dinvchi: Inverse-Chi distribution.

Description

(Scaled) inverse-Chi density, distribution, and quantile functions, random number generation and expectation and variance.

Usage

dinvchi(x, df, scale=1, log=FALSE)
  pinvchi(q, df, scale=1, lower.tail=TRUE, log.p=FALSE)
  qinvchi(p, df, scale=1, lower.tail=TRUE, log.p=FALSE)
  rinvchi(n, df, scale=1)
  einvchi(df, scale=1)
  vinvchi(df, scale=1)

Value

‘dinvchi()’ gives the density function,

‘pinvchi()’ gives the cumulative distribution function (CDF),

‘qinvchi()’ gives the quantile function (inverse CDF), and ‘rinvchi()’ generates random deviates. The ‘einvchi()’ and ‘vinvchi()’

functions return the corresponding distribution's expectation and variance, respectively.

Arguments

x, q: quantile.
p: probability.
n: number of observations.
df: degrees-of-freedom parameter ($>0$).
scale: scale parameter ($>0$).
log: logical; if TRUE, logarithmic density will be returned.
lower.tail: logical; if TRUE (default), probabilities are P(X <= x), otherwise, P(X > x).
log.p: logical; if TRUE, probabilities p are returned as log(p).

Author

Christian Roever christian.roever@med.uni-goettingen.de

Details

The (scaled) inverse-Chi distribution is defined as the distribution of the (scaled) inverse of the square root of a Chi-square-distributed random variable. It is a special case of the square-root inverted-gamma distribution (with $\alpha=\nu/2$ and $\beta=1/2$) (Bernardo and Smith; 1994). Its probability density function is given by $$p(x) \;=\; \frac{2^{(1-\nu/2)}}{s \, \Gamma(\nu/2)} \Bigl(\frac{s}{x}\Bigr)^{(\nu+1)} \exp\Bigl(-\frac{s^2}{2\,x^2}\Bigr)$$ where $\nu$ is the degrees-of-freedom and $s$ the scale parameter.

References

C. Roever, R. Bender, S. Dias, C.H. Schmid, H. Schmidli, S. Sturtz, S. Weber, T. Friede. On weakly informative prior distributions for the heterogeneity parameter in Bayesian random-effects meta-analysis. Research Synthesis Methods, 12(4):448-474, 2021. tools:::Rd_expr_doi("10.1002/jrsm.1475").

J.M. Bernardo, A.F.M. Smith. Bayesian theory, Appendix A.1. Wiley, Chichester, UK, 1994.

Examples

Run this code

#################################
# illustrate Chi^2 - connection;
# generate Chi^2-draws:
chi2 <- rchisq(1000, df=10)
# transform:
invchi <- sqrt(1 / chi2)
# show histogram:
hist(invchi, probability=TRUE, col="grey")
# show density for comparison:
x <- seq(0, 1, length=100)
lines(x, dinvchi(x, df=10, scale=1), col="red")
# compare theoretical and empirical moments:
rbind("theoretical" = c("mean" = einvchi(df=10, scale=1),
                        "var"  = vinvchi(df=10, scale=1)),
      "Monte Carlo" = c("mean" = mean(invchi),
                        "var"  = var(invchi)))

##############################################################
# illustrate the normal/Student-t - scale mixture connection;
# specify degrees-of-freedom:
df <- 5
# generate standard normal draws:
z <- rnorm(1000)
# generate random scalings:
sigma <- rinvchi(1000, df=df, scale=sqrt(df))
# multiply to yield Student-t draws:
t <- z * sigma
# check Student-t distribution via a Q-Q-plot:
qqplot(qt(ppoints(length(t)), df=df), t)
abline(0, 1, col="red")

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