NegativeBinomialDist (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
- - umontreal.iro.lecuyer.probdist.NegativeBinomialDist

All Implemented Interfaces:

Distribution

Direct Known Subclasses:

PascalDist
```
public class NegativeBinomialDist
extends DiscreteDistributionInt
```
Extends the class DiscreteDistributionInt for the negative binomial distribution with real parameters γ and p, where γ > 0 and 0 <= p <= 1. Its mass function is

p(x) = Γ(γ + x)/(x! Γ(γ))p^γ(1 - p)^x, for x = 0, 1, 2,…
where Γ is the gamma function.
If γ is an integer, p(x) can be interpreted as the probability of having x failures before the γ-th success in a sequence of independent Bernoulli trials with probability of success p. This special case is implemented as the Pascal distribution (see PascalDist).

Field Summary

Fields
Modifier and Type Field and Description

static double MAXN
- Fields inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
  EPSILON

Fields
Modifier and Type	Field and Description
`static double`	`MAXN`

Constructor Summary

Constructors
Constructor and Description
`NegativeBinomialDist(double gamma, double p)` Creates an object that contains the probability terms and the distribution function for the negative binomial distribution with parameters γ and p.

Method Summary

Methods
Modifier and Type	Method and Description
`double`	`barF(int x)` Returns bar(F)(x), the complementary distribution function.
`static double`	`cdf(double gamma, double p, int x)` Computes the distribution function.
`double`	`cdf(int x)` Returns the distribution function F evaluated at `x` (see).
`double`	`getGamma()` Returns the parameter γ of this object.
`static NegativeBinomialDist`	`getInstanceFromMLE(int[] x, int n)` Creates a new instance of a negative binomial distribution with parameters γ and p estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
`static NegativeBinomialDist`	`getInstanceFromMLE(int[] x, int n, double gamma)` Creates a new instance of a negative binomial distribution with parameters γ = `gamma` given and hat(p) estimated using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
`static double[]`	`getMaximumLikelihoodEstimate(int[] x, int n)` Deprecated.
`static double[]`	`getMaximumLikelihoodEstimate(int[] x, int n, double gamma)` Deprecated.
`double`	`getMean()` Returns the mean of the distribution function.
`static double`	`getMean(double gamma, double p)` Computes and returns the mean E[X] = γ(1 - p)/p of the negative binomial distribution with parameters γ and p.
`static double[]`	`getMLE(int[] x, int n)` Estimates the parameter (γ, p) of the negative binomial distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
`static double[]`	`getMLE(int[] x, int n, double gamma)` Estimates the parameter p of the negative binomial distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
`double`	`getP()` Returns the parameter p of this object.
`double[]`	`getParams()` Return a table containing the parameters of the current distribution.
`double`	`getStandardDeviation()` Returns the standard deviation of the distribution function.
`static double`	`getStandardDeviation(double gamma, double p)` Computes and returns the standard deviation of the negative binomial distribution with parameters γ and p.
`double`	`getVariance()` Returns the variance of the distribution function.
`static double`	`getVariance(double gamma, double p)` Computes and returns the variance Var[X] = γ(1 - p)/p² of the negative binomial distribution with parameters γ and p.
`static int`	`inverseF(double gamma, double p, double u)` Computes the inverse function without precomputing tables.
`int`	`inverseFInt(double u)` Returns the inverse distribution function F^-1(u), where 0 <= u <= 1.
`static double`	`prob(double gamma, double p, int x)` Computes the probability p(x).
`double`	`prob(int x)` Returns p(x), the probability of x, which should be a real number in the interval [0, 1].
`void`	`setParams(double gamma, double p)` Sets the parameter γ and p of this object.
`java.lang.String`	`toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
barF, cdf, getXinf, getXsup, inverseF

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Field Detail
  - MAXN
```
public static double MAXN
```
- Constructor Detail
  - NegativeBinomialDist
```
public NegativeBinomialDist(double gamma,
                    double p)
```
    Creates an object that contains the probability terms and the distribution function for the negative binomial distribution with parameters γ and p.
- Method Detail
  - prob
```
public double prob(int x)
```
    Description copied from class: DiscreteDistributionInt
    
    Returns p(x), the probability of x, which should be a real number in the interval [0, 1].
    
    Specified by:
    
    prob in class DiscreteDistributionInt
    
    Parameters:
    x - value at which the mass function must be evaluated
    
    Returns:
    the mass function evaluated at x
  - cdf
```
public double cdf(int x)
```
    Description copied from class: DiscreteDistributionInt
    
    Returns the distribution function F evaluated at x (see).
    
    Specified by:
    
    cdf in class DiscreteDistributionInt
    
    Parameters:
    x - value at which the distribution function must be evaluated
    
    Returns:
    the distribution function evaluated at x
  - barF
```
public double barF(int x)
```
    Description copied from class: DiscreteDistributionInt
    
    Returns bar(F)(x), the complementary distribution function. See the WARNING above.
    
