Fact-checked by Grok 2 weeks ago

Noncentral chi-squared distribution

The noncentral chi-squared distribution is a continuous probability distribution that generalizes the central chi-squared distribution, arising as the sum of squares of \nu independent Gaussian random variables, each with unit variance but possibly non-zero means \mu_i where the noncentrality parameter \lambda = \sum_{i=1}^\nu \mu_i^2 \geq 0.^[1] It is parameterized by the positive integer degrees of freedom \nu > 0 and the noncentrality parameter \lambda, with the central chi-squared distribution recovered as the special case \lambda = 0.^[2] The probability density function is given by

f(x; \nu, \lambda) = \frac{1}{2} \left( \frac{x}{\lambda} \right)^{(\nu-2)/4} \exp\left( -\frac{x + \lambda}{2} \right) I_{(\nu/2)-1} \left( \sqrt{\lambda x} \right),

for x > 0, where I_\alpha(z) denotes the modified Bessel function of the first kind of order \alpha.^[1] Equivalently, it admits a Poisson-mixture representation:

f(x; \nu, \lambda) = \sum_{k=0}^\infty \frac{e^{-\lambda/2} (\lambda/2)^k}{k!} \, f_{\chi^2_{\nu + 2k}}(x),

with f_{\chi^2_d}(x) the density of a central chi-squared random variable with d degrees of freedom.^[2] The mean is \nu + \lambda and the variance is $2(\nu + 2\lambda), both of which increase with \lambda relative to the central case.^[1] In statistical applications, the noncentral chi-squared distribution describes the sampling distribution of quadratic forms, such as chi-squared test statistics, under alternative hypotheses where the null assumption of zero means fails, enabling power calculations for tests of means and variances in normal models.^[3] It also arises in signal processing for modeling the squared envelope of non-zero-mean Gaussian noise and in thermodynamics for related quadratic functionals.^[4] Related noncentral distributions, including the noncentral t and F, extend these concepts to ratio statistics.^[2]

Definitions

Background and Motivation

The noncentral chi-squared distribution with k degrees of freedom and noncentrality parameter λ is the probability distribution of the sum of the squares of k independent normal random variables, each with mean μ_i and variance 1, where λ = ∑_{i=1}^k μ_i^2 measures the extent of deviation from zero means.^[1] When λ = 0, all means are zero, and the distribution reduces to the central chi-squared distribution with k degrees of freedom, which serves as the foundational case for many null hypothesis tests in statistics.^[1] This distribution was introduced by Ronald A. Fisher in 1928, who derived it while studying the sampling distribution of the multiple correlation coefficient and recognizing its role in non-null scenarios for variance analysis.^[5] Fisher's work highlighted its emergence from sums of squared normals under deviated expectations, laying the groundwork for broader applications in inferential statistics.^[5] The noncentral chi-squared distribution arises naturally in quadratic forms within the general linear model, where observations follow a multivariate normal distribution with non-zero means, such as in regression or analysis of variance settings.^[3] Intuitively, it models the behavior of test statistics when assessing deviations from a hypothesized mean vector, with λ quantifying the signal strength or effect size; this contrasts with the central chi-squared, which assumes no deviation under the null hypothesis and is used to evaluate type I error rates.^[6]

Probability Density Function

The noncentral chi-squared distribution arises as the distribution of the sum of squares of independent normal random variables with possibly nonzero means, generalizing the central chi-squared distribution.^[5] Its probability density function (PDF), denoted f(x; k, \lambda), is defined for x \geq 0, where k > 0 is the degrees of freedom (typically a positive integer but generalizable to real values) and \lambda \geq 0 is the noncentrality parameter measuring the squared sum of the means scaled by the variance.^[7] The standard closed-form expression for the PDF involves the modified Bessel function of the first kind and is given by

f(x; k, \lambda) = \frac{1}{2} e^{-\frac{(x + \lambda)}{2}} \left( \frac{x}{\lambda} \right)^{\frac{k-2}{4}} I_{\frac{k-2}{2}} \left( \sqrt{\lambda x} \right),

where I_{\nu}(z) denotes the modified Bessel function of the first kind of order \nu.^[7] This function is zero for x < 0 and integrates to 1 over [0, \infty), ensuring proper normalization.^[7] An alternative representation expresses the PDF as an infinite mixture of central chi-squared densities:

f(x; k, \lambda) = \sum_{j=0}^{\infty} e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} f_{k + 2j}(x),

where f_m(x) is the PDF of a central chi-squared distribution with m degrees of freedom, specifically f_m(x) = \frac{1}{2^{m/2} \Gamma(m/2)} x^{m/2 - 1} e^{-x/2} for x > 0.^[2] This Poisson-weighted series form highlights the distribution as a compound of central chi-squared components, with the mixing weights following a Poisson distribution with mean \lambda/2.^[2] For \lambda > 0, the PDF is positively skewed and right-tailed compared to the central case (\lambda = 0), with the mode shifting rightward and the mean increasing to k + \lambda.^[7]

