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Parameters k > 0 {displaystyle k>0,} degrees of freedom λ > 0 {displaystyle lambda >0,} non-centrality parameter Support x ∈ [ 0 ; + ∞ ) {displaystyle xin [0;+infty ),} PDF 1 2 e − ( x + λ ) / 2 ( x λ ) k / 4 − 1 / 2 I k / 2 − 1 ( λ x ) {displaystyle {rac {1}{2}}e^{-(x+lambda )/2}left({rac {x}{lambda }}ight)^{k/4-1/2}I_{k/2-1}({sqrt {lambda x}})} CDF 1 − Q k 2 ( λ , x ) {displaystyle 1-Q_{rac {k}{2}}left({sqrt {lambda }},{sqrt {x}}ight)} with Marcum Q-function Q M ( a , b ) {displaystyle Q_{M}(a,b)} Mean k + λ {displaystyle k+lambda ,} Variance 2 ( k + 2 λ ) {displaystyle 2(k+2lambda ),} |
In probability theory and statistics, the noncentral chi-squared or noncentral
Contents
Background
Let (
is distributed according to the noncentral chi-squared distribution. It has two parameters:
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
Definition
The probability density function (pdf) is given by
where
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
Alternatively, the pdf can be written as
where
Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:
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
Moments
The first few raw moments are:
The first few central moments are:
The nth cumulant is
Hence
Cumulative distribution function
Again using the relation between the central and noncentral chi-squared distributions, the cumulative distribution function (cdf) can be written as
where
The Marcum Q-function
Approximation
Sankaran discusses a number of closed form approximations for the cumulative distribution function. In an earlier paper, he derived and states the following approximation:
where
This and other approximations are discussed in a later text book.
To approximate the chi-squared distribution, the non-centrality parameter,
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
Differential equation
The pdf of the noncentral chi-squared distribution is a solution of the following differential equation:
Derivation of the pdf
The derivation of the probability density function is most easily done by performing the following steps:
- First, assume without loss of generality that
σ 1 = ⋯ = σ k = 1 . Then the joint distribution ofX 1 , … , X k - The spherical symmetry then implies that the distribution of
X = X 1 2 + ⋯ + X k 2 λ = μ 1 2 + ⋯ + μ k 2 μ 1 = λ μ 2 = ⋯ = μ k = 0 . - Now derive the density of
X = X 1 2
- 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.
- Finally, for the general case. We've assumed, without loss of generality, that
X 2 , … , X k X 2 2 + ⋯ + X k 2 X 1 2 X 1 2
Related distributions
Transformations
Sankaran (1963) discusses the transformations of the form
Also, a simpler transformation
Usability of these transformations may be hampered by the need to take the square roots of negative numbers.
Use in tolerance intervals
Two-sided normal regression tolerance intervals can be obtained based on the noncentral chi-squared distribution. This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.