SSJ  3.3.1 Stochastic Simulation in Java
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

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