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| GumbelDist () |
| | Constructor for the standard Gumbel distribution with parameters \(\beta\) = 1 and \(\delta\) = 0.
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| GumbelDist (double beta, double delta) |
| | Constructs a GumbelDist object with parameters \(\beta\) = beta and \(\delta\) = delta.
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double | density (double x) |
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| double | cdf (double x) |
| | Returns the distribution function \(F(x)\). More...
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| double | barF (double x) |
| | Returns \(\bar{F}(x) = 1 - F(x)\). More...
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| double | inverseF (double u) |
| | Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More...
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double | getMean () |
| | Returns the mean of the distribution function.
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double | getVariance () |
| | Returns the variance of the distribution function.
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double | getStandardDeviation () |
| | Returns the standard deviation of the distribution function.
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double | getBeta () |
| | Returns the parameter \(\beta\) of this object.
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double | getDelta () |
| | Returns the parameter \(\delta\) of this object.
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void | setParams (double beta, double delta) |
| | Sets the parameters \(\beta\) and \(\delta\) of this object.
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| double [] | getParams () |
| | Return a table containing the parameters of the current distribution. More...
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String | toString () |
| | Returns a String containing information about the current distribution.
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| abstract double | density (double x) |
| | Returns \(f(x)\), the density evaluated at \(x\). More...
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| double | barF (double x) |
| | Returns the complementary distribution function. More...
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| double | inverseBrent (double a, double b, double u, double tol) |
| | Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More...
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| double | inverseBisection (double u) |
| | Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More...
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| double | inverseF (double u) |
| | Returns the inverse distribution function \(x = F^{-1}(u)\). More...
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| double | getMean () |
| | Returns the mean. More...
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| double | getVariance () |
| | Returns the variance. More...
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| double | getStandardDeviation () |
| | Returns the standard deviation. More...
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| double | getXinf () |
| | Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...
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| double | getXsup () |
| | Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...
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| void | setXinf (double xa) |
| | Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...
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| void | setXsup (double xb) |
| | Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...
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static double | density (double beta, double delta, double x) |
| | Computes and returns the density function.
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static double | cdf (double beta, double delta, double x) |
| | Computes and returns the distribution function.
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static double | barF (double beta, double delta, double x) |
| | Computes and returns the complementary distribution function \(1 - F(x)\).
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static double | inverseF (double beta, double delta, double u) |
| | Computes and returns the inverse distribution function.
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| static double [] | getMLE (double[] x, int n) |
| | Estimates the parameters \((\beta,\delta)\) of the Gumbel distribution, assuming that \(\beta> 0\), and using the maximum likelihood method with the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...
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| static double [] | getMLEmin (double[] x, int n) |
| | Similar to getMLE, but for the case \(\beta< 0\). More...
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| static GumbelDist | getInstanceFromMLE (double[] x, int n) |
| | Creates a new instance of an Gumbel distribution with parameters \(\beta\) and \(\delta\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\), assuming that \(\beta> 0\). More...
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| static GumbelDist | getInstanceFromMLEmin (double[] x, int n) |
| | Similar to getInstanceFromMLE, but for the case \(\beta< 0\). More...
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| static double | getMean (double beta, double delta) |
| | Returns the mean, \(E[X] = \delta+ \gamma\beta\), of the Gumbel distribution with parameters \(\beta\) and \(\delta\), where \(\gamma= 0.5772156649015329\) is the Euler-Mascheroni constant. More...
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| static double | getVariance (double beta, double delta) |
| | Returns the variance \(\mbox{Var}[X] = \pi^2 \beta^2\!/6\) of the Gumbel distribution with parameters \(\beta\) and \(\delta\). More...
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| static double | getStandardDeviation (double beta, double delta) |
| | Returns the standard deviation of the Gumbel distribution with parameters \(\beta\) and \(\delta\). More...
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Extends the class ContinuousDistribution for the Gumbel distribution [98] (page 2), with location parameter \(\delta\) and scale parameter \(\beta\neq0\).
Using the notation \(z = (x-\delta)/\beta\), it has density
\[ f (x) = \frac{e^{-z} e^{-e^{-z}}}{|\beta|}, \qquad\mbox{for } -\infty< x < \infty \tag{densgumbel} \]
and distribution function
\[ F(x) = \left\{ \begin{array}{ll} e^{-e^{-z}}, \qquad & \mbox{for } \beta> 0 \\ 1 - e^{-e^{-z}}, \qquad & \mbox{for } \beta< 0. \end{array} \right. \]
| static double [] getMLE |
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double [] |
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Estimates the parameters \((\beta,\delta)\) of the Gumbel distribution, assuming that \(\beta> 0\), and using the maximum likelihood method with the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).
The estimates are returned in a two-element array, in regular order: [ \(\beta\), \(\delta\)]. The maximum likelihood estimators are the values \((\hat{\beta}, \hat{\delta})\) that satisfy the equations:
\begin{align*} \hat{\beta} & = \bar{x}_n - \frac{\sum_{i=1}^n x_i e^{- x_i/\hat{\beta}}}{\sum_{i=1}^n e^{- x_i / \hat{\beta}}} \\ \hat{\delta} & = -{\hat{\beta}} \ln\left( \frac{1}{n} \sum_{i=1}^n e^{-x_i/\hat{\beta}} \right), \end{align*}
where \(\bar{x}_n\) is the average of \(x[0],…,x[n-1]\).
- Parameters
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| x | the list of observations used to evaluate parameters |
| n | the number of observations used to evaluate parameters |
- Returns
- returns the parameters [ \(\hat{\delta}\), \(\hat{\beta}\)]