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| {{Probability distribution|
| | The author is known by the title of Figures Lint. Years in the past we moved to North Dakota. Hiring has been my profession for some time but I've currently applied for an additional 1. Body building is what my family members and I enjoy.<br><br>My website: [http://www.hard-ass-porn.com/user/MPolglaze at home std testing] |
| name =Noncentral chi-squared|
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| type =density|
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| pdf_image =[[File:Chi-Squared-(nonCentral)-pdf.png|325px]]|
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| cdf_image =[[File:Chi-Squared-(nonCentral)-cdf.png|325px]]|
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| parameters =<math>k > 0\,</math> degrees of freedom<br />
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| <math>\lambda > 0\,</math> non-centrality parameter|
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| support =<math>x \in [0; +\infty)\,</math>|
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| pdf =<math>\frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2}
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| I_{k/2-1}(\sqrt{\lambda x})</math>|
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| cdf =<math>1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)</math> with [[Marcum Q-function]] <math>Q_M(a,b)</math>
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| mean =<math>k+\lambda\,</math>|
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| median =|
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| mode =|
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| variance =<math>2(k+2\lambda)\,</math>|
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| skewness =<math>\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}</math>|
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| kurtosis =<math>\frac{12(k+4\lambda)}{(k+2\lambda)^2}</math>|
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| entropy =|
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| mgf =<math>\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}</math> for <math>2t<1</math>|
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| char =<math>\frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}</math>
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| }}
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| In [[probability theory]] and [[statistics]], the '''noncentral chi-squared''' or '''noncentral <math>\chi^2</math>''' distribution is a generalization of the [[chi-squared distribution]]. This distribution often arises in the [[statistical power|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 test]]s.
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| ==Background==
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| Let (<math>X_1</math>, <math>X_2, ..., </math><math>X_i, ...,</math> <math>X_k</math>) be ''k'' [[statistical independence|independent]], [[normal distribution|normally distributed]] random variables with means <math>\mu_i</math> and variances <math>\sigma_i^2</math>. Then the random variable
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| :<math>\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
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| is distributed according to the noncentral chi-squared distribution. It has two parameters: <math>k</math> which specifies the number of [[Degrees of freedom (statistics)|degrees of freedom]] (i.e. the number of <math>X_i</math>), and <math>\lambda</math> which is related to the mean of the random variables <math>X_i</math> by:
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| :<math>\lambda=\sum_{i=1}^k \left(\frac{\mu_i}{\sigma_i}\right)^2.</math>
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| <math>\lambda</math> is sometimes called the [[noncentrality parameter]]. Note that some references define <math>\lambda</math> in other ways, such as half of the above sum, or its square root.
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| This distribution arises in [[multivariate statistics]] as a derivative of the [[multivariate normal distribution]]. While the central [[chi-squared distribution]] is the squared [[Euclidean distance|norm]] of a [[random vector]] with <math>N(0_k,I_k)</math> distribution (i.e., the squared distance from the origin of a point taken at random from that distribution), the non-central <math>\chi^2</math> is the squared norm of a random vector with <math>N(\mu,I_k)</math> distribution. Here <math>0_k</math> is a zero vector of length ''k'', <math>\mu = (\mu_1, ..., \mu_k)</math> and <math>I_k</math> is the [[identity matrix]] of size ''k''.
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| ==Definition==
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| The [[probability density function]] (pdf) is given by
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| :<math>
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| f_X(x; k,\lambda) = \sum_{i=0}^\infty \frac{e^{-\lambda/2} (\lambda/2)^i}{i!} f_{Y_{k+2i}}(x),
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| </math>
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| where <math>Y_q</math> is distributed as chi-squared with <math>q</math> degrees of freedom.
