Inverse Chi Square Distribution

The inverse chi square (χ²) distribution, also known as the inverted chi square distribution, is a continuous probability distribution that models the distribution of the reciprocal (inverse) of a chi square variable. It is commonly used in Bayesian statistics.

Definition

The inverse chi square distribution with $ν$ degrees of freedom models the distribution of the reciprocal of a chi square variable.

The probability density function (PDF) is given by:

f (x) = \frac{2^{- ν / 2}}{Γ (ν / 2)} x^{- ν / 2 - 1} e^{- 1 / (2 x)}

The cumulative distribution function (CDF) can be expressed in terms of the gamma function:

F (x) = 1 - Γ (ν / 2, 1 / (2 x)) / Γ (ν / 2)

The distribution is defined for positive real values ( $x > 0$ ), with the degrees of freedom parameter $ν$ being strictly positive ( $ν > 0$ ).

Applications

Bayesian inference serves as a conjugate prior for the variance parameter in normal distributions.
Statistical quality control uses this distribution for variance component analysis.
Reliability engineering employs the distribution in lifetime modeling.

Properties

Statistical Properties

Property	Value
Mean	$\frac{1}{ν - 2}, ν > 2$
Mode	$\frac{1}{ν + 2}$
Variance	$\frac{2}{(ν - 2)^{2} (ν - 4)}, ν > 4$
Skewness	$\frac{4 \sqrt{2 (ν - 4)}}{ν - 6}, ν > 6$
Excess Kurtosis	$\frac{12 (5 ν - 22)}{(ν - 6) (ν - 8)}, ν > 8$
Entropy	$\frac{ν}{2} + \ln (2 Γ (\frac{ν}{2})) - (\frac{ν}{2} + 1) ψ (\frac{ν}{2})$

Relationships to Other Distributions

If X follows a chi-square distribution with $ν$ degrees of freedom, then 1/X follows an inverse chi-square distribution with $ν$ degrees of freedom.
The inverse chi-square distribution is a special case of the inverse gamma distribution, equivalent to an inverse gamma distribution with shape parameter $α = ν / 2$ and scale parameter $β = 1 / 2$ .
The scaled inverse chi-square distribution is obtained by scaling the inverse chi-square distribution by a factor.

The InverseChiSquareDistribution class

The inverse chi square distribution is implemented by the InverseChiSquareDistribution class. It has one constructor which takes the degrees of freedom as its only argument. The following constructs an inverse chi square distribution with 10 degrees of freedom:

var invChiSquare = new ChiSquareDistribution(10);

Visual Basic

Dim invChi = New InverseChiSquareDistribution(10)

Visual Basic

No code example is currently available or this language may not be supported.

let invChiSquare = InverseChiSquareDistribution(10.0)

The InverseChiSquareDistribution class has one specific property, DegreesOfFreedom, that returns the degrees of freedom of the distribution.

References

For more information on the inverse chi-squared distribution, refer to the following sources:

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 18. Wiley Series in Probability and Statistics.
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 29. Wiley Series in Probability and Statistics.