Derivation of the PDF

The noncentral chi-squared distribution with k degrees of freedom and noncentrality parameter \lambda \geq 0 arises as the distribution of X = \sum_{i=1}^k Z_i^2, where the Z_i are independent random variables with Z_i \sim \mathcal{N}(\mu_i, 1) and \lambda = \sum_{i=1}^k \mu_i^2. This setup assumes independence among the Z_i and unit variance for each.^[2] One standard derivation of the probability density function (PDF) begins with the moment-generating function (MGF) of X. The MGF is M_X(t) = \mathbb{E}[e^{tX}] = \prod_{i=1}^k \mathbb{E}[e^{t Z_i^2}] = (1 - 2t)^{-k/2} \exp\left( \frac{\lambda t}{1 - 2t} \right) for t < 1/2. To obtain the MGF for a single Z_i^2, compute \mathbb{E}[e^{t Z_i^2}] = \int_{-\infty}^\infty e^{t z^2} \cdot \frac{1}{\sqrt{2\pi}} \exp\left( -\frac{(z - \mu_i)^2}{2} \right) dz. Completing the square in the exponent yields \frac{(z - \mu_i)^2}{2} - t z^2 = \frac{1 - 2t}{2} \left( z - \frac{\mu_i}{1 - 2t} \right)^2 - \frac{\mu_i^2 t}{1 - 2t}, so the integral evaluates to (1 - 2t)^{-1/2} \exp\left( \frac{t \mu_i^2}{1 - 2t} \right). The product over i then gives the full MGF, as the noncentrality terms combine additively.^[2] This MGF can be expanded to reveal a mixture representation:

M_X(t) = e^{-\lambda/2} \sum_{j=0}^\infty \frac{(\lambda/2)^j}{j!} (1 - 2t)^{-(k + 2j)/2}.

The term (1 - 2t)^{-(k + 2j)/2} is the MGF of a central chi-squared distribution with k + 2j degrees of freedom. Thus, X follows the distribution of a central chi-squared random variable with degrees of freedom k + 2J, where J \sim \mathrm{Poisson}(\lambda/2). The PDF of X is therefore the Poisson-weighted mixture

f(x; k, \lambda) = e^{-\lambda/2} \sum_{j=0}^\infty \frac{(\lambda/2)^j}{j!} f_{k + 2j}(x), \quad x > 0,

where f_m(x) = \frac{x^{m/2 - 1} e^{-x/2}}{2^{m/2} \Gamma(m/2)} is the PDF of the central chi-squared distribution with m degrees of freedom. This series form follows directly from the uniqueness of the MGF.^[2] An alternative approach derives the closed-form PDF by inverting the MGF, which corresponds to taking the inverse Laplace transform (with appropriate substitution s = -t). This inversion yields an expression involving the modified Bessel function of the first kind:

\begin{aligned} f(x; k, \lambda) &= \frac{1}{2} e^{-(x + \lambda)/2} \left( \frac{x}{\lambda} \right)^{(k/2 - 1)/2} I_{k/2 - 1} \left( \sqrt{\lambda x} \right), \quad x > 0, \end{aligned}

where I_\nu(z) = \sum_{j=0}^\infty \frac{1}{j! \, \Gamma(j + \nu + 1)} \left( \frac{z}{2} \right)^{2j + \nu} is the modified Bessel function of order \nu = k/2 - 1. The equivalence to the series form arises because the Poisson mixture sum matches the series expansion of the Bessel function after substituting the central chi-squared PDFs and simplifying. This univariate derivation extends naturally to the noncentral Wishart distribution, which generalizes the quadratic form to a matrix case involving a p \times p covariance matrix from a k \times p multivariate normal sample, but the details differ and are not covered here.^[2]

Mathematical Properties

Moment-Generating Function

The moment-generating function (MGF) of a random variable X following a noncentral chi-squared distribution with k degrees of freedom and noncentrality parameter \lambda is given by

M_X(t) = \exp\left(\frac{\lambda t}{1 - 2t}\right) (1 - 2t)^{-k/2}, \quad t < \frac{1}{2}.