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| From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted [[mixture density|mixture]] of central chi-squared distributions. Suppose that a random variable ''J'' has a [[Poisson distribution]] with mean <math>\lambda/2</math>, and the [[conditional distribution]] of ''Z'' given <math>J=i</math> is chi-squared with ''k+2i'' degrees of freedom. Then the [[marginal distribution|unconditional distribution]] of ''Z'' is non-central chi-squared with ''k'' degrees of freedom, and non-centrality parameter <math>\lambda</math>.
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| Alternatively, the pdf can be written as
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| :<math>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})</math>
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| where <math>I_\nu(z)</math> is a modified [[Bessel function]] of the first kind given by
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| :<math> I_a(y) = (y/2)^a \sum_{j=0}^\infty \frac{ (y^2/4)^j}{j! \Gamma(a+j+1)} .</math>
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| Using the relation between [[Bessel functions]] and [[hypergeometric functions]], the pdf can also be written as:<ref>Muirhead (2005) Theorem 1.3.4</ref>
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| :<math>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}.</math>
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| Siegel (1979) discusses the case ''k''=0 specifically (zero degrees of freedom), in which case the distribution has a discrete component at zero.
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| == Properties ==
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| ===Moment generating function===
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| The [[moment generating function]] is given by
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| :<math>M(t;k,\lambda)=\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}.</math>
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| ===Moments===
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| The first few raw [[moment (mathematics)|moment]]s are:
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| :<math>\mu^'_1=k+\lambda</math>
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| :<math>\mu^'_2=(k+\lambda)^2 + 2(k + 2\lambda) </math>
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| :<math>\mu^'_3=(k+\lambda)^3 + 6(k+\lambda)(k+2\lambda)+8(k+3\lambda)</math>
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| :<math>\mu^'_4=(k+\lambda)^4+12(k+\lambda)^2(k+2\lambda)+4(11k^2+44k\lambda+36\lambda^2)+48(k+4\lambda)</math>
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| The first few central [[moment (mathematics)|moment]]s are:
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| :<math>\mu_2=2(k+2\lambda)\,</math>
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| :<math>\mu_3=8(k+3\lambda)\,</math>
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| :<math>\mu_4=12(k+2\lambda)^2+48(k+4\lambda)\,</math>
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| The ''n''th [[cumulant]] is
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| :<math>K_n=2^{n-1}(n-1)!(k+n\lambda).\,</math>
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| Hence
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| :<math>\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}. </math>
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| ===Cumulative distribution function===
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| Again using the relation between the central and noncentral chi-squared distributions, the [[cumulative distribution function]] (cdf) can be written as
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| :<math>P(x; k, \lambda ) = e^{-\lambda/2}\; \sum_{j=0}^\infty \frac{(\lambda/2)^j}{j!} Q(x; k+2j)</math>
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| where <math>Q(x; k)\,</math> is the cumulative distribution function of the central chi-squared distribution with ''k'' degrees of freedom which is given by
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| :<math>Q(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,</math>
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| :and where <math>\gamma(k,z)\,</math> is the [[incomplete Gamma function|lower incomplete Gamma function]].
