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Noncentral chi-square distribution

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Noncentral chi-square
Probability density function
Chi-Squared-(nonCentral)-pdf
Cumulative distribution function
Chi-Squared-(nonCentral)-cdf
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  :\sum_{j=0}^\infty e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} \frac{\gamma(j+k/2,x/2)}{\Gamma(j+k/2)}\,
Mean k+\lambda\,
Median
Mode
Variance 2(k+2\lambda)\,
Skewness \frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}
Kurtosis \frac{12(k+4\lambda)}{(k+2\lambda)^2}
Entropy
mgf \frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}
Char. func. \frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}

In probability theory and statistics, the noncentral chi-square or noncentral \chi^2 distribution is a generalization of the chi-square distribution. If X_i are k independent, normally distributed random variables with means \mu_i and variances \sigma_i^2, then the random variable

\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2

is distributed according to the noncentral chi-square distribution. The noncentral chi-square distribution 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_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2.

Note that some references define \lambda as one half of the above sum.

Properties Edit

The noncentral chi-square distribution is equivalent to a (central) chi-square distribution with k + 2P degrees of freedom, where P is a Poisson random variable with parameter \lambda/2. Thus, the probability distribution 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 Y_q is distributed as chi-square with q degrees of freedom.

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(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)}

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}

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

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

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

Q(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,

where \gamma(k,z) is the lower incomplete Gamma function.

Derivation of the pdfEdit

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

  1. Start with the joint PDF of two independent non-zero mean Gaussian distributions, X and Y.
  2. Convert the joint density f(X, Y) to polar: f(R, A) where R^2 = (X^2+Y^2), tan(A) = Y/X.
  3. Integrate over the angular variable A.
  4. Convert from R to r where r^2 = R. This will yield a series expansion in r one factor of which matches the modified Bessel function I_0.
  5. Take the Laplace (Fourier) transform term-by-term and the special case K = 2 and the MGF will result.
  6. For the general case, take the K = 2 MGF and raise it to the K/2 power.
  7. The final trick to hide the K-dependence in the numerator of the MGF is to note that \lambda is a function of K; that is,
\lambda_2=\sum_1^2 \left(\frac{\mu_i}{\sigma_i}\right)^2
\lambda_K=\sum_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2 = \lambda
and therefore, \lambda is not explicitly a function of K in the above table.

Related distributionsEdit

  • If X is chi-square distributed X \sim \chi_k^2 then X^2 is also non-central chi-square distributed: X^2 \sim NC\chi^2_k(0)
  • If J \sim Poisson(\lambda), then \chi_k^2(\lambda) \sim \chi_{k+2J}^2
Various chi and chi-square distributions
Name Statistic
chi-square distribution \sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2
noncentral chi-square 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}
Bvn-small Probability distributions [[[:Template:Tnavbar-plain-nodiv]]]
Univariate Multivariate
Discrete: BernoullibinomialBoltzmanncompound PoissondegeneratedegreeGauss-Kuzmingeometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniformYule-SimonzetaZipfZipf-Mandelbrot Ewensmultinomial
Continuous: BetaBeta primeCauchychi-squareDirac delta functionErlangexponentialexponential powerFfadingFisher's zFisher-TippettGammageneralized extreme valuegeneralized hyperbolicgeneralized inverse GaussianHotelling's T-squarehyperbolic secanthyper-exponentialhypoexponentialinverse chi-squareinverse gaussianinverse gammaKumaraswamyLandauLaplaceLévyLévy skew alpha-stablelogisticlog-normalMaxwell-BoltzmannMaxwell speednormal (Gaussian)ParetoPearsonpolarraised cosineRayleighrelativistic Breit-WignerRiceStudent's ttriangulartype-1 Gumbeltype-2 GumbeluniformVoigtvon MisesWeibullWigner semicircle DirichletKentmatrix normalmultivariate normalvon Mises-FisherWigner quasiWishart
Miscellaneous: Cantorconditionalexponential familyinfinitely divisiblelocation-scale familymarginalmaximum entropy phase-typeposterior priorquasisampling
</center>

ReferencesEdit

  • Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions, Dover. Section 26.4.25.
  • Johnson, N. L. and Kotz, S., (1970), Continuous Univariate Distributions, vol. 2, Houghton-Mifflin.

Related Links Edit

This distribution (and many others) is available in the free interactive statistical tables program, STATTAB. The cumulative distribution function, its inverse, and parameters of the distribution can be calculated with these packages. A free Fortran library for these distributions is in CDFLIB. The URL for download is [1]

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