< List of probability distributions < Noncentral chi-square distribution
What is the noncentral chi-square distribution?
The noncentral chi-square distribution is a generalized version of the chi-squared distribution. It is commonly encountered in power analysis for statistical tests, particularly when the null distribution — the distribution of a test statistic under the assumption that the null hypothesis is true — approximates a chi-squared distribution. Likelihood-ratio tests are one example of such tests [1].
The power of a statistical test gives us the probability of rejecting the null hypothesis when it is false. The power of a chi-square test can be calculated by considering the noncentral chi-squared distribution, which considers the difference between the population mean and the hypothesized value of the population mean.
The noncentral chi-square has applications in thermodynamics and signal processing, where it sometimes called the Rician Distribution or generalized Rayleigh Distribution [2].
Non-central chi-square distribution properties
![Graph of noncentral chi-square PDF [3].](https://probabilityhowto.com/wp-content/uploads/2023/08/600px-Chi-Squared-nonCentral-pdf-1.png)
The non-central chi-square distribution with degrees of freedom υ and non-centrality parameter δ is the sum of υ independent normal
distributions with a standard deviation of 1. It is formally defined as follows [4]:
Suppose X1, X2, …, Xn are independent random variables, where each Xi (i = 1, …, n) follows a normal distribution with mean µi and variance σ2. The noncentral chi-square distribution is the distribution of the random variable Y = (X12 + X22 + … + Xn2) / σ2, with n degrees of freedom and noncentrality parameter δ = (µ12 + µ22 + … + µn2) / σ2.
The non-centrality parameter is one half the sum of squares of the normal means.
The formula for the cumulative distribution function (CDF) is [5]:
where
- δ = non-centrality parameter,
- υ = degrees of freedom parameter,
- i = an integer from 0 to infinity,
- ! = a factorial — products of whole numbers up to the number of interest. For example, 3! (read “three factorial”) equals 3 x 2 x 1 = 6.
- Fc = central chi-square CDF.
The noncentrality parameter
The noncentrality parameter, δ, represents the difference between the population mean (µi) and the hypothesized value of the population mean (µ0). It is calculated the sum of the squared differences between µi and µ0.The chi-square distribution is a special case of the noncentral chi-square distribution with δ = 0. As a result, it is occasionally called a central chi-square distribution.
Central vs. noncentral chi-square distribution
The chi-squared distribution shows the distribution of the chi-squared test statistic assuming that the null hypothesis is true. The noncentral chi-squared distribution, on the other hand, is a generalization that represents the distribution of the chi-squared test statistic under the assumption that the alternative hypothesis is true. When the alternative hypothesis is true, the population mean is different from the hypothesized value of the population mean. This difference is reflected by the noncentrality parameter of the noncentral chi-squared distribution.
References
[1] Patnaik, P. B. (1949). “The Non-Central χ2- and F-Distribution and their Applications”. Biometrika. 36 (1/2): 202–232. doi:10.2307/2332542. ISSN 0006-3444.
[2] MathWorks. Noncentral Chi-Square Distribution. Retrieved August 5, 2023 from: https://www.mathworks.com/help/stats/noncentral-chi-square-distribution.html
[3] Thomas Steiner, CC BY-SA 2.5 https://creativecommons.org/licenses/by-sa/2.5, via Wikimedia Commons
[4] Shao, J. (2015). Lecture 13: Noncentral χ2-, t-, and F-distributions. Retrieved August 5, 2023 from: https://pages.stat.wisc.edu/~shao/stat609/stat609-13.pdf
[5] NIST. Probability Library Functions NCCCDF.