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| The [[Marcum Q-function]] <math>Q_M(a,b)</math> can also be used to represent the cdf.<ref>Nuttall, Albert H. (1975): ''[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1055327 Some Integrals Involving the Q<sub>M</sub> Function]'', ''IEEE Transactions on Information Theory'', 21(1), 95-96, {{ISSN|0018-9448}}</ref>
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| :<math>P(x; k, \lambda) = 1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)</math>
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| ====Approximation====
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| Sankaran <ref>Sankaran , M. (1963). [http://biomet.oxfordjournals.org/cgi/content/citation/50/1-2/199 Approximations to the non-central chi-squared distribution] ''[[Biometrika]]'', 50(1-2), 199–204</ref> discusses a number of [[Closed-form expression|closed form]] [[approximation]]s for the [[cumulative distribution function]]. In an earlier paper,<ref>Sankaran , M. (1959). "On the non-central chi-squared distribution", ''[[Biometrika]]'' 46, 235–237</ref> he derived and states the following approximation:
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| :<math> 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\} </math>
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| where
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| :<math> \Phi \lbrace \cdot \rbrace \, </math> denotes the [[cumulative distribution function]] of the [[standard normal]] distribution;
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| :<math> h = 1 - \frac{2}{3} \frac{(k+ \lambda) (k+ 3 \lambda)}{(k+ 2 \lambda) ^ 2} \, ;</math>
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| :<math> p = \frac{k+ 2 \lambda}{(k+ \lambda) ^ 2} ;</math>
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| :<math> m = (h - 1) (1 - 3 h) \, .</math>
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| This and other approximations are discussed in a later text book.<ref>Johnson et al. (1995) Section 29.8</ref>
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| To approximate the [[Chi-squared distribution]], the non-centrality parameter, <math> \lambda\, </math>, is set to zero, yielding
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| :<math> P(x; k, \lambda ) \approx \Phi \left\{ \frac{(\frac{x}{k} ) ^{1/3} - (1 - \frac{2}{9k}) } {\sqrt{\frac{2}{9k}} } \right\} , </math>
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| essentially approximating the normalized Chi-squared distribution ''X'' / ''k'' as the cube of a Gaussian.
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| <!-- : <math> Y^3 </math>, where <math>Y \sim \mathcal{N}(1-\frac{2}{9k},\sqrt{\frac{2}{9k}})</math> -->
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| For a given probability, the formula is easily inverted to provide the corresponding approximation for <math> x\, </math>.
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| <!-- ===Characteristic function===
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| The [[Characteristic function (probability theory)|characteristic function]] of the noncentral chi-squared distribution is <ref>{{cite web | url=http://www.planetmathematics.com/CharNonChi.pdf | title=Characteristic function of the noncentral chi-squared distribution | author=M.A. Sanders | accessdate=2009-03-07}}</ref>
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| :<math>
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| \chi(t;k)
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| =(1-i2t)^{-k/2}e^{\frac{it\sum_{j=1}^k(\mu_j/\sigma_j)^2}{1-i2t}}
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| =(1-i2t)^{-k/2}e^{\frac{it\lambda}{1-i2t}}.
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| </math> -->
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| ==Derivation of the pdf==
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| The derivation of the probability density function is most easily done by performing the following steps:
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| # First, assume without loss of generality that <math>\sigma_1=\ldots=\sigma_k=1</math>. Then the joint distribution of <math>X_1,\ldots,X_k</math> is spherically symmetric, up to a location shift.
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| # The spherical symmetry then implies that the distribution of <math>X=X_1^2+\ldots+X_k^2</math> depends on the means only through the squared length, <math>\lambda=\mu_1^2+\ldots+\mu_k^2</math>. Without loss of generality, we can therefore take <math>\mu_1=\sqrt{\lambda}</math> and <math>\mu_2=\dots=\mu_k=0</math>.
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| # Now derive the density of <math>X=X_1^2</math> (i.e. the ''k''=1 case). Simple transformation of random variables shows that
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| :::<math>\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}</math>
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| ::where <math>\phi(\cdot)</math> is the standard normal density.
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| # Expand the [[hyperbolic function|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+2''i'' in this case.
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| # Finally, for the general case. We've assumed, without loss of generality, that <math>X_2,\ldots,X_k</math> are standard normal, and so <math>X_2^2+\ldots+X_k^2</math> has a ''central'' chi-squared distribution with (''k''-1) degrees of freedom, independent of <math>X_1^2</math>. Using the poisson-weighted mixture representation for <math>X_1^2</math>, 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+2''i'')+(''k''-1) = ''k''+2''i'' as required.
