Noncentral chi squared distribution - Alchetron, the free social encyclopedia



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 χ 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

Let ( X 1 , X 2 , … , X i , … , X k ) be k independent, normally distributed random variables with means μ i and unit variances. Then the random variable

∑ i = 1 k X i 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 λ which is related to the mean of the random variables X i by:

λ = ∑ i = 1 k μ i 2 .

λ is sometimes called the noncentrality parameter. Note that some references define λ 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 χ 2 is the squared norm of a random vector with N ( μ , I k ) distribution. Here 0 k is a zero vector of length k, μ = ( μ 1 , … , μ k ) and I k is the identity matrix of size k.

Definition

The probability density function (pdf) is given by

f X ( x ; k , λ ) = ∑ i = 0 ∞ e − λ / 2 ( λ / 2 ) i i ! f Y k + 2 i ( 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 λ / 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 λ .

Alternatively, the pdf can be written as

f X ( x ; k , λ ) = 1 2 e − ( x + λ ) / 2 ( x λ ) k / 4 − 1 / 2 I k / 2 − 1 ( λ x )

where I ν ( y ) is a modified Bessel function of the first kind given by

I ν ( y ) = ( y / 2 ) ν ∑ j = 0 ∞ ( y 2 / 4 ) j j ! Γ ( ν + j + 1 ) .

Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:

f X ( x ; k , λ ) = e − λ / 2 0 F 1 ( ; k / 2 ; λ x / 4 ) 1 2 k / 2 Γ ( k / 2 ) 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.

Moment generating function

The moment generating function is given by

M ( t ; k , λ ) = exp ⁡ ( λ t 1 − 2 t ) ( 1 − 2 t ) k / 2 .

Moments

The first few raw moments are:

μ 1 ′ = k + λ μ 2 ′ = ( k + λ ) 2 + 2 ( k + 2 λ ) μ 3 ′ = ( k + λ ) 3 + 6 ( k + λ ) ( k + 2 λ ) + 8 ( k + 3 λ ) μ 4 ′ = ( k + λ ) 4 + 12 ( k + λ ) 2 ( k + 2 λ ) + 4 ( 11 k 2 + 44 k λ + 36 λ 2 ) + 48 ( k + 4 λ )

The first few central moments are:

μ 2 = 2 ( k + 2 λ ) μ 3 = 8 ( k + 3 λ ) μ 4 = 12 ( k + 2 λ ) 2 + 48 ( k + 4 λ )

The nth cumulant is

K n = 2 n − 1 ( n − 1 ) ! ( k + n λ ) .

Hence

μ n ′ = 2 n − 1 ( n − 1 ) ! ( k + n λ ) + ∑ j = 1 n − 1 ( n − 1 ) ! 2 j − 1 ( n − j ) ! ( k + j λ ) μ n − j ′ .

Cumulative distribution function

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

P ( x ; k , λ ) = e − λ / 2 ∑ j = 0 ∞ ( λ / 2 ) j j ! Q ( x ; k + 2 j )

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 ) = γ ( k / 2 , x / 2 ) Γ ( k / 2 ) and where γ ( k , z ) is the lower incomplete Gamma function.

The Marcum Q-function Q M ( a , b ) can also be used to represent the cdf.

P ( x ; k , λ ) = 1 − Q k 2 ( λ , x )

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:

P ( x ; k , λ ) ≈ Φ { ( x k + λ ) h − ( 1 + h p ( h − 1 − 0.5 ( 2 − h ) m p ) ) h 2 p ( 1 + 0.5 m p ) }

where

Φ { ⋅ } denotes the cumulative distribution function of the standard normal distribution; h = 1 − 2 3 ( k + λ ) ( k + 3 λ ) ( k + 2 λ ) 2 ; p = k + 2 λ ( k + λ ) 2 ; m = ( h − 1 ) ( 1 − 3 h ) .

This and other approximations are discussed in a later text book.

To approximate the chi-squared distribution, the non-centrality parameter, λ , is set to zero, yielding

P ( x ; k , λ ) ≈ Φ { ( x k ) 1 / 3 − ( 1 − 2 9 k ) 2 9 k } ,

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

The pdf of the noncentral chi-squared distribution is a solution of the following differential equation:

{ 4 x f ″ ( x ) + ( − 2 k + 4 x + 8 ) f ′ ( x ) + f ( x ) ( − k − λ + x + 4 ) = 0 f ( 1 ) 2 k / 2 e λ + 1 2 = 0 F ~ 1 ( ; k 2 ; λ 4 ) λ 0 F ~ 1 ( ; k 2 + 1 ; λ 4 ) + 2 ( k − 3 ) 0 F ~ 1 ( ; k 2 ; λ 4 ) = 2 k 2 + 2 e λ + 1 2 f ′ ( 1 ) }

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 of X 1 , … , X k is spherically symmetric, up to a location shift.
The spherical symmetry then implies that the distribution of X = X 1 2 + ⋯ + X k 2 depends on the means only through the squared length, λ = μ 1 2 + ⋯ + μ k 2 . Without loss of generality, we can therefore take μ 1 = λ and μ 2 = ⋯ = μ k = 0 .
Now derive the density of X = X 1 2 (i.e. the k = 1 case). Simple transformation of random variables shows that

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 are standard normal, and so X 2 2 + ⋯ + 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.

If V is chi-squared distributed V ∼ χ k 2 then V is also non-central chi-squared distributed: V ∼ χ ′ k 2 ( 0 )

If V 1 ∼ χ ′ k 1 2 ( λ ) and V 2 ∼ χ ′ k 2 2 ( 0 ) and V 1 is independent of V 2 then a noncentral F-distributed variable is developed as V 1 / k 1 V 2 / k 2 ∼ F k 1 , k 2 ′ ( λ )

If J ∼ P o i s s o n ( λ ) , then χ k + 2 J 2 ∼ χ ′ k 2 ( λ )

If V ∼ χ ′ 2 2 ( λ ) , then V takes the Rice distribution with parameter λ .

Normal approximation: if V ∼ χ ′ k 2 ( λ ) , then V − ( k + λ ) 2 ( k + 2 λ ) → N ( 0 , 1 ) in distribution as either k → ∞ or λ → ∞ .

Transformations

Sankaran (1963) discusses the transformations of the form z = [ ( X − b ) / ( k + λ ) ] 1 / 2 . He analyzes the expansions of the cumulants of z up to the term O ( ( k + λ ) − 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 λ

b = ( k − 1 ) / 3 makes the third cumulant of z approximately independent of λ

b = ( k − 1 ) / 4 makes the fourth cumulant of z approximately independent of λ

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 ( λ + ( k − 1 ) / 2 ) 1 / 2 and variance O ( ( k + λ ) − 2 ) .

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.

References

Noncentral chi-squared distribution Wikipedia

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