 Noncentral chisquared distribution

Noncentral chisquared Probability density function
Cumulative distribution function
parameters: degrees of freedom
noncentrality parameter
support: pdf: cdf: with Marcum Qfunction Q_{M}(a,b) mean: variance: skewness: ex.kurtosis: mgf: for 2t < 1 cf: In probability theory and statistics, the noncentral chisquared or noncentral χ^{2} distribution is a generalization of the chisquared distribution. This distribution 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 likelihood ratio tests.
Contents
Background
Let X_{i} be k independent, normally distributed random variables with means μ_{i} and variances . Then the random variable
is distributed according to the noncentral chisquared distribution. It has two parameters: k which specifies the number of degrees of freedom (i.e. the number of X_{i}), and λ which is related to the mean of the random variables X_{i} by:
λ is sometime called the noncentrality parameter. Note that some references define λ 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 N(0_{k},I_{k}) distribution (i.e., the squared distance from the origin of a point taken at random from that distribution), the noncentral χ^{2} is the squared norm of a random vector with N(μ,I_{k}) distribution. Here 0_{k} is a zero vector of length k, μ = (μ_{1},...,μ_{k}) and I_{k} is the identity matrix of size k.
Properties
Probability density function
The probability density function is given by
where Y_{q} is distributed as chisquared with q 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 λ / 2, 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 λ.
Alternatively, the pdf can be written as
where I_{ν}(z) 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.
Moment generating function
The moment generating function is given by
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 which is given by
 and where is the lower incomplete Gamma function.
The Marcum Qfunction Q_{M}(a,b)can also be used to represent the cdf.^{[2]}
Approximation
Sankaran ^{[3]} discusses a number of closed form approximations for the cumulative distribution function. In an earlier paper,^{[4]} 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.^{[5]}
To approximate the Chisquared distribution, the noncentrality parameter, , is set to zero.
For a given probability, the formula is easily inverted to provide the corresponding approximation for .
Derivation of the pdf
The derivation of the probability density function is most easily done by performing the following steps:
 First, assume without loss of generality that . Then the joint distribution of 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. 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 (k1) 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)+(k1) = k+2i as required.
Related distributions
 If V is chisquared distributed then V is also noncentral chisquared distributed:
 If and and V_{1} is independent of V_{2} then a noncentral Fdistributed variable is developed as
 If , then
 Normal approximation^{[6]}: if , then in distribution as either or .
Transformations
Sankaran (1963) discusses the transformations of the form z = [(X − b) / (k + λ)]^{1 / 2}. He analyzes the expansions of the cumulants of z up to the term O((k + λ) ^{− 4}) and shows that the following choices of b produce reasonable results:
 b = (k − 1) / 2 makes the second cumulant of z approximately independent of λ
 b = (k − 1) / 3 makes the third cumulant of z approximately independent of λ
 b = (k − 1) / 4 makes the fourth cumulant of z approximately independent of λ
Also, a simpler transformation z_{1} = (X − (k − 1) / 2)^{1 / 2} can be used as a variance stabilizing transformation that produces a random variable with mean (λ + (k − 1) / 2)^{1 / 2} and variance O((k + λ) ^{− 2}).
Usability of these transformations may be hampered by the need to take the square roots of negative numbers.
Various chi and chisquared distributions Name Statistic chisquared distribution noncentral chisquared distribution chi distribution noncentral chi distribution 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), 9596, ISSN 00189448
 ^ 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) Section 29.8
 ^ Muirhead (2005) pages 22–24 and problem 1.18.
References
 Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions, Dover. Section 26.4.25.
 Johnson, N. L., Kotz, S., Balakrishnan, N. (1970), Continuous Univariate Distributions, Volume 2, Wiley. ISBN 0471584940
 Muirhead, R. (2005) Aspects of Multivariate Statistical Theory, Wiley
 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, JSTOR 2238621
Categories: Continuous distributions
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