Noncentral chi-squared distribution

Noncentral chi-squared distribution is a probability distribution that generalizes the chi-squared distribution, allowing for the presence of a noncentrality parameter. It arises in statistics particularly in the context of inference under the null hypothesis versus an alternative hypothesis, and in the study of power of statistical tests.

Definition

The noncentral chi-squared distribution is defined by two parameters: the degrees of freedom k and the noncentrality parameter λ. The probability density function (pdf) of the noncentral chi-squared distribution for x > 0 is given by:

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

where I is the modified Bessel function of the first kind.

Properties

Mean and Variance

The mean and variance of the noncentral chi-squared distribution are given by:

Mean = k + λ,

Variance = 2(k + 2λ).

These properties indicate that the distribution's mean is shifted by the noncentrality parameter λ, and the variance increases with both k and λ.

Cumulative Distribution Function

The cumulative distribution function (CDF) is not as straightforward to express in a closed form as the pdf. However, it can be evaluated using numerical methods or approximations.

Applications

The noncentral chi-squared distribution is particularly important in statistical hypothesis testing, where it arises in the distribution of test statistics under the alternative hypothesis. It is used in:

• Analysis of variance (ANOVA) for noncentral F-tests,
• Power analysis to determine the sample size required to detect an effect of a given size with a given level of confidence,
• Likelihood-ratio tests for nested models.

Related Distributions

• When the noncentrality parameter λ = 0, the noncentral chi-squared distribution simplifies to the chi-squared distribution.
• The noncentral t-distribution and noncentral F-distribution are related distributions that also incorporate noncentrality parameters.

See Also

• Gamma distribution
• Exponential distribution
• Normal distribution

External Links

• NIST/SEMATECH e-Handbook of Statistical Methods