bayesmeta (version 3.4)

dhalfnormal: Half-normal, half-Student-t and half-Cauchy distributions.

Description

Half-normal, half-Student-t and half-Cauchy density, distribution, quantile functions, random number generation, and expectation and variance.

Usage

dhalfnormal(x, scale=1, log=FALSE)
  phalfnormal(q, scale=1)
  qhalfnormal(p, scale=1)
  rhalfnormal(n, scale=1)
  ehalfnormal(scale=1)
  vhalfnormal(scale=1)
  dhalft(x, df, scale=1, log=FALSE)
  phalft(q, df, scale=1)
  qhalft(p, df, scale=1)
  rhalft(n, df, scale=1)
  ehalft(df, scale=1)
  vhalft(df, scale=1)
  dhalfcauchy(x, scale=1, log=FALSE)
  phalfcauchy(q, scale=1)
  qhalfcauchy(p, scale=1)
  rhalfcauchy(n, scale=1)
  ehalfcauchy(scale=1)
  vhalfcauchy(scale=1)

Value

‘dhalfnormal()’ gives the density function,

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

‘qhalfnormal()’ gives the quantile function (inverse CDF), and ‘rhalfnormal()’ generates random deviates. The ‘ehalfnormal()’ and ‘vhalfnormal()’

functions return the corresponding half-normal distribution's expectation and variance, respectively. For the

‘dhalft()’, ‘dhalfcauchy()’ and related function it works analogously.

Arguments

x, q

quantile.

p

probability.

n

number of observations.

scale

scale parameter (\(>0\)).

df

degrees-of-freedom parameter (\(>0\)).

log

logical; if TRUE, logarithmic density will be returned.

Author

Christian Roever

Details

The half-normal distribution is simply a zero-mean normal distribution that is restricted to take only positive values. The scale parameter \(\sigma\) here corresponds to the underlying normal distribution's standard deviation: if \(X\sim\mathrm{Normal}(0,\sigma^2)\), then \(|X|\sim\mathrm{halfNormal}(\mathrm{scale}\!=\!\sigma)\). Its mean is \(\sigma \sqrt{2/\pi}\), and its variance is \(\sigma^2 (1-2/\pi)\). Analogously, the half-t distribution is a truncated Student-t distribution with df degrees-of-freedom, and the half-Cauchy distribution is again a special case of the half-t distribution with df=1 degrees of freedom.

Note that (half-) Student-t and Cauchy distributions arise as continuous mixture distributions of (half-) normal distributions. If $$Y|\sigma\;\sim\;\mathrm{Normal}(0,\sigma^2)$$ where the standard deviation is \(\sigma = \sqrt{k/X}\) and \(X\) is drawn from a \(\chi^2\)-distribution with \(k\) degrees of freedom, then the marginal distribution of \(Y\) is Student-t with \(k\) degrees of freedom.

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").

A. Gelman. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1(3):515-534, 2006. tools:::Rd_expr_doi("10.1214/06-BA117A").

F. C. Leone, L. S. Nelson, R. B. Nottingham. The folded normal distribution. Technometrics, 3(4):543-550, 1961. tools:::Rd_expr_doi("10.2307/1266560").

N. G. Polson, J. G. Scott. On the half-Cauchy prior for a global scale parameter. Bayesian Analysis, 7(4):887-902, 2012. tools:::Rd_expr_doi("10.1214/12-BA730").

S. Psarakis, J. Panaretos. The folded t distribution. Communications in Statistics - Theory and Methods, 19(7):2717-2734, 1990. tools:::Rd_expr_doi("10.1080/03610929008830342").

See Also

dnorm, dt, dcauchy, dlomax, drayleigh, TurnerEtAlPrior, RhodesEtAlPrior, bayesmeta.

Examples

#######################
# illustrate densities:
x <- seq(0,6,le=200)
plot(x, dhalfnormal(x), type="l", col="red", ylim=c(0,1),
     xlab=expression(tau), ylab=expression("probability density "*f(tau)))
lines(x, dhalft(x, df=3), col="green")
lines(x, dhalfcauchy(x), col="blue")
lines(x, dexp(x), col="cyan")
abline(h=0, v=0, col="grey")

# show log-densities (note the differing tail behaviour):
plot(x, dhalfnormal(x), type="l", col="red", ylim=c(0.001,1), log="y",
     xlab=expression(tau), ylab=expression("probability density "*f(tau)))
lines(x, dhalft(x, df=3), col="green")
lines(x, dhalfcauchy(x), col="blue")
lines(x, dexp(x), col="cyan")
abline(v=0, col="grey")