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| ==Related distributions==
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| *If <math>V</math> is [[chi-squared distribution|chi-squared]] distributed <math>V \sim \chi_k^2</math> then <math>V</math> is also non-central chi-squared distributed: <math>V \sim {\chi'}^2_k(0)</math>
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| *If <math>V_1 \sim {\chi'}_{k_1}^2(\lambda)</math> and <math>V_2 \sim {\chi'}_{k_2}^2(0)</math> and <math>V_1</math> is independent of <math>V_2</math> then a [[noncentral F distribution|noncentral ''F''-distributed]] variable is developed as <math>\frac{V_1/k_1}{V_2/k_2} \sim F'_{k_1,k_2}(\lambda)</math>
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| *If <math>J \sim Poisson(\frac{\lambda}{2})</math>, then <math>\chi_{k+2J}^2 \sim {\chi'}_k^2(\lambda)</math>
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| *Normal approximation:<ref>Muirhead (2005) pages 22–24 and problem 1.18.</ref> if <math>V \sim {\chi'}^2_k(\lambda)</math>, then <math>\frac{V-(k+\lambda)}{\sqrt{2(k+2\lambda)}}\to N(0,1)</math> in distribution as either <math>k\to\infty</math> or <math>\lambda\to\infty</math>.
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| ===Transformations===
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| Sankaran (1963) discusses the transformations of the form
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| <math>z=[(X-b)/(k+\lambda)]^{1/2}</math>.
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| He analyzes the expansions of the [[cumulants]] of <math>z</math> up to the term <math>O((k+\lambda)^{-4})</math> and shows that the following choices of <math>b</math> produce reasonable results:
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| * <math>b=(k-1)/2</math> makes the second cumulant of <math>z</math> approximately independent of <math>\lambda</math>
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| * <math>b=(k-1)/3</math> makes the third cumulant of <math>z</math> approximately independent of <math>\lambda</math>
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| * <math>b=(k-1)/4</math> makes the fourth cumulant of <math>z</math> approximately independent of <math>\lambda</math>
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| Also, a simpler transformation <math>z_1 = (X-(k-1)/2)^{1/2}</math> can be used as a [[variance stabilizing transformation]] that produces a random variable with mean <math>(\lambda + (k-1)/2)^{1/2}</math> and variance <math>O((k+\lambda)^{-2})</math>.
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| Usability of these transformations may be hampered by the need to take the square roots of negative numbers.
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| <center>
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| {| class="wikitable" align="center"
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| |+ '''Various chi and chi-squared distributions'''
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| |-
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| ! Name !! Statistic
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| |-
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| | [[chi-squared distribution]] || <math>\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2</math>
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| |-
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| | noncentral chi-squared distribution || <math>\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
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| |-
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| | [[chi distribution]] || <math>\sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
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| |-
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| | [[noncentral chi distribution]] || <math>\sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
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| |}
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| </center>
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| ==Notes==
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| <references/>
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| == References ==
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| * Abramowitz, M. and Stegun, I.A. (1972), ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]]'', Dover. [http://www.math.sfu.ca/~cbm/aands/page_942.htm Section 26.4.25.]
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| * Johnson, N. L., Kotz, S., Balakrishnan, N. (1970), ''Continuous Univariate Distributions, Volume 2'', Wiley. ISBN 0-471-58494-0
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| * Muirhead, R. (2005) ''Aspects of Multivariate Statistical Theory'' (2nd Edition). Wiley. ISBN 0-471-76985-1
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| * Siegel, A.F. (1979), "The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity", ''[[Biometrika]]'', 66, 381–386
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| * {{Citation
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| | title = Linear combinations of non-central chi-squared variates
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| | jstor = 2238621
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| | year = 1966
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| | author = Press, S.J.
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| | journal = The Annals of Mathematical Statistics
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| | pages = 480–487
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| | volume = 37
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| | issue = 2}}
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| == External Links ==
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| * [http://itfeature.com/probability/non-central-chi-square-distribution Non central chi squared distribution] - from itfeature.com
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Noncentral Chi-Squared Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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