Noncentral Chi DistributionEdit
The noncentral chi distribution is a fundamental object in probability theory and statistics that describes the magnitude of a k-dimensional Gaussian vector with a nonzero mean. Concretely, if Z1, Z2, ..., Zk are independent normal variables with means μi and unit variance, and the length of the mean vector is λ = sqrt(μ1^2 + μ2^2 + ... + μk^2), then the random variable X = sqrt(Z1^2 + Z2^2 + ... + Zk^2) has a noncentral chi distribution with degrees of freedom k and noncentrality parameter λ. When the noncentrality parameter is zero, this reduces to the central chi distribution, which arises from the magnitude of a purely zero-mean k-dimensional Gaussian vector.
The noncentral chi distribution is closely related to several other distributions. In particular, X^2 follows a noncentral chi-square distribution with the same degrees of freedom k and noncentrality parameter λ^2. The case k = 2 is of particular practical importance because it coincides with the Rician distribution, which is widely used to model the envelope of a complex Gaussian signal with a line-of-sight component Rician distribution. For λ = 0, X reduces to the central chi distribution with k degrees of freedom, a classical distribution in hypothesis testing and estimation.
Definition
Let X be defined as X = sqrt(∑_{i=1}^k Z_i^2) where Z_i ~ N(μ_i, 1) independently and the vector of means has length λ = sqrt(∑ μ_i^2). Then X is said to have a noncentral chi distribution with parameters k and λ. A convenient special case is when the Z_i are identically distributed with mean components that collectively yield λ^2 = ∑ μ_i^2.
The noncentral chi distribution is often introduced via its connection to the noncentral chi-square distribution: if Y ~ noncentral chi-square(k, λ^2), then X = sqrt(Y) has X ~ noncentral chi(k, λ). For many theoretical developments and practical computations, this relationship is a guiding principle.
Probability density function
The probability density function (pdf) of X ~ noncentral chi(k, λ) for x ≥ 0 is
f(x; k, λ) = x^{k-1} exp(-(x^2 + λ^2)/2) I_{k/2 - 1}(λ x),
where I_ν denotes the modified Bessel function of the first kind of order ν. This compact form specializes correctly to known cases; for example, when k = 2, f(x; 2, λ) = x exp(-(x^2 + λ^2)/2) I_0(λ x), which is the Rice distribution with the appropriate parameters.
A commonly used alternative representation is as a Poisson mixture of central chi densities:
f(x; k, λ) = ∑{n=0}^∞ e^{-λ^2/2} (λ^n / n!) f{k+n}(x),
where f_{m}(x) denotes the pdf of the central chi distribution with m degrees of freedom. This mixture form makes transparent the way nonzero mean adds weight to higher-dimension central chi components.
References to the Bessel function and its properties are central here; see the modified Bessel function of the first kind for more details on I_ν and its asymptotics.
Cumulative distribution function
The cumulative distribution function (CDF) of X can be expressed in terms of a Poisson-weighted mixture of the CDFs of central chi distributions:
F(x; k, λ) = ∑{n=0}^∞ e^{-λ^2/2} (λ^n / n!) F{k+n}(x),
where F_{m}(x) is the CDF of the central chi distribution with m degrees of freedom. This representation is useful for both theoretical considerations and numerical computation, since it reduces the noncentral problem to a sequence of central chi evaluations.
Moments and properties
- Existence: All moments of X exist and can be expressed in terms of series involving Bessel functions, often requiring numerical evaluation for general k and λ.
- Limiting cases: As λ → 0, the pdf and distribution converge to those of the central chi distribution with k degrees of freedom; as k grows large, asymptotic approximations can be obtained under appropriate scaling.
- Relationship to other distributions: The square X^2 has a noncentral chi-square distribution with the same k and λ^2. For k = 2, the distribution reduces to the Rice family of envelopes; for more general k, it generalizes those envelope models to higher dimensions.
Computational methods for the noncentral chi distribution typically rely on (i) the central chi components via the mixture representation or (ii) direct evaluation of the Bessel-function form, with careful handling of numerical stability for extreme parameter values. See Chi distribution for the central case and Noncentral chi-square distribution for the squared form.
Applications
- Signal processing and communications: The noncentral chi distribution describes the magnitude of a Gaussian vector with a deterministic component, which captures scenarios with a strong line-of-sight or direct-path signal amid noise. The special case k = 2 corresponds to common envelope models used in radar and wireless communications, linked to the Rician distribution.
- Hypothesis testing and multivariate analysis: In problems involving the norm of a Gaussian vector with a nonzero mean, the noncentral chi distribution naturally arises, informing power calculations and detector performance.
- Physics and engineering: The distribution appears in various models where a magnitude is formed from independent Gaussian components with nonzero means, providing a convenient closed form for analytic work and simulations.