Noncentral chi-squared distribution


Noncentral chi-squared distribution
Noncentral chi-squared
Probability density function
Chi-Squared-(nonCentral)-pdf.png
Cumulative distribution function
Chi-Squared-(nonCentral)-cdf.png
parameters: k > 0\, degrees of freedom

\lambda > 0\, non-centrality parameter

support: x \in [0; +\infty)\,
pdf: \frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2}
 I_{k/2-1}(\sqrt{\lambda x})
cdf: 1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right) with Marcum Q-function QM(a,b)
mean: k+\lambda\,
variance: 2(k+2\lambda)\,
skewness: \frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}
ex.kurtosis: \frac{12(k+4\lambda)}{(k+2\lambda)^2}
mgf: \frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}} for 2t < 1
cf: \frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}

In probability theory and statistics, the noncentral chi-squared or noncentral χ2 distribution is a generalization of the chi-squared distribution. This distribution often arises in the power analysis of statistical tests in which the null distribution is (perhaps asymptotically) a chi-squared distribution; important examples of such tests are the likelihood ratio tests.

Contents

Background

Let Xi be k independent, normally distributed random variables with means μi and variances \sigma_i^2. Then the random variable

\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2

is distributed according to the noncentral chi-squared distribution. It has two parameters: k which specifies the number of degrees of freedom (i.e. the number of Xi), and λ which is related to the mean of the random variables Xi by:

\lambda=\sum_{i=1}^k \left(\frac{\mu_i}{\sigma_i}\right)^2.

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 chi-squared distribution is the squared norm of a random vector with N(0k,Ik) distribution (i.e., the squared distance from the origin of a point taken at random from that distribution), the non-central χ2 is the squared norm of a random vector with N(μ,Ik) distribution. Here 0k is a zero vector of length k, μ = (μ1,...,μk) and Ik is the identity matrix of size k.

Properties

Probability density function

The probability density function is given by


f_X(x; k,\lambda) = \sum_{i=0}^\infty \frac{e^{-\lambda/2} (\lambda/2)^i}{i!} f_{Y_{k+2i}}(x),

where Yq is distributed as chi-squared with q degrees of freedom.

From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted mixture of central chi-squared 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 chi-squared with k+2i degrees of freedom. Then the unconditional distribution of Z is non-central chi-squared with k degrees of freedom, and non-centrality parameter λ.

Alternatively, the pdf can be written as

f_X(x;k,\lambda)=\frac{1}{2} e^{-(x+\lambda)/2} \left (\frac{x}{\lambda}\right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})

where Iν(z) is a modified Bessel function of the first kind given by

 I_a(y) = (y/2)^a \sum_{j=0}^\infty \frac{ (y^2/4)^j}{j! \Gamma(a+j+1)} .

Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:[1]

f_X(x;k,\lambda)={{\rm e}^{-\lambda/2}} _0F_1(;k/2;\lambda x/4)\frac{1}{2^{k/2}\Gamma(k/2)} {\rm e}^{-x/2} x^{k/2-1}.

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

M(t;k,\lambda)=\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}.

The first few raw moments are:

\mu^'_1=k+\lambda
\mu^'_2=(k+\lambda)^2 + 2(k + 2\lambda)
\mu^'_3=(k+\lambda)^3 + 6(k+\lambda)(k+2\lambda)+8(k+3\lambda)
\mu^'_4=(k+\lambda)^4+12(k+\lambda)^2(k+2\lambda)+4(11k^2+44k\lambda+36\lambda^2)+48(k+4\lambda)

The first few central moments are:

\mu_2=2(k+2\lambda)\,
\mu_3=8(k+3\lambda)\,
\mu_4=12(k+2\lambda)^2+48(k+4\lambda)\,

The nth cumulant is

K_n=2^{n-1}(n-1)!(k+n\lambda).\,

Hence

\mu^'_n = 2^{n-1}(n-1)!(k+n\lambda)+\sum_{j=1}^{n-1} \frac{(n-1)!2^{j-1}}{(n-j)!}(k+j\lambda )\mu^'_{n-j}.

Cumulative distribution function

Again using the relation between the central and noncentral chi-squared distributions, the cumulative distribution function (cdf) can be written as

P(x; k, \lambda ) = e^{-\lambda/2}\; \sum_{j=0}^\infty  \frac{(\lambda/2)^j}{j!} Q(x; k+2j)

where Q(x; k)\, is the cumulative distribution function of the central chi-squared distribution which is given by

Q(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,
and where \gamma(k,z)\, is the lower incomplete Gamma function.

