- Noncentral chi-square distribution
Probability distribution
name =Noncentral chi-square
type =density
pdf_
cdf_
parameters = degrees of freedom
non-centrality parameter
support =
pdf =
cdf =:| mean =
median =
mode =
variance =
skewness =
kurtosis =
entropy =
mgf = for
char =In
probability theory andstatistics , the noncentral chi-square or noncentral distribution is a generalization of thechi-square distribution . If are "k" independent, normally distributed random variables with means and variances , then the random variable:
is distributed according to the noncentral chi-square distribution. The noncentral chi-square distribution has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), and which is related to the mean of the random variables by:
:
Note that some references define as one half of the above sum.
Properties
The probability density function is given by:where is distributed as chi-square with 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 , and theconditional distribution of "Z" given 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 .Alternatively, the pdf can be written as:
where is a modified
Bessel function of the first kind given by:
The
moment generating function is given by:
The first few raw moments are:
::::
The first few central moments are:
:::
The "n"th
cumulant is:
Hence:
Again using the relation between the central and noncentral chi-square distributions, the
cumulative distribution function (cdf) can be written as:
where is the cumulative distribution function of the central chi-squared distribution which is given by
:
where is the lower incomplete Gamma function.
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 . Then the joint distribution of is spherically symmetric, up to a location shift.
# The spherical symmetry then implies that the distribution of depends on the means only through the squared length, . Without loss of generality, we can therefore take and .
# Now derive the density of (i.e. "k=1" case). Simple transformation of random variables shows that :
where is the standard normal density.
# 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, wlog, that are standard normal, and so has a "central" chi-squared distribution with "(k-1)" degrees of freedom, independent of . Using the poisson-weighted mixture representation for , 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.Related distributions
*If is chi-square distributed then is also non-central chi-square distributed:
*If , then
Software
Many statistical software packages and libraries include functions for computing noncentral chisquare densities and probabilities. The table below gives commands for the following example problems:
# Density with "x=5.0", "k=3", (Answer: 0.097257)
# Cumulative probability with "x=5.0", "k=3", (Answer: 0.649285)
# Quantile: Find "x" in with "k=3", , "a=0.5" (Answer: 3.668745)
# Random numbers: Generate 100 random observations from the distribution with "k=3",Note: Any software that produces the answers 0.101384, 0.490071, 5.09848 for the first three problems is including a factor of 0.5 in the definition of the noncentrality parameter. This is standard in statistics texts (e.g. [R. Christensen, "Plane Answers to Complex Questions" (3rd edition, 2002), Springer, NY, p.424.] ), but apparently not among programmers who don't read before writing their code.
References
* 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.
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