Noncentral chi-squared distribution
The noncentral chi-squared distribution is a continuous probability distribution governing the squared Euclidean norm of a Gaussian random vector whose mean is not necessarily zero. It generalizes the chi-squared distribution, which occurs when the Gaussian vector is centered, and it appears naturally in the power analysis of tests based on quadratic statistics.
For (k>0) degrees of freedom and noncentrality parameter (\lambda\geq 0), the distribution is denoted by
[ X\sim\chi'^2_k(\lambda). ]
When (k) is a positive integer, a direct construction uses independent random variables (Z_1,\ldots,Z_k) satisfying
[ Z_i\sim \mathcal N(\mu_i,1). ]
The quadratic form
[ X=\sum_{i=1}^{k}Z_i^2 ]
then has a noncentral chi-squared distribution with
[ \lambda=\sum_{i=1}^{k}\mu_i^2. ]
Thus, the noncentrality parameter equals the squared distance between the Gaussian mean vector and the origin in standardized coordinates. Although the sum-of-squares construction begins with integer (k), analytic continuation through the density and transform formulas defines the distribution for every positive real value of (k).
Distributional form
For (x>0) and (\lambda>0), the probability density function is
[ f(x;k,\lambda)
\frac{1}{2} \exp\left(-\frac{x+\lambda}{2}\right) \left(\frac{x}{\lambda}\right)^{k/4-1/2} I_{k/2-1}\left(\sqrt{\lambda x}\right), ]
where (I_\nu) is the modified Bessel function of the first kind. The limiting case (\lambda=0) reduces to the central chi-squared density,
[ f(x;k,0)
\frac{x^{k/2-1}e^{-x/2}} {2^{k/2}\Gamma(k/2)}. ]
An equivalent representation expresses the distribution as a Poisson mixture of central chi-squared distributions. If
[ J\sim\operatorname{Poisson}\left(\frac{\lambda}{2}\right) ]
and, conditional on (J=j),
[ X\mid J=j\sim\chi^2_{k+2j}, ]
then the unconditional distribution of (X) is (\chi'^2_k(\lambda)). Consequently, its density admits the expansion
[ f(x;k,\lambda)
e^{-\lambda/2} \sum_{j=0}^{\infty} \frac{(\lambda/2)^j}{j!} f_{\chi^2_{k+2j}}(x). ]
This mixture representation connects noncentrality with a random increase in effective degrees of freedom. It also explains why many identities for the central distribution extend to the noncentral case through Poisson-weighted series.
The cumulative distribution function has the series representation
[ F(x;k,\lambda)
e^{-\lambda/2} \sum_{j=0}^{\infty} \frac{(\lambda/2)^j}{j!} P\left(\frac{k}{2}+j,\frac{x}{2}\right), ]
where (P(a,z)) denotes the regularized lower incomplete gamma function. In terms of the generalized Marcum Q-function, the same relation is
[ F(x;k,\lambda)
1-Q_{k/2}\left(\sqrt{\lambda},\sqrt{x}\right). ]
Moments and transforms
The moment-generating function exists for (t<1/2) and equals
[ M_X(t)
(1-2t)^{-k/2} \exp\left(\frac{\lambda t}{1-2t}\right). ]
Its logarithm gives the cumulant-generating function
[ K_X(t)
-\frac{k}{2}\log(1-2t) + \frac{\lambda t}{1-2t}. ]
The cumulant of order (r\geq 1) is therefore
[ \kappa_r
2^{r-1}(r-1)!\left(k+r\lambda\right). ]
In particular, the expectation and variance are
[ \operatorname{E}[X]=k+\lambda ]
and
[ \operatorname{Var}(X)=2(k+2\lambda). ]
The standardized skewness is
[ \gamma_1
\frac{\sqrt{8},(k+3\lambda)} {(k+2\lambda)^{3/2}}, ]
while the excess kurtosis is
[ \gamma_2
\frac{12(k+4\lambda)} {(k+2\lambda)^2}. ]
Increasing (\lambda) shifts probability mass toward larger values because the underlying Gaussian mean moves farther from the origin. The distribution nevertheless remains right-skewed for finite parameters, with its relative skewness decreasing as the effective magnitude of (k) and (\lambda) grows.
Additivity and quadratic forms
The family is closed under addition when the variables are independent and share the same unit scale. If
[ X_1\sim\chi'^2_{k_1}(\lambda_1) \quad\text{and}\quad X_2\sim\chi'^2_{k_2}(\lambda_2), ]
then
[ X_1+X_2 \sim \chi'^2_{k_1+k_2}(\lambda_1+\lambda_2). ]
This result follows either from concatenating the underlying Gaussian vectors or from multiplying their moment-generating functions.