^[8] This function encapsulates the distribution's moments and facilitates derivations of higher-order properties, such as cumulants and convolutions with other distributions.^[2] The derivation proceeds from the representation of X as the sum of k independent noncentral chi-squared random variables each with 1 degree of freedom and respective noncentrality parameters \mu_1^2, \dots, \mu_k^2, where \lambda = \sum_{i=1}^k \mu_i^2. Each component X_i = Z_i^2, with Z_i \sim N(\mu_i, 1), has MGF

M_{X_i}(t) = \exp\left(\frac{\mu_i^2 t}{1 - 2t}\right) (1 - 2t)^{-1/2}, \quad t < \frac{1}{2}.

^[2] The independence implies that the MGF of X = \sum_{i=1}^k X_i is the product

M_X(t) = \prod_{i=1}^k M_{X_i}(t) = \exp\left( \sum_{i=1}^k \frac{\mu_i^2 t}{1 - 2t} \right) (1 - 2t)^{-k/2} = \exp\left(\frac{\lambda t}{1 - 2t}\right) (1 - 2t)^{-k/2}, $$ confirming the overall form.[](https://www.math.wm.edu/~leemis/chart/UDR/PDFs/NoncentralchisquareC.pdf) Key properties of this MGF include its analytic continuation to complex arguments $t$ with $\operatorname{Re}(t) < 1/2$, ensuring uniqueness in the moment-determining region.[](https://pages.stat.wisc.edu/~shao/stat609/stat609-13.pdf) The cumulant-generating function is $\log M_X(t) = \frac{\lambda t}{1 - 2t} - \frac{k}{2} \log(1 - 2t)$, which directly yields the cumulants via differentiation.[](https://www.math.wm.edu/~leemis/chart/UDR/PDFs/NoncentralchisquareC.pdf) Notably, setting $\lambda = 0$ reduces the MGF to $(1 - 2t)^{-k/2}$, the form for the [central chi-squared distribution](/page/central_chi-squared_distribution), highlighting the noncentrality's role in shifting the distribution.[](https://pages.stat.wisc.edu/~shao/stat609/stat609-13.pdf) ### Moments and Cumulants The moments of the noncentral chi-squared distribution with $k$ degrees of freedom and noncentrality parameter $\lambda$ can be derived from its moment-generating function. The first raw moment, or mean, is $E[X] = k + \lambda$.[](https://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html)[](https://pages.stat.wisc.edu/~shao/stat609/stat609-13.pdf) The second central moment, or variance, is $\mathrm{Var}(X) = 2(k + 2\lambda)$.[](https://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html)[](https://pages.stat.wisc.edu/~shao/stat609/stat609-13.pdf) The skewness, defined as the third standardized central moment, is

\gamma_1 = \sqrt{8} \frac{k + 3\lambda}{(k + 2\lambda)^{3/2}}.

[1] For higher raw moments, a general expression is

E[X^r] = 2^r e^{-\lambda/2} \Gamma\left(r + \frac{k}{2}\right) \ {}_1F_1\left(r + \frac{k}{2}; \frac{k}{2}; \frac{\lambda}{2}\right),

where ${}_1F_1$ denotes the [confluent hypergeometric function of the first kind](/page/confluent_hypergeometric_function_of_the_first_kind).[](https://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html) For example, the second raw moment is $E[X^2] = (k + \lambda)^2 + 2(k + 2\lambda)$.[](https://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html) The cumulants $\kappa_r$ are given by

\kappa_r = 2^{r-1} (r-1)! (k + r \lambda), \quad r \geq 1.

The first two cumulants match the mean and variance: $\kappa_1 = k + \lambda$ and $\kappa_2 = 2(k + 2\lambda)$. The noncentrality parameter $\lambda$ increases both the mean and variance relative to the central chi-squared case ($\lambda = 0$), with the variance inflated by an additional $4\lambda$. This also heightens the skewness, amplifying the distribution's asymmetry, particularly for small $k$. For large $k$ or $\lambda$, the higher cumulants grow such that the distribution becomes approximately normal by the central limit theorem, as the standardized moments approach those of a Gaussian. ### Cumulative Distribution Function The cumulative distribution function (CDF) of a random variable $X$ following the noncentral chi-squared distribution with $k$ degrees of freedom and noncentrality parameter $\lambda \geq 0$ is defined as

F(x; k, \lambda) = \Pr(X \leq x) = \int_0^x f(t; k, \lambda) , dt

for $x \geq 0$, where $f(\cdot; k, \lambda)$ denotes the corresponding probability density function.[](https://www.itl.nist.gov/div898/software/dataplot/refman2/ch8/ncccdf.pdf) This integral representation highlights the CDF as the accumulation of probability mass from 0 to $x$, but it lacks a simple closed-form expression in terms of elementary functions.[](https://link.springer.com/article/10.1007/s00009-017-0874-1) An alternative series expansion provides a practical representation for computation:

