Weibull Distribution — Weibull • Distributacalcul

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