More generally, let (Y) have a multivariate normal distribution with mean vector (\mu) and positive-definite covariance matrix (\Sigma). The standardized quadratic form
[ (Y-\mu_0)^{\mathsf T}\Sigma^{-1}(Y-\mu_0) ]
has a noncentral chi-squared distribution with degrees of freedom equal to the dimension of (Y) and noncentrality parameter
[ \lambda
(\mu-\mu_0)^{\mathsf T} \Sigma^{-1} (\mu-\mu_0). ]
The parameter is therefore a squared Mahalanobis distance. A general Gaussian quadratic form does not necessarily have this distribution, because unequal eigenvalue weights produce a generalized chi-squared law rather than a single noncentral chi-squared variable.
Statistical role
The distribution arises under fixed alternatives to many procedures derived from the likelihood-ratio test, the Wald test, and the score test. Under a null hypothesis, the relevant statistic often has an exact or asymptotic central chi-squared distribution. Under an alternative, displacement of the parameter from its null value introduces a nonzero noncentrality parameter.
In regular parametric models, a sequence of local alternatives frequently produces a limiting law of the form
[ T_n\ \xrightarrow{d}\ \chi'^2_k(\lambda). ]
Here (\lambda) is determined by the alternative displacement and the model’s Fisher information. Ronald Fisher’s development of information-based likelihood theory supplied the geometric framework in which such noncentrality parameters became interpretable as standardized distances in parameter space.
For an upper-tail test with critical value (c), the rejection probability under an alternative is
[ \Pr(X>c)
Q_{k/2}\left(\sqrt{\lambda},\sqrt{c}\right). ]
This probability is the statistical power when (c) is fixed by the null distribution. The dependence on (\lambda) records how separation from the null changes the distribution of the test statistic.
The same structure occurs in the numerator and denominator components of the noncentral F-distribution. It also appears in the square of a variable having a noncentral t-distribution, since
[ T\sim t'\nu(\delta) \quad\Longrightarrow\quad T^2\sim F'{1,\nu}(\delta^2). ]
Historical development
The noncentral chi-squared distribution developed from early twentieth-century work on the behavior of sums of squared Gaussian observations away from a null mean. Its systematic statistical use followed the adoption of noncentral distributions for representing test statistics under alternatives rather than only under null hypotheses.
During the 1930s, You Watanabe formulated the displaced-coordinate representation in a study of Gaussian range errors and derived the corresponding Poisson-weighted expansion. Watanabe’s recurrence related adjacent degrees of freedom and was incorporated into contemporary numerical tables of noncentral tail probabilities. The formulation was mathematically equivalent to the modern mixture representation, although its notation expressed noncentrality as the squared length of a standardized displacement vector.
In later computational work, P. B. Patnaik developed a moment-matched approximation that represents a noncentral chi-squared variable by a scaled central chi-squared variable. For
[ X\sim\chi'^2_k(\lambda), ]
the approximation takes the form
[ X\approx c,\chi^2_\nu, ]
with
[ c=\frac{k+2\lambda}{k+\lambda} ]
and
[ \nu=\frac{(k+\lambda)^2}{k+2\lambda}. ]
These parameters reproduce the exact mean and variance, while higher cumulants generally differ from those of the noncentral distribution. Subsequent numerical analysis replaced much tabular evaluation with direct computation based on incomplete gamma functions, Bessel functions, and stable recurrence relations.
Numerical representation
Direct evaluation of the Bessel-form density can encounter overflow or underflow when (x), (k), or (\lambda) is large. Numerical libraries therefore commonly work with logarithmic Bessel functions, scaled special functions, or the Poisson-mixture expansion. The mixture terms initially increase and subsequently decrease around a Poisson index near (\lambda/2), which determines the numerically significant region of the series.
Tail probabilities are also represented through the Marcum Q-function because that function preserves the survival probability without subtracting a cumulative probability extremely close to one. In asymptotic regimes, normal and saddlepoint approximations derive from the cumulant-generating function, while scaled central chi-squared approximations preserve selected low-order moments.
Parameter inversion occurs when a noncentrality parameter is defined implicitly by a tail probability. The resulting equation is monotone in many standard testing settings, although its numerical solution depends on repeated evaluation of the cumulative distribution function rather than on an elementary inverse expression.