F(x; k, \lambda) = \sum_{j=0}^\infty e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} , F_{k+2j}(x),

where $F_m(x)$ is the CDF of the central chi-squared distribution with $m$ degrees of freedom.[](https://www.itl.nist.gov/div898/software/dataplot/refman2/ch8/ncccdf.pdf) This Poisson-weighted mixture arises from the distributional equivalence of the noncentral chi-squared to a central chi-squared with randomly shifted degrees of freedom, where the shift follows a [Poisson distribution](/page/Poisson_distribution) with mean $\lambda/2$. The series converges for all $x \geq 0$ and $\lambda \geq 0$, though truncation is typically required for numerical evaluation.[](https://www.itl.nist.gov/div898/software/dataplot/refman2/ch8/ncccdf.pdf) The CDF possesses standard properties of continuous distributions: it is continuous and strictly monotonically increasing from $F(0; k, \lambda) = 0$ to $\lim_{x \to \infty} F(x; k, \lambda) = 1$. Due to the absence of a closed form, evaluation generally relies on the integral or series forms, with the choice depending on parameter values and computational context.[](https://link.springer.com/article/10.1007/s00009-017-0874-1) The quantile function, or inverse CDF, $Q(p; k, \lambda) = F^{-1}(p; k, \lambda)$, is the value $x_p$ such that $F(x_p; k, \lambda) = p$ for $p \in (0,1)$; it has no closed-form expression and must be obtained numerically, often via root-finding methods like bisection on the CDF.[](https://www.itl.nist.gov/div898/software/dataplot/refman2/ch8/nccppf.pdf) This function is essential for determining critical values in statistical tests, where $x_p$ defines the threshold for a given significance level $p$. Computation can be sensitive for small $p$, as the lower tail of the distribution concentrates near 0, particularly when $\lambda$ is small.[](https://www.itl.nist.gov/div898/software/dataplot/refman2/ch8/nccppf.pdf) The survival function, complementary to the CDF, is given by $\overline{F}(x; k, \lambda) = 1 - F(x; k, \lambda) = \Pr(X > x)$, which quantifies right-tail probabilities and is directly applicable in hypothesis testing scenarios.[](https://www.itl.nist.gov/div898/software/dataplot/refman2/ch8/ncccdf.pdf) This form inherits the series representation by replacing $F_{k+2j}(x)$ with its complement $1 - F_{k+2j}(x)$. ### Approximations and Numerical Computation One common approach for approximating the cumulative distribution function (CDF) of the noncentral chi-squared distribution involves a truncated series expansion as a Poisson-weighted sum of central chi-squared CDFs: $$ F(x; k, \lambda) \approx \sum_{j=0}^{m} e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} F_{k+2j}(x), $$ where $ F_{k+2j}(x) $ is the CDF of a central chi-squared distribution with $ k + 2j $ degrees of freedom, and the truncation index $ m $ is chosen such that the remainder term is below a desired tolerance, often $ m \approx \lambda/2 + \sqrt{2\lambda} $ for large noncentrality parameter $ \lambda $. This method provides error bounds of order $ O(e^{-\lambda/2} (\lambda/2)^{m+1}/(m+1)!) $, which tighten for moderate $ \lambda $ but require careful truncation for large $ \lambda > 100 $ to avoid numerical instability.[](https://cran.r-project.org/web/packages/DPQ/vignettes/Noncentral-Chisq.pdf) For large degrees of freedom $ k $ or noncentrality $ \lambda $, a normal approximation is effective, where the noncentral chi-squared random variable $ X \sim \chi^2_k(\lambda) $ is approximated by $ X \approx \mathcal{N}(k + \lambda, 2(k + 2\lambda)) $. This follows from the mean $ \mathbb{E}[X] = k + \lambda $ and variance $ \mathrm{Var}(X) = 2(k + 2\lambda) $, with refined local limit theorems improving the approximation error to $ O(r^{-2}) $ for the [survival function](/page/Survival_function) when $ r = k/2 $.[](https://arxiv.org/pdf/2201.07407) The Wilson-Hilferty cube-root transformation enhances quantile accuracy, particularly in the tails, by approximating $ (X/(k + \lambda))^{1/3} \approx \mathcal{N}(1 - 2/9(k + \lambda), 2/9(k + \lambda)) $, an extension of the central case that reduces skewness effects for moderate $ k $ and $ \lambda $. This method outperforms the plain normal approximation for percentage points, with relative errors under 1% for $ k \geq 10 $ and $ \lambda \leq 50 $.