Wald function

Wald (inverse Gaussian) distribution

Wald (inverse Gaussian) distribution

Density, distribution function and random generation for the Wald distribution.

dwald(x, mu, lambda, log = FALSE)

pwald(q, mu, lambda, lower.tail = TRUE, log.p = FALSE)

rwald(n, mu, lambda)

Arguments

  • x, q: vector of quantiles.

  • mu, lambda: location and shape parameters. Scale must be positive.

  • log, log.p: logical; if TRUE, probabilities p are given as log(p).

  • lower.tail: logical; if TRUE (default), probabilities are P[X≤x]P[X \le x]

    otherwise, P[X>x]P[X > x].

  • n: number of observations. If length(n) > 1, the length is taken to be the number required.

  • p: vector of probabilities.

Details

Probability density function

f(x)=λ2πx3exp⁡(−λ(x−μ)22μ2x)f(x)=sqrt(λ/(2∗π∗x3))∗exp((−λ∗(x−μ)2)/(2∗μ2∗x)) f(x) = \sqrt{\frac{\lambda}{2\pi x^3}} \exp\left( \frac{-\lambda(x-\mu)^2}{2\mu^2 x} \right)f(x) = sqrt(\lambda/(2*\pi*x^3)) * exp((-\lambda*(x-\mu)^2)/(2*\mu^2*x))

Cumulative distribution function

F(x)=Φ(λx(xμ−1))+exp⁡(2λμ)Φ(λx(xμ+1))F(x)=Φ(sqrt(λ/μ)∗(x/μ−1))−exp((2∗λ)/μ)∗Φ(sqrt(λ/μ)∗(x/μ+1)) F(x) = \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu}-1 \right) \right) +\exp\left(\frac{2\lambda}{\mu} \right) \Phi\left(\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}+1 \right) \right)F(x) = \Phi(sqrt(\lambda/\mu)*(x/\mu-1)) - exp((2*\lambda)/\mu) *\Phi(sqrt(\lambda/\mu)*(x/\mu+1))

Random generation is done using the algorithm described by Michael, Schucany and Haas (1976).

Examples

x <- rwald(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dwald(x, 5, 16), 0, 50, col = "red", add = TRUE)
hist(pwald(x, 5, 16))
plot(ecdf(x))
curve(pwald(x, 5, 16), 0, 50, col = "red", lwd = 2, add = TRUE)

References

Michael, J.R., Schucany, W.R., and Haas, R.W. (1976). Generating Random Variates Using Transformations with Multiple Roots. The American Statistician, 30(2): 88-90.

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  • Maintainer: Tymoteusz Wolodzko
  • License: GPL-2
  • Last published: 2023-11-30

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