invweibullDist | R Documentation |

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

```
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} \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`

.

`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.

`Weibull`

,
`gamma`

.

```
#(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)
```

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