[](https://academic.oup.com/biomet/article-pdf/47/3-4/411/651133/47-3-4-411.pdf) Other approximations include the saddlepoint method, which excels for tail probabilities by expanding around the saddlepoint of the [moment-generating function](/page/Moment-generating_function), achieving relative errors below 0.1% in extreme tails for quadratic forms underlying the noncentral chi-squared. Imhof's method computes the CDF via numerical integration of the [characteristic function](/page/Characteristic_function), offering high precision (errors < 10^{-8}) for a wide range of parameters but at higher computational cost than series methods; post-2000 variants, such as those incorporating adaptive quadrature, address limitations in earlier implementations like Garwood's table-based algorithm from 1936.[](https://ecommons.cornell.edu/bitstream/1813/31905/1/BU-1311-M.pdf)[](https://www.tqmp.org/RegularArticles/vol18-3/p255/p255.pdf) Numerical computation in software libraries relies on these approximations and exact recursions. For instance, R's `pchisq` function implements Ding's Algorithm AS 275, which uses a recursive evaluation of the PDF terms to compute the CDF via integration, avoiding overflow in modified Bessel functions by forward and backward recursions with starting values from asymptotic expansions. Similarly, [SciPy](/page/SciPy)'s `stats.ncx2` employs a hybrid approach combining series summation for small $ \lambda $ and normal approximations for large $ \lambda $, with relative accuracies around 10^{-15} for most parameters. Recent advances as of 2025 include GPU-accelerated implementations for series summation in high-dimensional settings, leveraging parallel computation of Poisson terms to handle $ \lambda > 10^4 $ and $ k > 10^3 $ in seconds, as in optimized libraries for statistical simulations.[](https://cran.r-project.org/web/packages/DPQ/vignettes/Noncentral-Chisq.pdf)[](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ncx2.html)[](https://arxiv.org/html/2404.05062v3) For [quantiles](/page/Quantile), iterative methods such as [bisection](/page/Bisection) search on the approximated CDF provide robust solutions, converging in under 20 iterations for precision 10^{-10}. Direct formulas like Patnaik's approximation model the distribution as a scaled central chi-squared with adjusted [degrees of freedom](/page/Degrees_of_freedom) $ k' = \frac{(k + \lambda)^2}{k + 2\lambda} $ and [scale](/page/Scale) $ \theta = \frac{k + 2\lambda}{k + \lambda} $, yielding quantile errors under 2% for moderate $ \lambda < 20 $ and offering a simple alternative to inversion.[](https://search.r-project.org/CRAN/refmans/DPQ/html/pnchisqAppr.html) ## Related Distributions ### Central Limit and Special Cases The noncentral chi-squared distribution with parameters $k$ (degrees of freedom) and $\lambda$ (noncentrality parameter) encompasses several special cases that connect it directly to the central chi-squared distribution. When $\lambda = 0$, the distribution simplifies to the central chi-squared distribution with $k$ degrees of freedom, as there is no shift due to nonzero means in the underlying normal variables.[](https://real-statistics.com/chi-square-and-f-distributions/noncentral-chi-square-distribution/) For the specific case of $k = 1$, the random variable $X \sim \chi^2_1(\lambda)$ follows the distribution of the square of a normal random variable with mean $\sqrt{\lambda}$ and unit variance, i.e., $X = [Z]^2$ where $Z \sim N(\sqrt{\lambda}, 1)$.[](https://utstat.utoronto.ca/brunner/oldclass/appliedf16/lectures/2101f16NonCentralChisq.pdf) A key representation of the noncentral chi-squared distribution is as a Poisson mixture of central chi-squared distributions. Specifically, if $J \sim \mathrm{[Poisson](/page/Poisson)}(\lambda/2)$, then $X \stackrel{d}{=} \chi^2_{k + 2J}$, where $\chi^2_{k + 2J}$ denotes a central chi-squared random variable with $k + 2J$ [degrees of freedom](/page/Degrees_of_freedom). This mixture arises because the noncentrality introduces a random number of additional "effective" degrees of freedom, each contributing 2 to the total, weighted by the [Poisson](/page/Poisson) probabilities.[](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Chisquare.html) In limiting regimes, the noncentral chi-squared distribution exhibits normal behavior consistent with the central limit theorem. For fixed $\lambda$ and large $k$, the standardized variable converges in distribution to the standard normal:

