MultinomialDist Class Reference

Implements the abstract class DiscreteDistributionIntMulti for the multinomial distribution with parameters \(n\) and ( \(p_1\), …, \(p_d\)). More...

Inheritance diagram for MultinomialDist:

Collaboration diagram for MultinomialDist:

Public Member Functions
	MultinomialDist (int n, double p[])
	Creates a MultinomialDist object with parameters \(n\) and ( \(p_1\),…, \(p_d\)) such that \(\sum_{i=1}^d p_i = 1\). More...
double	prob (int x[])
double	cdf (int x[])
double []	getMean ()
double [][]	getCovariance ()
double [][]	getCorrelation ()
int	getN ()
	Returns the parameter \(n\) of this object.
double []	getP ()
	Returns the parameters ( \(p_1\),…, \(p_d\)) of this object.
void	setParams (int n, double p[])
	Sets the parameters \(n\) and ( \(p_1\),…, \(p_d\)) of this object.
Public Member Functions inherited from DiscreteDistributionIntMulti
abstract double	prob (int[] x)
	Returns the probability mass function \(p(x_1, x_2, …, x_d)\), which should be a real number in \([0,1]\). More...
double	cdf (int x[])
	Computes the cumulative probability function \(F\) of the distribution evaluated at x, assuming the lowest values start at 0, i.e. More...
int	getDimension ()
	Returns the dimension \(d\) of the distribution.
abstract double []	getMean ()
	Returns the mean vector of the distribution, defined as \(\mu_i = E[X_i]\).
abstract double [][]	getCovariance ()
	Returns the variance-covariance matrix of the distribution, defined as \(\sigma_{ij} = E[(X_i - \mu_i)(X_j - \mu_j)]\).
abstract double [][]	getCorrelation ()
	Returns the correlation matrix of the distribution, defined as \(\rho_{ij} = \sigma_{ij}/\sqrt{\sigma_{ii}\sigma_{jj}}\).

Static Public Member Functions
static double	prob (int n, double p[], int x[])
	Computes the probability mass function ( fMultinomial ) of the multinomial distribution with parameters \(n\) and ( \(p_1\),…, \(p_d\)) evaluated at \(x\).
static double	cdf (int n, double p[], int x[])
	Computes the function \(F\) of the multinomial distribution with parameters \(n\) and ( \(p_1\),…, \(p_d\)) evaluated at \(x\).
static double []	getMean (int n, double[] p)
	Computes the mean \(E[X_i] = np_i\) of the multinomial distribution with parameters \(n\) and ( \(p_1\),…, \(p_d\)).
static double [][]	getCovariance (int n, double[] p)
	Computes the covariance matrix of the multinomial distribution with parameters \(n\) and ( \(p_1\),…, \(p_d\)).
static double [][]	getCorrelation (int n, double[] p)
	Computes the correlation matrix of the multinomial distribution with parameters \(n\) and ( \(p_1\),…, \(p_d\)).
static double []	getMLE (int x[][], int m, int d, int n)
	Estimates and returns the parameters [ \(\hat{p_i}\),…, \(\hat{p_d}\)] of the multinomial distribution using the maximum likelihood method. More...

Protected Attributes
int	n
double	p []
Protected Attributes inherited from DiscreteDistributionIntMulti
int	dimension

Detailed Description

Implements the abstract class DiscreteDistributionIntMulti for the multinomial distribution with parameters \(n\) and ( \(p_1\), …, \(p_d\)).

The probability mass function is [97] 

\[ P[X = (x_1,…,x_d)] = {n!} \prod_{i=1}^d\frac{p_i^{x_i}}{x_i!}, \tag{fMultinomial} \]

where \(\sum_{i=1}^d x_i = n\) and \(\sum_{i=1}^d p_i = 1\).

Constructor & Destructor Documentation

MultinomialDist()

MultinomialDist	(	int	n,
		double	p[]
	)

Creates a MultinomialDist object with parameters \(n\) and ( \(p_1\),…, \(p_d\)) such that \(\sum_{i=1}^d p_i = 1\).

We have \(p_i = \) p[i-1].

Member Function Documentation

getMLE()

static double [] getMLE ( int  x[][], int  m, int  d, int  n  )

static

Estimates and returns the parameters [ \(\hat{p_i}\),…, \(\hat{p_d}\)] of the multinomial distribution using the maximum likelihood method.

It uses the \(m\) observations of \(d\) components in table \(x[i][j]\), \(i = 0, 1, …, m-1\) and \(j = 0, 1, …, d-1\). The equations of the maximum likelihood are defined as

\begin{align*} \hat{p}_i = \frac{\bar{X_i}}{N}. \end{align*}

Parameters

x	the list of observations used to evaluate parameters
m	the number of observations used to evaluate parameters
d	the dimension of each observation
n	the number of independant trials for each series

Returns

returns the parameters [ \(\hat{p_i}\),…, \(\hat{p_d}\)]

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The documentation for this class was generated from the following file:

• MultinomialDist.java