Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

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

1 2 3 4 | ```
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)
``` |

`x, q, p, n` |
Same as |

`scale, shape` |
Scale and shape parameters, same as |

`log, log.p, lower.tail` |
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*
underlie on `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
`gamma`

.

`dinvweibull()`

returns the density, `pinvweibull()`

computes the
distribution function, `qinvweibull()`

gives the quantiles, and
`rinvweibull()`

generates random numbers from the Inverse Weibull
distribution.

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

.

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.

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
#(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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