Probability density function  
Cumulative distribution function  
Parameters  degrees of freedom noncentrality parameter 

Support  
CDF  with Marcum Qfunction 
Mean  
Variance  
Skewness  
Ex. kurtosis  
MGF  
CF 
In probability theory and statistics, the noncentral chisquared distribution (or noncentral distribution) is a generalization of the chisquared distribution. It often arises in the power analysis of statistical tests in which the null distribution is (perhaps asymptotically) a chisquared distribution; important examples of such tests are the likelihoodratio tests.
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✪ Statistical Distributions: Noncentral Chisquare Distribution

✪ Statistical Distributions: Central & Noncentral Chi square df=1 Distributions

✪ An Introduction to the ChiSquare Distribution

✪ Chi  Square Distribution

✪ Chisquare distribution introduction  Probability and Statistics  Khan Academy
Transcription
Contents
Background
Let be k independent, normally distributed random variables with means and unit variances. Then the random variable
is distributed according to the noncentral chisquared distribution. It has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), and which is related to the mean of the random variables by:
is sometimes called the noncentrality parameter. Note that some references define in other ways, such as half of the above sum, or its square root.
This distribution arises in multivariate statistics as a derivative of the multivariate normal distribution. While the central chisquared distribution is the squared norm of a random vector with distribution (i.e., the squared distance from the origin of a point taken at random from that distribution), the noncentral is the squared norm of a random vector with distribution. Here is a zero vector of length k, and is the identity matrix of size k.
Definition
The probability density function (pdf) is given by
where is distributed as chisquared with degrees of freedom.
From this representation, the noncentral chisquared distribution is seen to be a Poissonweighted mixture of central chisquared distributions. Suppose that a random variable J has a Poisson distribution with mean , and the conditional distribution of Z given J = i is chisquared with k + 2i degrees of freedom. Then the unconditional distribution of Z is noncentral chisquared with k degrees of freedom, and noncentrality parameter .
Alternatively, the pdf can be written as
where the noncentrality parameter in this formular is the square root of sum of square and is a modified Bessel function of the first kind given by
Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:^{[1]}
Siegel (1979) discusses the case k = 0 specifically (zero degrees of freedom), in which case the distribution has a discrete component at zero.
Properties
Moment generating function
The momentgenerating function is given by
Moments
The first few raw moments are:
The first few central moments are:
The nth cumulant is
Hence
Cumulative distribution function
Again using the relation between the central and noncentral chisquared distributions, the cumulative distribution function (cdf) can be written as
where is the cumulative distribution function of the central chisquared distribution with k degrees of freedom which is given by
 and where is the lower incomplete gamma function.
The Marcum Qfunction can also be used to represent the cdf.^{[2]}
Approximation (including for quantiles)
AbdelAty ^{[3]} derives (as "first approx.") a noncentral WilsonHilferty approximation:
is approximately normally distributed, i.e.,
which is quite accurate and well adapting to the noncentrality. Also, becomes for , the (central) chisquared case.
Sankaran ^{[4]} discusses a number of closed form approximations for the cumulative distribution function. In an earlier paper,^{[5]} he derived and states the following approximation:
where
 denotes the cumulative distribution function of the standard normal distribution;
This and other approximations are discussed in a later text book.^{[6]}
For a given probability, these formulas are easily inverted to provide the corresponding approximation for , to compute approximate quantiles.
Derivation of the pdf
The derivation of the probability density function is most easily done by performing the following steps:
 Since have unit variances, their joint distribution is spherically symmetric, up to a location shift.
 The spherical symmetry then implies that the distribution of depends on the means only through the squared length, . Without loss of generality, we can therefore take and .
 Now derive the density of (i.e. the k = 1 case). Simple transformation of random variables shows that
 where is the standard normal density.
 Expand the cosh term in a Taylor series. This gives the Poissonweighted mixture representation of the density, still for k = 1. The indices on the chisquared random variables in the series above are 1 + 2i in this case.
 Finally, for the general case. We've assumed, without loss of generality, that are standard normal, and so has a central chisquared distribution with (k − 1) degrees of freedom, independent of . Using the poissonweighted mixture representation for , and the fact that the sum of chisquared random variables is also chisquared, completes the result. The indices in the series are (1 + 2i) + (k − 1) = k + 2i as required.
Related distributions
 If is chisquared distributed then is also noncentral chisquared distributed:
 If and and is independent of then a noncentral Fdistributed variable is developed as
 If , then
 If , then takes the Rice distribution with parameter .
 Normal approximation:^{[7]} if , then in distribution as either or .
 If and , where are independent and , then where .
 In general, for a finite set of , the sum of these noncentral chisquare distributed random variables has the distribution where . This can be seen using moment generating functions as follows: by the independence of the random variables. It remains to plug in the MGF for the noncentral chi squared distributions into the product and compute the new MGF  this is left as an exercise. Alternatively it can be seen via the interpretation in the background section above as sums of squares of independent normally distributed random variables with variances of 1 and the specified means.
Transformations
Sankaran (1963) discusses the transformations of the form . He analyzes the expansions of the cumulants of up to the term and shows that the following choices of produce reasonable results:
 makes the second cumulant of approximately independent of
 makes the third cumulant of approximately independent of
 makes the fourth cumulant of approximately independent of
Also, a simpler transformation can be used as a variance stabilizing transformation that produces a random variable with mean and variance .
Usability of these transformations may be hampered by the need to take the square roots of negative numbers.
Name  Statistic 

chisquared distribution  
noncentral chisquared distribution  
chi distribution  
noncentral chi distribution 
Occurrences
Use in tolerance intervals
Twosided normal regression tolerance intervals can be obtained based on the noncentral chisquared distribution.^{[8]} This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.
Notes
 ^ Muirhead (2005) Theorem 1.3.4
 ^ Nuttall, Albert H. (1975): Some Integrals Involving the Q_{M} Function, IEEE Transactions on Information Theory, 21(1), 95–96, ISSN 00189448
 ^ AbdelAty, S. (1954). Approximate Formulae for the Percentage Points and the Probability Integral of the NonCentral χ2 Distribution Biometrika 41, 538–540. doi:10.2307/2332731
 ^ Sankaran , M. (1963). Approximations to the noncentral chisquared distribution Biometrika, 50(12), 199–204
 ^ Sankaran , M. (1959). "On the noncentral chisquared distribution", Biometrika 46, 235–237
 ^ Johnson et al. (1995) Continuous Univariate Distributions Section 29.8
 ^ Muirhead (2005) pages 22–24 and problem 1.18.
 ^ Derek S. Young (August 2010). "tolerance: An R Package for Estimating Tolerance Intervals". Journal of Statistical Software. 36 (5): 1–39. ISSN 15487660. Retrieved 19 February 2013., p.32
References
 Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, Dover. Section 26.4.25.
 Johnson, N. L., Kotz, S., Balakrishnan, N. (1995), Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0471584940
 Muirhead, R. (2005) Aspects of Multivariate Statistical Theory (2nd Edition). Wiley. ISBN 0471769851
 Siegel, A. F. (1979), "The noncentral chisquared distribution with zero degrees of freedom and testing for uniformity", Biometrika, 66, 381–386
 Press, S.J. (1966), "Linear combinations of noncentral chisquared variates", The Annals of Mathematical Statistics, 37 (2): 480–487, doi:10.1214/aoms/1177699531, JSTOR 2238621
External links
 Non central chi squared distribution – from itfeature.com