Weibull distribution with shape parameter $$\tau$$ and rate parameter $$\beta$$.

expValWeibull(shape, rate = 1/scale, scale = 1/rate)

varWeibull(shape, rate = 1/scale, scale = 1/rate)

kthMomentWeibull(k, shape, rate = 1/scale, scale = 1/rate)

expValLimWeibull(d, shape, rate = 1/scale, scale = 1/rate)

expValTruncWeibull(
d,
shape,
rate = 1/scale,
scale = 1/rate,
less.than.d = TRUE
)

stopLossWeibull(d, shape, rate = 1/scale, scale = 1/rate)

meanExcessWeibull(d, shape, rate = 1/scale, scale = 1/rate)

VatRWeibull(kap, shape, rate = 1/scale, scale = 1/rate)

TVatRWeibull(kap, shape, rate = 1/scale, scale = 1/rate)

## Arguments

shape shape parameter $$\tau$$, must be positive rate parameter $$\beta$$, must be positive. alternative parameterization to the rate parameter, scale = 1 / rate. kth-moment. cut-off value. logical; if TRUE (default) truncated mean for values <= d, otherwise, for values > d. probability.

## Value

Function :

• expValWeibull gives the expected value.

• varWeibull gives the variance.

• kthMomentWeibull gives the kth moment.

• expValLimWeibull gives the limited mean.

• expValTruncWeibull gives the truncated mean.

• stopLossWeibull gives the stop-loss.

• meanExcessWeibull gives the mean excess loss.

• VatRWeibull gives the Value-at-Risk.

• TVatRWeibull gives the Tail Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

## Details

The Weibull distribution with shape parameter $$\tau$$ and rate parameter $$\beta$$ has density: $$f\left(x\right) = \beta \tau \left( \beta x \right) ^{\tau -1} % \mathrm{e}^{-\left( \beta x\right) ^{\tau }}$$ for $$x \in \mathcal{R}^+$$, $$\beta > 0$$, $$\tau > 0$$

## Examples

# With scale parameter
expValWeibull(shape = 2, scale = 5)#> [1] 4.431135
# With rate parameter
expValWeibull(shape = 2, rate = 0.2)#> [1] 4.431135
# With scale parameter
varWeibull(shape = 2, scale = 5)#> [1] 5.365046
# With rate parameter
varWeibull(shape = 2, rate = 0.2)#> [1] 5.365046
# With scale parameter
kthMomentWeibull(k = 2, shape = 2, scale = 5)#> [1] 25
# With rate parameter
kthMomentWeibull(k = 2, shape = 2, rate = 0.2)#> [1] 25
# With scale parameter
expValLimWeibull(d = 2, shape = 2, scale = 5)#> [1] 6.135422
# With rate parameter
expValLimWeibull(d = 2, shape = 2, rate = 0.2)#> [1] 6.135422
# With scale parameter
expValTruncWeibull(d = 2, shape = 2, scale = 5)#> [1] 4.431135
# With rate parameter
expValTruncWeibull(d = 2, shape = 2, rate = 0.2)#> [1] 4.431135
# Mean of values greater than d
expValTruncWeibull(d = 2, shape = 2, rate = 0.2, less.than.d = FALSE)#> [1] 1.869292e-42
# With scale parameter
stopLossWeibull(d = 2, shape = 3, scale = 4)#> [1] -1.764994
# With rate parameter
stopLossWeibull(d = 2, shape = 3, rate = 0.25)#> [1] -1.764994
# With scale parameter
meanExcessWeibull(d = 2, shape = 3, scale = 4)#> [1] -2
# With rate parameter
meanExcessWeibull(d = 2, shape = 3, rate = 0.25)#> [1] -2
# With scale parameter
VatRWeibull(kap = .2, shape = 3, scale = 4)#> [1] 2.426171
# With rate parameter
VatRWeibull(kap = .2, shape = 3, rate = 0.25)#> [1] 2.426171
# With scale parameter
TVatRWeibull(kap = .2, shape = 3, scale = 4)#> [1] 4.017289
# With rate parameter
TVatRWeibull(kap = .2, shape = 3, rate = 0.25)#> [1] 4.017289