The Marcum Q-function QM(a,b)can also be used to represent the cdf.[2]

P(x; k, \lambda) = 1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)

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:

 P(x; k, \lambda ) \approx \Phi \lbrace \frac{(\frac{x}  {k + \lambda}) ^ h - (1 + h  p  (h - 1 - 0.5 (2 - h)  m  p))}  {h  \sqrt{  2p}  (1 + 0.5 m  p)} \rbrace

where

 \Phi \lbrace \cdot \rbrace \, denotes the cumulative distribution function of the standard normal distribution;
 h = 1 - \frac{2}{3} \frac{(k+ \lambda)  (k+ 3  \lambda)}{(k+ 2  \lambda) ^ 2} \, ;
 p = \frac{k+ 2  \lambda}{(k+ \lambda) ^ 2} ;
 m = (h - 1)  (1 - 3  h) \, .

This and other approximations are discussed in a later text book.[5]

To approximate the Chi-squared distribution, the non-centrality parameter,  \lambda\, , is set to zero.

For a given probability, the formula is easily inverted to provide the corresponding approximation for  x\, .


Derivation of the pdf

The derivation of the probability density function is most easily done by performing the following steps:

  1. First, assume without loss of generality that \sigma_1=\ldots=\sigma_k=1. Then the joint distribution of X_1,\ldots,X_k is spherically symmetric, up to a location shift.
  2. The spherical symmetry then implies that the distribution of X=X_1^2+\ldots+X_k^2 depends on the means only through the squared length, \lambda=\mu_1^2+\ldots+\mu_k^2. Without loss of generality, we can therefore take \mu_1=\sqrt{\lambda} and \mu_2=\dots=\mu_k=0.
  3. Now derive the density of X=X_1^2 (i.e. k=1 case). Simple transformation of random variables shows that :\begin{align}f_X(x,1,\lambda) &= \frac{1}{2\sqrt{x}}\left( \phi(\sqrt{x}-\sqrt{\lambda}) + \phi(\sqrt{x}+\sqrt{\lambda}) \right )\\ &= \frac{1}{\sqrt{2\pi x}} e^{-(x+\lambda)/2} \cosh(\sqrt{\lambda x}),\\ \end{align}
    where \phi(\cdot) is the standard normal density.
  4. Expand the cosh term in a Taylor series. This gives the Poisson-weighted mixture representation of the density, still for k=1. The indices on the chi-squared random variables in the series above are 1+2i in this case.
  5. Finally, for the general case. We've assumed, without loss of generality, that X_2,\ldots,X_k are standard normal, and so X_2^2+\ldots+X_k^2 has a central chi-squared distribution with (k-1) degrees of freedom, independent of X_1^2. Using the poisson-weighted mixture representation for X_1^2, and the fact that the sum of chi-squared random variables is also chi-squared, completes the result. The indices in the series are (1+2i)+(k-1) = k+2i as required.

Related distributions

  • If V is chi-squared distributed V \sim \chi_k^2 then V is also non-central chi-squared distributed: V \sim {\chi'}^2_k(0)
  • If J \sim Poisson(\frac{\lambda}{2}), then \chi_{k+2J}^2 \sim {\chi'}_k^2(\lambda)
  • Normal approximation[6]: if V \sim {\chi'}^2_k(\lambda), then \frac{V-(k+\lambda)}{\sqrt{2(k+2\lambda)}}\to N(0,1) in distribution as either k\to\infty or \lambda\to\infty.

Transformations

Sankaran (1963) discusses the transformations of the form z = [(Xb) / (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 z1 = (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 chi-squared distributions
Name Statistic
chi-squared distribution \sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2
noncentral chi-squared distribution \sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2
chi distribution \sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}
noncentral chi distribution \sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}

Notes

  1. ^ Muirhead (2005) Theorem 1.3.4
  2. ^ Nuttall, Albert H. (1975): Some Integrals Involving the QM Function, IEEE Transactions on Information Theory, 21(1), 95-96, ISSN 0018-9448
  3. ^ Sankaran , M. (1963). Approximations to the non-central chi-squared distribution Biometrika, 50(1-2), 199–204
  4. ^ Sankaran , M. (1959). "On the non-central chi-squared distribution", Biometrika 46, 235–237
  5. ^ Johnson et al. (1995) Section 29.8
  6. ^ 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 0-471-58494-0
  • Muirhead, R. (2005) Aspects of Multivariate Statistical Theory, Wiley
  • Siegel, A.F. (1979), "The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity", Biometrika, 66, 381–386
  • Press, S.J. (1966), "Linear combinations of non-central chi-squared variates", The Annals of Mathematical Statistics 37 (2): 480–487, JSTOR 2238621 

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