invweibullUC: The Inverse Weibull Distribution

In VGAMextra: Additions and Extensions of the 'VGAM' Package

The Inverse Weibull Distribution

Description

Density, distribution function, quantile function and random numbers generator for the Inverse Weibull Distribution.

Usage

    dinvweibull(x, scale = 1, shape, log = FALSE)
    pinvweibull(q, scale = 1, shape, lower.tail = TRUE, log.p = FALSE)
    qinvweibull(p, scale = 1, shape, lower.tail = TRUE, log.p = FALSE)
    rinvweibull(n, scale = 1, shape)

Arguments

x, q, p, n	Same as Weibull.
scale, shape	Scale and shape parameters, same as Weibull. Both must be positive.
log, log.p, lower.tail	Same as Weibull.

Details

The Inverse Weibull density with parameters scale = b and shape = s, is

f(y) = s b^s y^{-s-1} \exp{[-(y/b)^{-s}}],

for y > 0, b > 0, and s > 0.

The Weibull distribution and the Inverse Weibull distributions are related as follows:

Let X be a Weibull random variable with paramaters scale =b and shape =s. Then, the random variable Y = 1/X has the Inverse Weibull density with parameters scale = 1/b and shape = s. Thus, algorithms of [dpqr]-Inverse Weibull underlie on Weibull.

Let Y be a r.v. distributed as Inverse Weibull (b, s). The k^{th} moment exists for -\infty < k < s and is given by

E[Y^k] = b^{k} \ \Gamma(1 - k/s).

The mean (if s > 1) and variance (if s > 2) are

E[Y] = b \ \Gamma(1 - 1/s); \ \ \ Var[Y] = b^{2} \ [\Gamma(1 - 2/s) - (\Gamma(1 - 1/s))^2].

Here, \Gamma(\cdot) is the gamma function as in gamma.

Value

dinvweibull() returns the density, pinvweibull() computes the distribution function, qinvweibull() gives the quantiles, and rinvweibull() generates random numbers from the Inverse Weibull distribution.

Warning

The order of the arguments of [dpqr]-Inverse Weibull does not match those in Weibull.

Note

Small values of scale or shape will provide Inverse Weibull values too close to zero. Then, function rinvweibull() with such characteristics will return either values too close to zero or values represented as zero in computer arithmetic.

The Inverse Weibull distribution, which is that of X where 1/X has the Weibull density, is known as the log-Gompertz distribution. Thus, in order to emphazise the continuity concept of the Inverse Weibull density, if x = 0, then dinvweibull returns zero, which is the limit of such a density when 'x' tends to zero.

Author(s)

V. Miranda and T. W. Yee.

References

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Wiley Series in Probability and Statistics. Hoboken, New Jersey, USA.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. ch.6, p.255. Dover, New York, USA.

See Also

Weibull, gamma.

Examples

  #(1) ______________
  n        <- 20
  scale    <- exp(2)
  shape    <- exp(1)
  data.1   <- runif(n, 0, 1)
  data.q   <- qinvweibull(-data.1, scale = scale, shape = shape, log.p = TRUE)  
  data.p   <- -log(pinvweibull(data.q, scale = scale, shape = shape)) 
  arg.max  <- max(abs(data.p - data.1))     # Should be zero
  

  #(2)_________________
   scale  <- exp(1.0)
    shape <- exp(1.2)
    xx    <- seq(0, 10.0, len = 201)
    yy    <- dinvweibull(xx, scale = scale, shape = shape)
    qtl   <- seq(0.1, 0.9, by =0.1)
    d.qtl <- qinvweibull(qtl, scale = scale, shape = shape)
    plot(xx, yy, type = "l", col = "red", 
         main = "Red is density, blue is cumulative distribution function",
         sub  = "Brown dashed lines represent the 10th, ... 90th percentiles",
         las = 1, xlab = "x", ylab = "", ylim = c(0,1))
    abline(h = 0, col= "navy", lty = 2)
    lines(xx, pinvweibull(xx, scale = scale, shape = shape), col= "blue")
    lines(d.qtl, dinvweibull(d.qtl, scale = scale, shape = shape), 
          type ="h", col = "brown", lty = 3)
        

VGAMextra documentation built on Nov. 2, 2023, 5:59 p.m.