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- [section:chi_squared_dist Chi Squared Distribution]
- ``#include <boost/math/distributions/chi_squared.hpp>``
- namespace boost{ namespace math{
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- 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 [chi][sub i] are [nu]
- independent, normally distributed
- random variables with means [mu][sub i] and variances [sigma][sub i][super 2],
- then the random variable:
- [equation chi_squ_ref1]
- 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 [nu] that specifies the number of degrees of
- freedom. The following graph illustrates how the distribution changes
- for different values of [nu]:
- [graph chi_squared_pdf]
- [h4 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.
- [variablelist
- [[difference_from_variance][The difference from the assumed nominal variance
- that is to be detected: Note that the sign of this value is critical, see below.]]
- [[alpha][The maximum acceptable risk of rejecting the null hypothesis when it is
- in fact true.]]
- [[beta][The maximum acceptable risk of falsely failing to reject the null hypothesis.]]
- [[variance][The nominal variance being tested against.]]
- [[hint][An optional hint on where to start looking for a result: the current sample
- size would be a good choice.]]
- ]
- 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
- [@http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm the NIST Engineering Statistics Handbook, Section 7.2.3.2].
- [h4 Non-member Accessors]
- All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
- that are generic to all distributions are supported: __usual_accessors.
- (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 \[0, +[infin]\].
- [h4 Examples]
- Various [link math_toolkit.stat_tut.weg.cs_eg worked examples]
- are available illustrating the use of the Chi Squared Distribution.
- [h4 Accuracy]
- The Chi-Squared distribution is implemented in terms of the
- [link math_toolkit.sf_gamma.igamma incomplete gamma functions]:
- please refer to the accuracy data for those functions.
- [h4 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/.
- [table
- [[Function][Implementation Notes]]
- [[pdf][Using the relation: pdf = __gamma_p_derivative(v / 2, x / 2) / 2 ]]
- [[cdf][Using the relation: p = __gamma_p(v / 2, x / 2) ]]
- [[cdf complement][Using the relation: q = __gamma_q(v / 2, x / 2) ]]
- [[quantile][Using the relation: x = 2 * __gamma_p_inv(v / 2, p) ]]
- [[quantile from the complement][Using the relation: x = 2 * __gamma_q_inv(v / 2, p) ]]
- [[mean][v]]
- [[variance][2v]]
- [[mode][v - 2 (if v >= 2)]]
- [[skewness][2 * sqrt(2 / v) == sqrt(8 / v)]]
- [[kurtosis][3 + 12 / v]]
- [[kurtosis excess][12 / v]]
- ]
- [h4 References]
- * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm NIST Exploratory Data Analysis]
- * [@http://en.wikipedia.org/wiki/Chi-square_distribution Chi-square distribution]
- * [@http://mathworld.wolfram.com/Chi-SquaredDistribution.html Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource.]
- [endsect] [/section:chi_squared_dist Chi Squared]
- [/ chi_squared.qbk
- Copyright 2006 John Maddock and Paul A. Bristow.
- Distributed under the Boost Software License, Version 1.0.
- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt).
- ]
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