStatistics`"]

Probability density function:

Wolfram Language code: PDF[MultivariateTDistribution[{{1, ρ}, {ρ, 1}}, m], {x, y}]

Wolfram Language code: Plot3D[PDF[MultivariateTDistribution[{{1, 1 / 2}, {1 / 2, 1}}, 10], {x, y}], {x, -3, 3}, {y, -3, 3}]

Scope (3)

Wolfram Language code: Needs["MultivariateStatistics`"]

Generate a set of pseudorandom vectors that follow a trivariate distribution:

Wolfram Language code: RandomReal[MultivariateTDistribution[{{1, 1 / 2, -1 / 3}, {1 / 2, 1, 0}, {-1 / 3, 0, 1}}, 5], 10]

Wolfram Language code: Needs["MultivariateStatistics`"]

Wolfram Language code: Skewness[MultivariateTDistribution[{{1, 1 / 2, -1 / 3}, {1 / 2, 1, 0}, {-1 / 3, 0, 1}}, 5]]

Wolfram Language code: Needs["MultivariateStatistics`"]

Wolfram Language code: Kurtosis[MultivariateTDistribution[{{1, 1 / 2, -1 / 3}, {1 / 2, 1, 0}, {-1 / 3, 0, 1}}, 5]]

Applications (1)

Wolfram Language code: Needs["MultivariateStatistics`"]

Equal probability contours for a bivariate distribution:

Wolfram Language code: ContourPlot[PDF[MultivariateTDistribution[{{1, 1 / 2}, {1 / 2, 1}}, 10], {x, y}], {x, -2.5, 2.5}, {y, -2.5, 2.5}]

Properties & Relations (1)

Wolfram Language code: Needs["MultivariateStatistics`"]

The probability density function integrates to unity:

Wolfram Language code: NIntegrate[PDF[MultivariateTDistribution[{{1, 1 / 2}, {1 / 2, 1}}, 10], {x, y}], {x, -∞, ∞}, {y, -∞, ∞}]

Possible Issues (2)

Wolfram Language code: Needs["MultivariateStatistics`"]

MultivariateTDistribution is not defined when Σ is not a symmetric positive definite matrix:

Wolfram Language code: PDF[MultivariateTDistribution[{{1, I / 2}, {1 / 2, 1}}, 10], {x, y}]

MultivariateTDistribution is not defined when m is not positive:

Wolfram Language code: PDF[MultivariateTDistribution[{{1, 1 / 2}, {1 / 2, 1}}, -3], {x, y}]

Wolfram Language code: Needs["MultivariateStatistics`"]

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

Wolfram Language code: Variance[MultivariateTDistribution[{{1, ρ}, {ρ, 1}}, m]] /. {ρ -> 1, m -> I}

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