Package umontreal.ssj.probdistmulti

Multivariate Probability Distributions. More...

Packages
package	norta
	This package implements the correlation matching algorithms proposed in [13]  for the situation where one wants to use the NORTA method to fit a multivariate distribution with discrete marginals.

Classes
class	BiNormalDist
	Extends the class ContinuousDistribution2Dim for the bivariate normal distribution [98]  (page 84). More...
class	BiNormalDonnellyDist
	Extends the class BiNormalDist for the bivariate normal distribution [98]  (page 84) using a translation of Donnelly’s Fortran code in [53] . More...
class	BiNormalGenzDist
	Extends the class BiNormalDist for the bivariate normal distribution [98]  (page 84) using Genz’s algorithm as described in [67] . More...
class	BiStudentDist
	Extends the class ContinuousDistribution2Dim for the standard bivariate Student’s \(t\) distribution [98]  (page 132). More...
class	ContinuousDistribution2Dim
	Classes implementing 2-dimensional continuous distributions should inherit from this class. More...
class	ContinuousDistributionMulti
	Classes implementing continuous multi-dimensional distributions should inherit from this class. More...
class	DirichletDist
	Implements the abstract class ContinuousDistributionMulti for the Dirichlet distribution with parameters ( \(\alpha_1\),…, \(\alpha_d\)), \(\alpha_i > 0\). More...
class	DiscreteDistributionIntMulti
	Classes implementing multi-dimensional discrete distributions over the integers should inherit from this class. More...
class	MultinomialDist
	Implements the abstract class DiscreteDistributionIntMulti for the multinomial distribution with parameters \(n\) and ( \(p_1\), …, \(p_d\)). More...
class	MultiNormalDist
	Implements the abstract class ContinuousDistributionMulti for the multinormal distribution with mean vector \(\boldsymbol{\mu}\) and covariance matrix \(\boldsymbol{\Sigma}\). More...
class	NegativeMultinomialDist
	Implements the class DiscreteDistributionIntMulti for the negative multinomial distribution with parameters \(n > 0\) and ( \(p_1, …, p_d\)) such that all \(0<p_i<1\) and \(\sum_{i=1}^d p_i < 1\). More...

Detailed Description

Multivariate Probability Distributions.

This package contains Java classes providing methods to compute mass, density, distribution and complementary distribution functions for some multi-dimensional discrete and continuous probability distributions. It does not generate random numbers for multivariate distributions; for that, see the package umontreal.ssj.randvarmulti.

Definitions

We recall that the distribution function of a continuous random vector \(X= \{x_1, x_2, …, x_d\}\) with density \(f(x_1, x_2, …, x_d)\) over the \(d\)-dimensional space \(R^d\) is

\begin{align} F(x_1, x_2, …, x_d) & = P[X_1\le x_1, X_2\le x_2, …, X_d\le x_d] \\ & = \int_{-\infty}^{x_1}\int_{-\infty}^{x_2} \cdots\int_{-\infty}^{x_d} f(s_1, s_2, …, s_d)\; ds_1 ds_2 …ds_d \tag{FDist} \end{align}

while that of a discrete random vector \(X\) with mass function \(\{p_1, p_2, …, p_d\}\) over a fixed set of real numbers is

\begin{align} F(x_1, x_2, …, x_d) & = P[X_1\le x_1, X_2\le x_2, …, X_d\le x_d] \\ & = \sum_{i_1\le x_1}\sum_{i_2\le x_2} \cdots\sum_{i_d\le x_d} p(x_1, x_2, …, x_d), \tag{FDistDisc} \end{align}

where \(p(x_1, x_2, …, x_d) = P[X_1 = x_1, X_2 = x_2, …, X_d = x_d]\). For a discrete distribution over the set of integers, one has

\begin{align} F (x_1, x_2, …, x_d) & = P[X_1\le x_1, X_2\le x_2, …, X_d\le x_d] \\ & = \sum_{s_1=-\infty}^{x_1} \sum_{s_2=-\infty}^{x_2} \cdots\sum_{s_d=-\infty}^{x_d} p(s_1, s_2, …, s_d), \tag{FDistDiscInt} \end{align}

where \(p(s_1, s_2, …, s_d) = P[X_1=s_1, X_2=s_2, …, X_d=s_d]\).

We define \(\bar{F}\), the complementary distribution function of \(X\), as

\[ \bar{F} (x_1, x_2, …, x_d) = P[X_1\ge x_1, X_2\ge x_2, …, X_d\ge x_d]. \]