Inverse Chi-Squared Distribution Functions

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Inverse Chi-Squared distribution.

For degrees of freedom \(\nu\) and scale \(\xi\), the PDF is

$$f(x;\nu,\xi) = \frac{(\nu\xi/2)^{\nu/2}}{\Gamma(\nu/2)} x^{-1-\nu/2} \exp\left(-\frac{\nu\xi}{2x}\right)$$

The CDF is:

$$F(x; \nu,\xi)=\frac{\Gamma(\nu/2,\nu\xi/2x)}{\Gamma(\nu/2)}$$

The unscaled case corresponds to \(\xi = 1/\nu\).

Usage

inverse_chi_squared_distribution(df = 1, scale = 1/df)

inverse_chi_squared_pdf(x, df = 1, scale = 1/df)

inverse_chi_squared_lpdf(x, df = 1, scale = 1/df)

inverse_chi_squared_cdf(x, df = 1, scale = 1/df)

inverse_chi_squared_lcdf(x, df = 1, scale = 1/df)

inverse_chi_squared_quantile(p, df = 1, scale = 1/df)

Arguments

df

Degrees of freedom (df > 0; default is 1).

scale

Scale parameter (default is 1/df).

x

Quantile value (x >= 0).

p

Probability (0 <= p <= 1).

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Inverse Chi-Squared distribution with 10 degrees of freedom, scale = 1
dist <- inverse_chi_squared_distribution(10, 1)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.02925269
logcdf(dist, 0.5)
#> [1] -3.531784
pdf(dist, 0.5)
#> [1] 0.3783327
logpdf(dist, 0.5)
#> [1] -0.9719812
hazard(dist, 0.5)
#> [1] 0.3897335
chf(dist, 0.5)
#> [1] 0.02968908
mean(dist)
#> [1] 1.25
median(dist)
#> [1] 1.070455
mode(dist)
#> [1] 0.8333333
range(dist)
#> [1]  0.000000e+00 1.797693e+308
quantile(dist, 0.2)
#> [1] 0.7439393
standard_deviation(dist)
#> [1] 0.7216878
support(dist)
#> [1]  0.000000e+00 1.797693e+308
variance(dist)
#> [1] 0.5208333
skewness(dist)
#> [1] 3.464102
kurtosis(dist)
#> [1] 45
kurtosis_excess(dist)
#> [1] 42

# Convenience functions
inverse_chi_squared_pdf(2, 10, 1)
#> [1] 0.1670024
inverse_chi_squared_lpdf(2, 10, 1)
#> [1] -1.789747
inverse_chi_squared_cdf(2, 10, 1)
#> [1] 0.891178
inverse_chi_squared_lcdf(2, 10, 1)
#> [1] -0.1152111
inverse_chi_squared_quantile(0.5, 10, 1)
#> [1] 1.070455