The Pascal distribution is a special case of the negative binomial distribution [118] (page 324) with parameters \(n\) and \(p\), where \(n\) is a positive integer and \(0\le p\le1\). More...
Inheritance diagram for PascalDist:
Collaboration diagram for PascalDist:
Public Member Functions | |
| PascalDist (int n, double p) | |
| Creates an object that contains the probability terms ( fmass-pascal ) and the distribution function for the Pascal distribution with parameter \(n\) and \(p\). | |
| int | getN1 () |
| Returns the parameter \(n\) of this object. | |
| void | setParams (int n, double p) |
| Sets the parameter \(n\) and \(p\) of this object. | |
| String | toString () |
Returns a String containing information about the current distribution. | |
 Public Member Functions inherited from NegativeBinomialDist | |
| NegativeBinomialDist (double n, double p) | |
| Creates an object that contains the probability terms ( fmass-negbin ) and the distribution function for the negative binomial distribution with parameters \(n\) and \(p\). | |
| double | prob (int x) |
| double | cdf (int x) |
| double | barF (int x) |
| int | inverseFInt (double u) |
| double | getMean () |
| Returns the mean of the distribution function. | |
| double | getVariance () |
| Returns the variance of the distribution function. | |
| double | getStandardDeviation () |
| Returns the standard deviation of the distribution function. | |
| double | getGamma () |
| Returns the parameter \(n\) of this object. | |
| double | getN () |
| Returns the parameter \(n\) of this object. | |
| double | getP () |
| Returns the parameter \(p\) of this object. | |
| void | setParams (double n, double p) |
| Sets the parameter \(n\) and \(p\) of this object. More... | |
| double [] | getParams () |
| Return a table containing the parameters of the current distribution. More... | |
| String | toString () |
Returns a String containing information about the current distribution. | |
| Public Member Functions inherited from DiscreteDistributionInt | |
| abstract double | prob (int x) |
| Returns \(p(x)\), the probability of \(x\). More... | |
| double | cdf (double x) |
| Returns the distribution function \(F\) evaluated at \(x\) (see ( FDistDisc )). More... | |
| abstract double | cdf (int x) |
| Returns the distribution function \(F\) evaluated at \(x\) (see ( FDistDisc )). More... | |
| double | barF (double x) |
| Returns \(\bar{F}(x)\), the complementary distribution function. More... | |
| double | barF (int x) |
| Returns \(\bar{F}(x)\), the complementary distribution function. More... | |
| int | getXinf () |
| Returns the lower limit \(x_a\) of the support of the probability mass function. More... | |
| int | getXsup () |
| Returns the upper limit \(x_b\) of the support of the probability mass function. More... | |
| double | inverseF (double u) |
| Returns the inverse distribution function \(F^{-1}(u)\), where. More... | |
| int | inverseFInt (double u) |
| Returns the inverse distribution function \(F^{-1}(u)\), where. More... | |
Static Public Member Functions | |
| static double [] | getMLE (int[] x, int m) |
| Estimates the parameter \((n, p)\) of the Pascal distribution using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| static PascalDist | getInstanceFromMLE (int[] x, int m) |
| Creates a new instance of a Pascal distribution with parameters \(n\) and \(p\) estimated using the maximum likelihood method based on the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| Static Public Member Functions inherited from NegativeBinomialDist | |
| static double | prob (double n, double p, int x) |
| Computes the probability \(p(x)\) defined in ( fmass-negbin ). | |
| static double | cdf (double n, double p, int x) |
| Computes the distribution function. | |
| static double | barF (double n, double p, int x) |
| Returns \(\bar{F}(x) = P[X \ge x]\), the complementary distribution function. | |
| static int | inverseF (double n, double p, double u) |
| Computes the inverse function without precomputing tables. | |
| static double [] | getMLE (int[] x, int m, double n) |
| Estimates the parameter \(p\) of the negative binomial distribution using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| static NegativeBinomialDist | getInstanceFromMLE (int[] x, int m, double n) |
| Creates a new instance of a negative binomial distribution with parameters \(n\) given and \(\hat{p}\) estimated using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| static double [] | getMLE1 (int[] x, int m, double p) |
