com.imsl.stat.distributions

Class NormalPD

java.lang.Object

com.imsl.stat.distributions.ProbabilityDistribution

com.imsl.stat.distributions.NormalPD

All Implemented Interfaces:

ClosedFormMaximumLikelihoodInterface, PDFGradientInterface, PDFHessianInterface, Serializable, Cloneable

public class NormalPD
extends ProbabilityDistribution
implements PDFHessianInterface, ClosedFormMaximumLikelihoodInterface, Serializable, Cloneable

The normal (Gaussian) probability distribution.

See Also:

Example, Serialized Form

Constructor Summary

Constructors

Constructor and Description
NormalPD() Constructor for the normal probability distribution.

Method Summary

Modifier and Type	Method and Description
double[]	getClosedFormMLE(double[] x) Returns the closed form maximum likelihood estimates.
double[]	getClosedFormMlStandardError(double[] x) Returns the standard errors of the closed form maximum likelihood estimates.
double[]	getMLEs(double[] x) Deprecated. Use NormalPD.getClosedFormMLE(double[]) instead.
double[]	getParameterLowerBounds() Returns the lower bounds of the parameters.
double[]	getParameterUpperBounds() Returns the upper bounds of the parameters.
double[]	getPDFGradient(double x, double... params) Returns the analytic gradient of the pdf.
double[][]	getPDFHessian(double x, double... params) Returns the analytic Hessian matrix of the pdf.
double	pdf(double x, double... params) Returns the value of the normal probability density function.

Methods inherited from class com.imsl.stat.distributions.ProbabilityDistribution

getNumberOfParameters, getPDFGradientApproximation, getPDFHessianApproximation, getRangeOfX, setRangeOfX

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

NormalPD

public NormalPD()

Constructor for the normal probability distribution.

Method Detail

getParameterLowerBounds

public double[] getParameterLowerBounds()

Returns the lower bounds of the parameters.

Specified by:

getParameterLowerBounds in class ProbabilityDistribution

Returns:

a double array containing the lower bounds for \(\mu\in\mathbb{R}\) and \(\sigma\gt0\)

getParameterUpperBounds

public double[] getParameterUpperBounds()

Returns the upper bounds of the parameters.

Specified by:

getParameterUpperBounds in class ProbabilityDistribution

Returns:

a double array containing the upper bounds for \(\mu\in\mathbb{R}\) and \(\sigma\gt0\)

pdf

public double pdf(double x,
                  double... params)

Returns the value of the normal probability density function.

The probability density function for a normal distribution is given by $$f(x; \mu,\sigma) = \frac{1}{\sqrt{2\pi}\sigma} {\exp}^{ -\frac{{(x - \mu)}^2}{{2 {\sigma}^2}} } $$ where \(\mu\) and \(\sigma >0\) are the mean and standard deviation of the random variable.

Specified by:

pdf in class ProbabilityDistribution

Parameters:

x - a double, the value (quantile) at which to evaluate the pdf

params - a double array containing values of the parameters,\(\mu\) and \(\sigma\). The parameters can also be given in the form pdf(x,a,b), where a=\(\mu\) and b=\(\sigma\) are scalars.

Returns:

a double, the probability density at x given the parameter values

getPDFGradient

public double[] getPDFGradient(double x,
                               double... params)

Returns the analytic gradient of the pdf.

Specified by:

getPDFGradient in interface PDFGradientInterface

Parameters:

x - a double, the value at which to evaluate the gradient

params - a double array containing the parameters

Returns:

a double array containing the partial derivatives of the pdf with respect to the parameters

getPDFHessian

public double[][] getPDFHessian(double x,
                                double... params)

Returns the analytic Hessian matrix of the pdf.

Specified by:

getPDFHessian in interface PDFHessianInterface

Parameters:

x - a double, the value at which to evaluate the Hessian

params - a double array containing the parameters

Returns:

a double matrix containing the second partial derivatives of the pdf with respect to the parameters

getClosedFormMLE

public double[] getClosedFormMLE(double[] x)

Returns the closed form maximum likelihood estimates.

Specified by:

getClosedFormMLE in interface ClosedFormMaximumLikelihoodInterface

Parameters:

x - a double array containing the data

Returns:

a double array containing the maximum likelihood estimates for \(\mu\) and \(\sigma\)

getMLEs

public double[] getMLEs(double[] x)

Deprecated. Use NormalPD.getClosedFormMLE(double[]) instead.

Returns the mean and standard deviation of the sample data.

These are the maximum likelihood estimates for the mean and standard deviation of the Normal distribution, given the data.

Parameters:

x - a double array containing the data

Returns:

a double array containing the mean and standard deviation

getClosedFormMlStandardError

public double[] getClosedFormMlStandardError(double[] x)

Returns the standard errors of the closed form maximum likelihood estimates.

Specified by:

getClosedFormMlStandardError in interface ClosedFormMaximumLikelihoodInterface

Parameters:

x - a double array containing the data

Returns:

a double array containing the standard errors for the estimates of \(\mu,\sigma\)