\frac{X - (k + \lambda)}{\sqrt{2(k + 2\lambda)}} \to N(0, 1)

as $k \to \infty$. Similarly, for fixed $k$ and large $\lambda$, the same standardization applies, with the variance term $2(k + 2\lambda) \approx 4\lambda$ dominating, yielding an approximate normal distribution centered near $\lambda$ with standard deviation approximately $2\sqrt{\lambda}$. These approximations are particularly useful for large-sample inference where exact computations are infeasible.[](https://arxiv.org/pdf/2201.07407)[](https://rseri.me/publication/j020/J020.pdf) The parameters $k$ and $\lambda$ have clear interpretations in terms of the underlying model. The degrees of freedom $k$ correspond to the number of independent standard normal components in the quadratic form defining the distribution, analogous to the central case. The noncentrality parameter $\lambda$ quantifies the squared signal strength, specifically $\lambda = \sum_{i=1}^k \mu_i^2$, where $\mu_i$ are the nonzero means of the normals, measuring the deviation from the [null hypothesis](/page/Null_hypothesis) of zero means.[](https://www.sciencedirect.com/topics/mathematics/noncentral-chi) ### Transformations to Other Distributions One important monotonic transformation of the noncentral chi-squared distribution involves the square root, which links it to the noncentral chi distribution. Let $ X \sim \chi_k^2(\lambda) $ denote a noncentral chi-squared random variable with $ k $ degrees of freedom and noncentrality parameter $ \lambda \geq 0 $. Then, $ \sqrt{X} $ follows a noncentral chi distribution with the same $ k $ degrees of freedom and noncentrality parameter $ \sqrt{\lambda} $. This distribution represents the Euclidean norm of a $ k $-dimensional normal random vector with independent components $ N(\mu_i, 1) $, where $ \sum_{i=1}^k \mu_i^2 = \lambda $. In the special case $ k=2 $, the noncentral chi distribution reduces to the Rice distribution (also known as the Rician distribution), which models the amplitude of a signal with a line-of-sight component in the presence of Gaussian noise.[](http://www.ensc.sfu.ca/people/faculty/ho/ENSC805/Rayl-Rice-chi-sq.pdf) Ratio transformations provide another class of links to F-related distributions. If $ X \sim \chi_{k_1}^2(\lambda) $ and $ Y \sim \chi_{k_2}^2 $ are [independent](/page/Independent), with $ Y $ following a central chi-squared distribution with $ k_2 $ [degrees of freedom](/page/Degrees_of_freedom), then the scaled ratio

F = \frac{X / k_1}{Y / k_2}

follows a noncentral F distribution with [degrees of freedom](/page/Degrees_of_freedom) $ k_1 $, $ k_2 $, and noncentrality parameter $ \lambda $. This arises naturally in the analysis of variance under non-null hypotheses. If the denominator $ Y $ is also noncentral, say $ Y \sim \chi_{k_2}^2(\lambda_2) $, the resulting distribution is doubly noncentral F with noncentrality parameters $ \lambda $ and $ \lambda_2 $. Logarithmic and power transformations are often considered for variance stabilization of the noncentral chi-squared, especially when the variance increases with the mean, but they do not produce distributions with simple closed forms. For instance, the logarithm $ \log(X) $ can approximate a [normal distribution](/page/Normal_distribution) under certain conditions, though exact mappings are unavailable. More precise variance-stabilizing transformations have been derived, such as those based on the [delta method](/page/Delta_method) or direct optimization for the noncentral case, which linearize the variance-mean relationship.[](https://www.researchgate.net/publication/265741201_Bar-Lev_SK_and_Boukai_B_2007_Variance_stabilizing_transformations_for_the_noncentral_chi-square_distribution_Journal_of_Probability_and_Statistical_Science_52_113-122) These are particularly useful in [signal processing](/page/Signal_processing) and [imaging](/page/Imaging), where noncentral chi-squared noise models speckle or magnitude data. Unlike the central chi-squared distribution, which is equivalent to a [gamma distribution](/page/Gamma_distribution) $ \Gamma(k/2, 2) $, the noncentral chi-squared lacks a direct inverse transformation to a single gamma variate. Instead, it admits a compound representation as a Poisson-weighted mixture of central chi-squared distributions:

