lognorm function

Lognormal distribution

Lognormal distribution

Computes the pdf, cdf, value at risk and expected shortfall for the lognormal distribution given by [REMOVE_ME]\displaystylef(x)=1σxϕ(logxμσ),\displaystyleF(x)=Φ(logxμσ),VaRp(X)=exp[μ+σΦ1(p)],ESp(X)=exp(μ)p0pexp[σΦ1(v)]dv[REMOVEME2] \begin{array}{ll}&\displaystylef (x) = \frac {1}{\sigma x} \phi \left( \frac {\log x - \mu}{\sigma} \right),\\&\displaystyleF (x) = \Phi \left( \frac {\log x - \mu}{\sigma} \right),\\&\displaystyle{\rm VaR}_p (X) = \exp \left[ \mu + \sigma \Phi^{-1} (p) \right],\\&\displaystyle{\rm ES}_p (X) = \frac {\exp (\mu)}{p} \int_0^p \exp \left[ \sigma \Phi^{-1} (v) \right] dv\end{array} [REMOVE_ME_2]

for x>0x > 0, 0<p<10 < p < 1, <μ<-\infty < \mu < \infty, the location parameter, and σ>0\sigma > 0, the scale parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the lognormal distribution given by

\displaystylef(x)=1σxϕ(logxμσ),\displaystyleF(x)=Φ(logxμσ),VaRp(X)=exp[μ+σΦ1(p)],ESp(X)=exp(μ)p0pexp[σΦ1(v)]dv \begin{array}{ll}&\displaystylef (x) = \frac {1}{\sigma x} \phi \left( \frac {\log x - \mu}{\sigma} \right),\\&\displaystyleF (x) = \Phi \left( \frac {\log x - \mu}{\sigma} \right),\\&\displaystyle{\rm VaR}_p (X) = \exp \left[ \mu + \sigma \Phi^{-1} (p) \right],\\&\displaystyle{\rm ES}_p (X) = \frac {\exp (\mu)}{p} \int_0^p \exp \left[ \sigma \Phi^{-1} (v) \right] dv\end{array}

for x>0x > 0, 0<p<10 < p < 1, <μ<-\infty < \mu < \infty, the location parameter, and σ>0\sigma > 0, the scale parameter.

dlognorm(x, mu=0, sigma=1, log=FALSE) plognorm(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE) varlognorm(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE) eslognorm(p, mu=0, sigma=1)

Arguments

  • x: scaler or vector of values at which the pdf or cdf needs to be computed
  • p: scaler or vector of values at which the value at risk or expected shortfall needs to be computed
  • mu: the value of the location parameter, can take any real value, the default is zero
  • sigma: the value of the scale parameter, must be positive, the default is 1
  • log: if TRUE then log(pdf) are returned
  • log.p: if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
  • lower.tail: if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Returns

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Author(s)

Saralees Nadarajah

Examples

x=runif(10,min=0,max=1) dlognorm(x) plognorm(x) varlognorm(x) eslognorm(x)
  • Maintainer: Leo Belzile
  • License: GPL (>= 2)
  • Last published: 2023-04-22

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