Scaled inverse chi-squared distribution

Scaled Inverse Chi-Squared Distribution

The Scaled Inverse Chi-Squared Distribution is a probability distribution that is widely used in Bayesian statistics, particularly in the context of Bayesian inference and hierarchical models. It is a two-parameter family of continuous probability distributions and is closely related to the Inverse chi-squared distribution and the Inverse gamma distribution.

Definition

The scaled inverse chi-squared distribution is defined by two parameters: the degrees of freedom \( \nu > 0 \) and the scale parameter \( \sigma^2 > 0 \). The probability density function (pdf) of the scaled inverse chi-squared distribution for \( x > 0 \) is given by:

\[ f(x|\nu,\sigma^2) = \frac{(\frac{\nu\sigma^2}{2})^{\frac{\nu}{2}}}{\Gamma(\frac{\nu}{2})} x^{-\frac{\nu}{2}-1} \exp\left(-\frac{\nu\sigma^2}{2x}\right) \]

where \( \Gamma \) is the Gamma function.

Properties

Mean

If \( \nu > 2 \), the mean of the scaled inverse chi-squared distribution is:

\[ \text{Mean} = \frac{\nu\sigma^2}{\nu-2} \]

Variance

If \( \nu > 4 \), the variance is:

\[ \text{Variance} = \frac{2(\nu\sigma^2)^2}{(\nu-2)^2(\nu-4)} \]

Mode

The mode of the distribution is:

\[ \text{Mode} = \frac{\nu\sigma^2}{\nu+2} \]

Applications

The scaled inverse chi-squared distribution is particularly useful in Bayesian statistics as a Conjugate prior for the variance of a normally distributed population when the mean is known. It is also used in the Bayesian analysis of variance (ANOVA) models and in the estimation of random effects in hierarchical linear models.

Related Distributions

- The Inverse chi-squared distribution is a special case of the scaled inverse chi-squared distribution with \( \sigma^2 = 1 \). - The Inverse gamma distribution is a generalization of the scaled inverse chi-squared distribution. - When the scale parameter \( \sigma^2 \) is known, the scaled inverse chi-squared distribution is related to the Chi-squared distribution through a transformation.

See Also

• Bayesian statistics
• Gamma function
• Conjugate prior
• Hierarchical linear models