X \stackrel{d}{=} \chi_{k + 2J}^2,

References

[1]
Noncentral Chi-Squared Distribution -- from Wolfram MathWorld
The noncentral chi-squared distribution with noncentrality parameter lambda is given by where I_n(x) is a modified Bessel function of the first kind.
[2]
[PDF] Lecture 13: Noncentral χ
Lecture 13 covers noncentral chi-square, t-, and F-distributions, and the noncentral chi-square distribution is defined.
[3]
Noncentral Chi - an overview | ScienceDirect Topics
The noncentral χ 2 and F distributions are used in evaluating “power properties” of tests of significance of various hypotheses about means and variances of ...
[4]
Noncentral Chi-Square Distribution - MATLAB & Simulink - MathWorks
The noncentral chi-square distribution is a more general case of the chi-square distribution, with applications in thermodynamics and signal processing.Definition · Background · Examples
[5]
The general sampling distribution of the multiple correlation coefficient
The solution introduces an extensive group of distributions, occurring naturally in the most diverse types of statistical investigation.
[6]
Noncentral Chi-square Distribution - Real Statistics Using Excel
The mean of the noncentral chi-square distribution is k + λ. The variance is 2(k+2λ). When λ = 0 the noncentral chi-square distribution is equal to the ...
[7]
[PDF] Handbook on probability distributions - Rice Statistics
Moment generating function for the non central chi-squared distribution exists ... ) denotes the modified Bessel's function. The distribution function can ...
[8]
[PDF] Theorem If X i are mutually independent noncentral chi-square(δi,ni ...
i=1 ni/2. = et Pm i=1 δi/(1−2t). (1 - 2t)Pm i=1 ni/2 t < 1/2, which is the moment generating function of a noncentral chi-square random variable with. Pm i=1 ...
[9]
[PDF] NCCCDF
Mar 21, 1997 · DESCRIPTION. The non-central chi-square distribution with degrees of freedom υ and non-centrality parameter δ is the sum of υ independent ...Missing: definition | Show results with:definition
[10]
On New Formulas for the Cumulative Distribution Function of the ...
Mar 3, 2017 · The main aim of this article is to derive three new formulas for the cumulative distribution function of the noncentral chi-square ...
[11]
[PDF] NCCPPF
Mar 21, 1997 · The non-central chi-square distribution with degrees of freedom υ and non-centrality parameter δ is the sum of υ independent normal.
[12]
[PDF] Non-central Chi-Squared Probabilities – Algorithms in R
For integer n, this is the distribution of the sum of squares of n normals each with variance one, λ being the sum of squares of the normal means; further,. E(X) ...
[13]
[PDF] Refined normal approximations for the central and noncentral chi ...
Abstract. In this paper, we prove a local limit theorem for the chi-square distribution with r > 0 degrees of freedom and noncentrality parameter λ ≥ 0.
[14]
https://academic.oup.com/biomet/article-pdf/47/3-4/411/651133/47-3-4-411.pdf
[15]
[PDF] Explaining the Saddlepoint Approximation - Cornell eCommons
Dec 16, 1995 · Example: Noncentral chi-squared. An interesting application of the saddlepoint approximation occurs in the case of the noncentral chi-squared.
[16]
[PDF] Computation of cumulative density of noncentral chi-square ...
The paper presents Visual Basic for Application code in Microsoft Excel to compute the cumulative distribution function for noncentral chi-square distributions.
[17]
scipy.stats.ncx2 — SciPy v1.16.2 Manual
A non-central chi-squared continuous random variable. As an instance of the rv_continuous class, ncx2 object inherits from it a collection of generic methods.
[18]
New methods to compute the generalized chi-square distribution
Feb 26, 2025 · We present four new mathematical methods, two exact and two approximate, along with open-source software, to compute the cdf, pdf and inverse ...Missing: 2020-2025 | Show results with:2020-2025
[19]
(Approximate) Probabilities of Non-Central Chi-squared... - R
Patnaik(1949)'s approximation to the non-central via central chi-squared. Is also the formula 26.4.27 26.4.27 26.4.27 in Abramowitz & Stegun, p.942. Johnson ...
[20]
[PDF] Non-Central Chi-squared=1See last slide for copyright information.
If X ∼ N(µ,1) then Y = X2 is said to have a non-central chi-squared distribution with degrees of freedom one and non-centrality parameter λ = µ2. Write Y ∼ χ2(1 ...Missing: formula | Show results with:formula<|separator|>
[21]
The (non-central) Chi-Squared Distribution - R
... Johnson, Kotz, and Balakrishnan (1995, chapter 29). In that (noncentral, zero df ) case, the distribution is a mixture of a point mass at x = 0 x = 0 x=0 ...Missing: formula | Show results with:formula
[22]
