The Inverse (non-central) Chi-Squared Distribution

Description

Density, distribution function, quantile function and random generation for the inverse chi-squared distribution.

Usage

dinvchisq(x, df, ncp = 0, log = FALSE)

pinvchisq(q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)

qinvchisq(p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)

rinvchisq(n, df, ncp = 0)

Arguments

Details

The functions ⁠(d/p/q/r)invchisq()⁠ simply wrap those of the standard ⁠(d/p/q/r)chisq()⁠ R implementation, so look at, say, stats::dchisq() for details.

See Also

stats::dchisq(); these functions just wrap the ⁠(d/p/q/r)chisq()⁠ functions.

Examples

s <- seq(0, 3, .01)
plot(s, dinvchisq(s, 3), type = 'l')

f <- function(x) dinvchisq(x, 3)
q <- 2
integrate(f, 0, q)
(p <- pinvchisq(q, 3))
qinvchisq(p, 3) # = q
mean(rinvchisq(1e5, 3) <= q)




f <- function(x) dinvchisq(x, 3, ncp = 2)
q <- 1.5
integrate(f, 0, q)
(p <- pinvchisq(q, 3, ncp = 2))
qinvchisq(p, 3, ncp = 2) # = q
mean(rinvchisq(1e7, 3, ncp = 2) <= q)

The Inverse Exponential Distribution

Description

Density, distribution function, quantile function and random generation for the inverse exponential distribution.

Usage

dinvexp(x, rate = 1, log = FALSE)

pinvexp(q, rate = 1, lower.tail = TRUE, log.p = FALSE)

qinvexp(p, rate = 1, lower.tail = TRUE, log.p = FALSE)

rinvexp(n, rate = 1)

Arguments

Details

The functions ⁠(d/p/q/r)invexp()⁠ simply wrap those of the standard ⁠(d/p/q/r)exp()⁠ R implementation, so look at, say, stats::dexp() for details.

See Also

stats::dexp(); these functions just wrap the ⁠(d/p/q/r)exp()⁠ functions.

Examples

s <- seq(0, 10, .01)
plot(s, dinvexp(s, 2), type = 'l')

f <- function(x) dinvexp(x, 2)
q <- 3
integrate(f, 0, q)
(p <- pinvexp(q, 2))
qinvexp(p, 2) # = q
mean(rinvexp(1e5, 2) <= q)

pinvgamma(q, 1, 2)

The Inverse Gamma Distribution

Description

Density, distribution function, quantile function and random generation for the inverse gamma distribution.

Usage

dinvgamma(x, shape, rate = 1, scale = 1/rate, log = FALSE)

pinvgamma(q, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

qinvgamma(p, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

rinvgamma(n, shape, rate = 1, scale = 1/rate)

Arguments

Details

The inverse gamma distribution with parameters shape and rate has density

f(x) = \frac{rate^{shape}}{\Gamma(shape)} x^{-1-shape} e^{-rate/x}

it is the inverse of the standard gamma parameterization in R. If X \sim InvGamma(shape, rate),

E[X] = \frac{rate}{shape-1}

when shape > 1 and

Var(X) = \frac{rate^2}{(shape - 1)^2(shape - 2)}

for shape > 2.

The functions ⁠(d/p/q/r)invgamma()⁠ simply wrap those of the standard ⁠(d/p/q/r)gamma()⁠ R implementation, so look at, say, stats::dgamma() for details.

See Also

stats::dgamma(); these functions just wrap the ⁠(d/p/q/r)gamma()⁠ functions.

Examples

s <- seq(0, 5, .01)
plot(s, dinvgamma(s, 7, 10), type = 'l')

f <- function(x) dinvgamma(x, 7, 10)
q <- 2
integrate(f, 0, q)
(p <- pinvgamma(q, 7, 10))
qinvgamma(p, 7, 10) # = q
mean(rinvgamma(1e5, 7, 10) <= q)

shape <- 3; rate <- 7
x <- rinvgamma(1e6, shape, rate)
mean(x) # = rate / (shape - 1)
var(x)  # = rate^2 / ( (shape - 1)^2 * (shape - 2) )

shape <- 7; rate <- 2.01
x <- rinvgamma(1e6, shape, rate)
mean(x) # = rate / (shape - 1)
var(x)  # = rate^2 / ( (shape - 1)^2 * (shape - 2) )

qnorm(log(.25), log.p = TRUE)
qnorm(.25)

qinvgamma(log(.25), shape = shape, rate = rate, log.p = TRUE)
qinvgamma(.25, shape = shape, rate = rate)

## Not run:  `rinvgamma()` warns when shape <= .01

rinvgamma(10, .01, rate) # warns


## End(Not run)