Noncentral chi-squared distribution

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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 Q_M(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 \chi^2 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.

Background[edit]

Let (X_1, X_2, \ldots, X_i, \ldots, X_k) be k independent, normally distributed random variables with means \mu_i 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 X_i), and \lambda which is related to the mean of the random variables X_i by:

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

\lambda is sometimes called the noncentrality parameter. Note that some references define \lambda 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(0_k,I_k) distribution (i.e., the squared distance from the origin of a point taken at random from that distribution), the non-central \chi^2 is the squared norm of a random vector with N(\mu,I_k) distribution. Here 0_k is a zero vector of length k, \mu = (\mu_1, \ldots, \mu_k) and I_k is the identity matrix of size k.

Definition[edit]

The probability density function (pdf) 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 Y_q 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 \lambda/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 \lambda.

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 I_\nu(y) is a modified Bessel function of the first kind given by

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

Properties[edit]

Moment generating function[edit]

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}}.

Moments[edit]

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[edit]

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 with k degrees of freedom 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 Q_M(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[edit]

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 \left\{ \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)} \right\}

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, yielding

 P(x; k, \lambda ) \approx \Phi \left\{ \frac{(\frac{x}{k} ) ^{1/3} - (1 - \frac{2}{9k}) }  {\sqrt{\frac{2}{9k}}  } \right\} ,

essentially approximating the normalized Chi-squared distribution X / k as the cube of a Gaussian.

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


Differential equation[edit]


\left\{4 x f''(x)+(-2 k+4 x+8) f'(x)+f(x) (-k-\lambda +x+4)=0,f(1)
   2^{k/2} e^{\frac{\lambda +1}{2}}=\,
   _0\tilde{F}_1\left(;\frac{k}{2};\frac{\lambda }{4}\right),\lambda  \,
   _0\tilde{F}_1\left(;\frac{k}{2}+1;\frac{\lambda }{4}\right)+2 (k-3) \,
   _0\tilde{F}_1\left(;\frac{k}{2};\frac{\lambda
   }{4}\right)=2^{\frac{k}{2}+2} e^{\frac{\lambda +1}{2}} f'(1)\right\}

Derivation of the pdf[edit]

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=\cdots=\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+\cdots+X_k^2 depends on the means only through the squared length, \lambda=\mu_1^2+\cdots+\mu_k^2. Without loss of generality, we can therefore take \mu_1=\sqrt{\lambda} and \mu_2=\cdots=\mu_k=0.
  3. Now derive the density of X=X_1^2 (i.e. the 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.
  1. 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.
  2. 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+\cdots+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[edit]

  • 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 \mathrm{Poisson}\left(\frac{\lambda}{2}\right), then \chi_{k+2J}^2 \sim {\chi'}_k^2(\lambda)
  • If V\sim{\chi'}^2_2(\lambda), then \sqrt{V} takes the Rice distribution with parameter \sqrt{\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[edit]

Sankaran (1963) discusses the transformations of the form z=[(X-b)/(k+\lambda)]^{1/2}. He analyzes the expansions of the cumulants of z up to the term O((k+\lambda)^{-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 \lambda
  • b=(k-1)/3 makes the third cumulant of z approximately independent of \lambda
  • b=(k-1)/4 makes the fourth cumulant of z approximately independent of \lambda

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 (\lambda + (k-1)/2)^{1/2} and variance O((k+\lambda)^{-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}

Occurrences[edit]

Use in tolerance intervals[edit]

Two-sided normal regression tolerance intervals can be obtained based on the noncentral chi-squared distribution.[7] This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.

Notes[edit]

  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.
  7. ^ Derek S. Young (August 2010). "tolerance: An R Package for Estimating Tolerance Intervals". Journal of Statistical Software 36 (5): 1–39. ISSN 1548-7660. Retrieved 19 February 2013. , p.32

References[edit]

External Links[edit]