[PDF] A Tight Bound on the Distance Between a Noncentral Chi Square ...
Remark that for either k → ∞ or s → ∞, the limit of the noncentral chi square distribution is exactly the normal distribution, so that these bounds can be ...
[23]
[PDF] Rayleigh, Rice and Chi-Squared Distributions
4.2.5 The Non-Central Chi-Squared Distribution. There are counterpart results in the case of Rice fading, where the centre of the Gaussian distribution of g ...
[24]
(PDF) Bar-Lev, S.K. and Boukai, B. (2007). Variance stabilizing ...
Variance stabilizing transformations for the noncentral chi-square distribution. Journal of Probability and Statistical Science, 5(2), 113-122. January 2007.
[25]
[PDF] Understanding statistical power using noncentral probability ...
Like the noncentral chi-squared distribution, an increase in the value of the noncentrality parameter diminishes the overlap between the null and alternative ...
[26]
Practical guide to calculate sample size for chi-square test in ...
May 26, 2025 · Formula for sample size calculation ncp = n w 2 is the non-centrality parameter of the non-central chi-squared distribution. χ 1 - α , df 2 is ...Missing: density function
[27]
Approximations to the Non-Central Chi-Square Distribution - jstor
The objective was to make the distribution of y' more nearly normal than that of y. Again, results were given for a' first (normal) approximation' and a' second ...
[28]
Power/Sample Size Calculations for Generalized Linear Models - jstor
Therefore, the result relies on the quality of the chi-square approximation at a sequence of points that move progressively farther out in the tail of the ...
[29]
(PDF) On the Exact Two-Sided Tolerance Intervals for Univariate ...
Aug 9, 2025 · In this paper we present a brief overview of the theoretical background and approaches for computing the tolerance factors based on samples from ...
[30]
[PDF] On the Exact Two-Sided Tolerance Intervals for Univariate Normal ...
Jun 1, 2014 · chi-square distribution with 1 degree of freedom and the non-centrality parameter. √ δ2z2. For m = 1, the tolerance factor given by the solution ...
[31]
Confidence, Prediction, and Tolerance Regions for the Multivariate ...
Apr 10, 2012 · Formulas for confidence, prediction, and tolerance regions for the multivariate normal distribution for the various cases of known and unknown mean vector and ...Missing: squared | Show results with:squared
[32]
[PDF] tolerance regions for a multivariate normal
To construct the tolerance ellipsoidal regions under the situations in sections 2 and 5, we have to calculate the upper q=1-p point of the noncentral chi-square ...Missing: squared | Show results with:squared
[33]
None
### Summary of Noncentral Chi-Squared Distribution, Square Root, Rice Distribution, and Transformations
[34]
[PDF] Fundamentals of Radar Signal Processing, Second Edition (McGraw ...
... monotonic increasing function. It is therefore useful to see what ... noncentral chi-squared density under H1: Page 477. (6.73). (6.74). Since z′ is the ...
[35]
Parametric Rao Test for Multichannel Adaptive Signal Detection
Jul 31, 2007 · Parametric Rao Test for Multichannel Adaptive Signal Detection | IEEE Journals & Magazine | IEEE Xplore ... noncentral Chi-squared distribution ...
[36]
Applied noncentral Chi-squared distribution in CFAR detection of ...
Oct 15, 2015 · In this paper, we use a noncentral chi-squared distribution to approximate the projected image of subspace based anomaly detectors. Firstly, the ...Missing: spectroscopy | Show results with:spectroscopy
[37]
[PDF] Search and Discovery Statistics in HEP Lecture 2 - CERN Indico
μ| ′μ ) follows a noncentral Chi squared distribution ... discovering the Higgs Boson) with a given analysis, a given ... discovery. But even that formula can ...
[38]
A Closed-Form Second-Order Reliability Method Using Noncentral ...
Aug 9, 2025 · The method is based on an asymptotic expansion of the sum of noncentral chi-squared variables taken from the literature. The two most widely ...
[39]
[PDF] Aggregation of Nonparametric Estimators for Volatility Matrix ∗
estimate of the portfolio variance: aT b. ΣA,ta = ωtaT b. ΣS,ta + (1 − ωt)aT ... noncentral chi-squared distribution with degrees of freedom 4a. Q. 11 and ...
[40]
Identifying and Mitigating Machine Learning Biases for the ... - arXiv
Oct 4, 2025 · The histograms should resemble a noncentral chi-squared distribution for both classes given the behaviour of real noise PSD s. This is true ...Missing: post- | Show results with:post-
[41]
[PDF] Physics-Driven Spatiotemporal Modeling for AI-Generated Video ...
Ai-generated video detection via spatial- temporal anomaly learning. In ... is the noncentral chi-squared distribution with noncentrality parameter φ [61].