#include <boost/math/distributions/chi_squared.hpp>

namespace boost{ namespace math{

template <class RealType = double,
          class Policy   = policies::policy<> >
class chi_squared_distribution;

typedef chi_squared_distribution<> chi_squared;

template <class RealType, class Policy>
class chi_squared_distribution
{
public:
   typedef RealType  value_type;
   typedef Policy    policy_type;

   // Constructor:
   chi_squared_distribution(RealType i);

   // Accessor to parameter:
   RealType degrees_of_freedom()const;

   // Parameter estimation:
   static RealType find_degrees_of_freedom(
      RealType difference_from_mean,
      RealType alpha,
      RealType beta,
      RealType sd,
      RealType hint = 100);
};

}} // namespaces

The Chi-Squared distribution is one of the most widely used distributions in statistical tests. If χ_i   are ν   independent, normally distributed random variables with means μ_i   and variances σ_i², then the random variable:

is distributed according to the Chi-Squared distribution.

The Chi-Squared distribution is a special case of the gamma distribution and has a single parameter ν   that specifies the number of degrees of freedom. The following graph illustrates how the distribution changes for different values of ν:

Member Functions

chi_squared_distribution(RealType v);

Constructs a Chi-Squared distribution with v degrees of freedom.

Requires v > 0, otherwise calls domain_error.

RealType degrees_of_freedom()const;

Returns the parameter v from which this object was constructed.

static RealType find_degrees_of_freedom(
   RealType difference_from_variance,
   RealType alpha,
   RealType beta,
   RealType variance,
   RealType hint = 100);

Estimates the sample size required to detect a difference from a nominal variance in a Chi-Squared test for equal standard deviations.

Note that this calculation works with variances and not standard deviations.

The sign of the parameter difference_from_variance is important: the Chi Squared distribution is asymmetric, and the caller must decide in advance whether they are testing for a variance greater than a nominal value (positive difference_from_variance) or testing for a variance less than a nominal value (negative difference_from_variance). If the latter, then obviously it is a requirement that variance + difference_from_variance > 0, since no sample can have a negative variance!

This procedure uses the method in Diamond, W. J. (1989). Practical Experiment Designs, Van-Nostrand Reinhold, New York.

See also section on Sample sizes required in the NIST Engineering Statistics Handbook, Section 7.2.3.2.

Non-member Accessors

All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.

(We have followed the usual restriction of the mode to degrees of freedom >= 2, but note that the maximum of the pdf is actually zero for degrees of freedom from 2 down to 0, and provide an extended definition that would avoid a discontinuity in the mode as alternative code in a comment).

The domain of the random variable is [0, +∞].

Examples

Various worked examples are available illustrating the use of the Chi Squared Distribution.

Accuracy

The Chi-Squared distribution is implemented in terms of the incomplete gamma functions: please refer to the accuracy data for those functions.

Implementation

In the following table v is the number of degrees of freedom of the distribution, x is the random variate, p is the probability, and q = 1-p.

References