    Overrides:
    
    barF in class DiscreteDistributionInt
    
    Parameters:
    x - value at which the complementary distribution function must be evaluated
    
    Returns:
    the complementary distribution function evaluated at x
  - inverseFInt
```
public int inverseFInt(double u)
```
    Description copied from class: DiscreteDistributionInt
    
    Returns the inverse distribution function F^-1(u), where 0 <= u <= 1. The default implementation uses binary search.
    
    Overrides:
    
    inverseFInt in class DiscreteDistributionInt
    
    Parameters:
    u - value in the interval (0, 1) for which the inverse distribution function is evaluated
    
    Returns:
    the inverse distribution function evaluated at u
  - getMean
```
public double getMean()
```
    Description copied from interface: Distribution
    
    Returns the mean of the distribution function.
  - getVariance
```
public double getVariance()
```
    Description copied from interface: Distribution
    
    Returns the variance of the distribution function.
  - getStandardDeviation
```
public double getStandardDeviation()
```
    Description copied from interface: Distribution
    
    Returns the standard deviation of the distribution function.
  - prob
```
public static double prob(double gamma,
          double p,
          int x)
```
    Computes the probability p(x).
  - cdf
```
public static double cdf(double gamma,
         double p,
         int x)
```
    Computes the distribution function.
  - inverseF
```
public static int inverseF(double gamma,
           double p,
           double u)
```
    Computes the inverse function without precomputing tables. This method computes the CDF at the mode (maximum term) and performs a linear search from that point.
  - getMLE
```
public static double[] getMLE(int[] x,
              int n,
              double gamma)
```
    Estimates the parameter p of the negative binomial distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The parameter γ = gamma is assumed known. The estimate hat(p) is returned in element 0 of the returned array. The maximum likelihood estimator hat(p) satisfies the equation hat(p) = γ/(γ + bar(x)_n), where bar(x)_n is the average of x[0],…, x[n - 1].
    
    Parameters:
    x - the list of observations used to evaluate parameters
    n - the number of observations used to evaluate parameters
    gamma - the first parameter of the negative binomial
    
    Returns:
    returns the parameters [hat(p)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(int[] x,
                                               int n,
                                               double gamma)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static NegativeBinomialDist getInstanceFromMLE(int[] x,
                                      int n,
                                      double gamma)
```
    Creates a new instance of a negative binomial distribution with parameters γ = gamma given and hat(p) estimated using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
    gamma - the first parameter of the negative binomial
  - getMLE
```
public static double[] getMLE(int[] x,
              int n)
```
    Estimates the parameter (γ, p) of the negative binomial distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a two-element array, in regular order: [γ, p].
    
    Parameters:
    x - the list of observations used to evaluate parameters
    n - the number of observations used to evaluate parameters
    
    Returns:
    returns the parameters [ hat(γ), hat(p)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(int[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static NegativeBinomialDist getInstanceFromMLE(int[] x,
                                      int n)
```
    Creates a new instance of a negative binomial distribution with parameters γ and p estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
  - getMean
```
public static double getMean(double gamma,
             double p)
```
    Computes and returns the mean E[X] = γ(1 - p)/p of the negative binomial distribution with parameters γ and p.
    
    Returns:
    the mean of the negative binomial distribution E[X] = γ(1 - p)/p
  - getVariance
```
public static double getVariance(double gamma,
                 double p)
```
    Computes and returns the variance Var[X] = γ(1 - p)/p² of the negative binomial distribution with parameters γ and p.
    
    Returns:
    the variance of the negative binomial distribution Var[X] = γ(1 - p)/p²
  - getStandardDeviation
```
public static double getStandardDeviation(double gamma,
                          double p)
```
    Computes and returns the standard deviation of the negative binomial distribution with parameters γ and p.
    
    Returns:
    the standard deviation of the negative binomial distribution
  - getGamma
```
public double getGamma()
```
    Returns the parameter γ of this object.
  - getP
```
public double getP()
```
    Returns the parameter p of this object.
  - setParams
```
public void setParams(double gamma,
             double p)
```
    Sets the parameter γ and p of this object.
  - getParams
```
public double[] getParams()
```
    Return a table containing the parameters of the current distribution. This table is put in regular order: [γ, p].
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class NegativeBinomialDist

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt

Constructor Summary

Method Summary

Methods inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt

Methods inherited from class java.lang.Object

Field Detail

MAXN

Constructor Detail

NegativeBinomialDist

Method Detail

prob

cdf

barF

inverseFInt

getMean

getVariance

getStandardDeviation

prob

cdf

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getGamma

getP

setParams

getParams

toString