Density, distribution function, quantile function and random numbers generator for the Inverse Weibull Distribution.
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Scale and shape parameters, same as
The Inverse Weibull density with parameters scale = b and shape = s, is
f(y) = s * (b^s) * y^(-s-1) * e^[(-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
Let Y be a r.v. distributed as Inverse Weibull (b , s). The k-th moment exists for -Inf < k < s and is given by
E[Y^k] = (b^k) * Γ(1 - k/s).
The mean (if s > 1) and variance (if s >2) are
E[Y] = b * Γ(1 - 1/s), Var[Y] = (b^2) * [Γ(1 - 2/s) - (Γ(1 - 1/s))^2].
Here, gamma() is the gamma function as in
dinvweibull() returns the density,
pinvweibull() computes the
qinvweibull() gives the quantiles, and
rinvweibull() generates random numbers from the Inverse Weibull
The order of the arguments of [dpqr]-Inverse Weibull does not match
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
returns zero, which is the limit of such a density when 'x'
tends to zero.
V. Miranda and T. W. Yee.
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.
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#(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)
Loading required package: stats4 Loading required package: VGAM Loading required package: splines ===== VGAMextra 0.0-2 ===== Additions and extensions of the package VGAM. For more on VGAMextra, visit https://www.stat.auckland.ac.nz/~vmir178/ For a short description, fixes/bugs, and new features type vgamextraNEWS().
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