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DISTRIBUTIONS / CONTINUOUS / NON CENTRAL CHI SQUARE

Non Central Chi Square distribution

Portrait

The noncentral chi-square is a sum of squared normal variables whose means need not be zero. lambda measures total displacement and n the degrees of freedom.

Historical trail

It arose from quadratic statistics under alternatives, where the underlying normal vector has non-zero mean. It is fundamental for test power and signal models.

Two useful connections

  • The central case is Gamma and ordinary chi-square. Noncentral chi-square connects to noncentral F and noncentral t when ratios and square roots of these sums enter test statistics.
  • As lambda → 0 it connects smoothly to the central family, but estimating noncentrality from one sample can be difficult.

A use case

In a test, lambda represents effect or signal under the alternative, not additional noise. It gives rejection probabilities when the null is false.

Diagnostic advice

Do not confuse lambda with a Poisson rate or with loc: they change different mechanisms.

Squares retain the trace of a signal

In central Chi Square, the Normal variables being squared have zero means. If they contain a signal, those means become nonzero and the sum of squares becomes noncentral. The parameter lambda collects the standardized energy in those shifts.

The law can be represented as a Poisson mixture of central Chi Square variables with added degrees of freedom. This aids computation and shows how the central case returns as lambda reaches zero. In signal detection and testing, it describes the statistic under an alternative and permits power calculations. It is not a shifted Chi Square: noncentrality changes the mechanism before summation, not the origin afterwards.

Decision guide

A good candidate when: the variable is a sum of squares of Normal variables with nonzero means, as in statistical power or signal energy.

Compare it with: central Chi-square when noncentrality is compatible with zero. Do not interpret noncentrality as a simple horizontal shift.

References

Non Central Chi Square Distribution: equations and calculator

Distribution defintion

X∼NonCentralChiSquare(λ,n)X\sim\mathrm{NonCentralChiSquare}\left(\lambda,n\right)

Distribution domain

x∈[0,+∞)x\in [0,+\infty)

Parameters domain and parameters constraints

λ∈R+,n∈R+\lambda\in\mathbb{R}^{+},n\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−Qn2(λ,x)F_{X}\left(x\right)=1-Q_{\frac{n}{2}}\left(\sqrt{\lambda},\sqrt{x}\right)

Probability density function

fX(x)=12e−(x+λ)/2(xλ)n/4−1/2In/2−1(λx)f_{X}\left(x\right)=\frac{1}{2}e^{-(x+\lambda)/2} \left(\frac{x}{\lambda}\right)^{n/4-1/2}I_{n/2-1}(\sqrt{\lambda x})

Percent point function/Sample

SampleX=∑i=1n(λn+Φ−1(ui))2\text{Sample}_{X}=\sum_{i=1}^{n}\left(\sqrt{\frac{\lambda}{n}}+\Phi^{-1}\left(u_{i}\right)\right)^{2}

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=2k−1(k−1)!(n+kλ)+∑j=1k−1(k−1)!2j−1(k−j)!(n+jλ)μk−j′\mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k}f_{X}\left(x\right)dx=2^{k-1}(k-1)!(n+k\lambda)+\sum_{j=1}^{k-1}\frac{(k-1)!2^{j-1}}{(k-j)!}(n+j\lambda )\mu'_{k-j}

Parametric mean

Mean(X)=μ1′=n+λ\mathrm{Mean}(X)=\mu'_{1}=n+\lambda

Parametric variance

Variance(X)=μ2′−μ1′2=2(n+2λ)\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=2(n+2\lambda)

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=23/2(n+3λ)(n+2λ)3/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2^{3/2}(n+3\lambda)}{(n+2\lambda)^{3/2}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=12(n+4λ)(n+2λ)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{12(n+4\lambda)}{(n+2\lambda)^2}

Parametric median

Median(X)=FX−1(12)\mathrm{Median}(X)=F^{-1}_{X}\left(\frac{1}{2}\right)

Parametric mode

Mode(X)=arg⁡max⁡xfX(x)\mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right)

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution.
ui:Uniform[0,1] random varibleu_{i}:\text{Uniform[0,1] random varible}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
Iα(x):Modified Bessel function of the first kind of order α∈NI_{\alpha}\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha\in\mathbb{N}