| Estimates the parameter \(n\) of the negative binomial distribution using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| static NegativeBinomialDist | getInstanceFromMLE1 (int[] x, int m, double p) |
| Creates a new instance of a negative binomial distribution with parameters \(p\) given and \(\hat{n}\) estimated using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| static double [] | getMLE (int[] x, int m) |
| Estimates the parameter \((n, p)\) of the negative binomial distribution using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| static NegativeBinomialDist | getInstanceFromMLE (int[] x, int m) |
| Creates a new instance of a negative binomial distribution with parameters \(n\) and \(p\) estimated using the maximum likelihood method based on the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| static double | getMLEninv (int[] x, int m) |
| Estimates and returns the parameter \(\nu= 1/\hat{n}\) of the negative binomial distribution using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More... | |
| static double | getMean (double n, double p) |
| Computes and returns the mean \(E[X] = n(1 - p)/p\) of the negative binomial distribution with parameters \(n\) and \(p\). More... | |
| static double | getVariance (double n, double p) |
| Computes and returns the variance \(\mbox{Var}[X] = n(1 - p)/p^2\) of the negative binomial distribution with parameters \(n\) and \(p\). More... | |
| static double | getStandardDeviation (double n, double p) |
| Computes and returns the standard deviation of the negative binomial distribution with parameters \(n\) and \(p\). More... | |
Additional Inherited Members | |
| Static Public Attributes inherited from NegativeBinomialDist | |
| static double | MAXN = 100000 |
| If the maximum term is greater than this constant, then the tables will not be precomputed. | |
| Static Public Attributes inherited from DiscreteDistributionInt | |
| static double | EPSILON = 1.0e-16 |
| Environment variable that determines what probability terms can be considered as negligible when building precomputed tables for distribution and mass functions. More... | |
| Protected Attributes inherited from NegativeBinomialDist | |
| double | n |
| double | p |
| Protected Attributes inherited from DiscreteDistributionInt | |
| double | cdf [] = null |
| double | pdf [] = null |
| int | xmin = 0 |
| int | xmax = 0 |
| int | xmed = 0 |
| int | supportA = Integer.MIN_VALUE |
| int | supportB = Integer.MAX_VALUE |
| Static Protected Attributes inherited from DiscreteDistributionInt | |
| static final double | EPS_EXTRA = 1.0e-6 |
Detailed Description
The Pascal distribution is a special case of the negative binomial distribution [118] (page 324) with parameters \(n\) and \(p\), where \(n\) is a positive integer and \(0\le p\le1\).
\[ p(x) = \binom{n + x - 1}{x} p^n (1 - p)^x, \qquad\mbox{for } x = 0, 1, 2, …\tag{fmass-pascal} \]
This \(p(x)\) can be interpreted as the probability of having \(x\) failures before the \(n\)th success in a sequence of independent Bernoulli trials with probability of success \(p\). For \(n=1\), this gives the geometric distribution.
Member Function Documentation
◆ getInstanceFromMLE()
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static |
Creates a new instance of a Pascal distribution with parameters \(n\) and \(p\) estimated using the maximum likelihood method based on the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\).
- Parameters
-
x the list of observations to use to evaluate parameters m the number of observations to use to evaluate parameters
◆ getMLE()
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static |
Estimates the parameter \((n, p)\) of the Pascal distribution using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\).
The estimates are returned in a two-element array, in regular order: [ \(n\), \(p\)]. The maximum likelihood estimators are the values \((\hat{n}\), \(\hat{p})\) that satisfy the equations
\begin{align*} \frac{\hat{n}(1 - \hat{p})}{\hat{p}} & = \bar{x}_m \\ \ln(1 + \hat{p}) & = \sum_{j=1}^{\infty} \frac{F_j}{(\hat{n} + j - 1)} \end{align*}
where \(\bar{x}_m\) is the average of \(x[0],…,x[m-1]\), and \(F_j = \sum_{i=j}^{\infty} f_i\) = proportion of \(x\)’s which are greater than or equal to \(j\) [97] (page 132).
- Parameters
-
x the list of observations used to evaluate parameters m the number of observations used to evaluate parameters
- Returns
- returns the parameters [ \(\hat{n}\), \(\hat{p}\)]
The documentation for this class was generated from the following file:
- PascalDist.java
