Distributacalcul (version 0.4.0)

Weibull: Weibull Distribution

Description

Weibull distribution with shape parameter \(\tau\) and rate parameter \(\beta\).

Usage

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)

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.

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.

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)

# With rate parameter
expValWeibull(shape = 2, rate = 0.2)

# With scale parameter
varWeibull(shape = 2, scale = 5)

# With rate parameter
varWeibull(shape = 2, rate = 0.2)

# With scale parameter
kthMomentWeibull(k = 2, shape = 2, scale = 5)

# With rate parameter
kthMomentWeibull(k = 2, shape = 2, rate = 0.2)

# With scale parameter
expValLimWeibull(d = 2, shape = 2, scale = 5)

# With rate parameter
expValLimWeibull(d = 2, shape = 2, rate = 0.2)

# With scale parameter
expValTruncWeibull(d = 2, shape = 2, scale = 5)

# With rate parameter
expValTruncWeibull(d = 2, shape = 2, rate = 0.2)

# Mean of values greater than d
expValTruncWeibull(d = 2, shape = 2, rate = 0.2, less.than.d = FALSE)

# With scale parameter
stopLossWeibull(d = 2, shape = 3, scale = 4)

# With rate parameter
stopLossWeibull(d = 2, shape = 3, rate = 0.25)

# With scale parameter
meanExcessWeibull(d = 2, shape = 3, scale = 4)

# With rate parameter
meanExcessWeibull(d = 2, shape = 3, rate = 0.25)

# With scale parameter
VatRWeibull(kap = .2, shape = 3, scale = 4)

# With rate parameter
VatRWeibull(kap = .2, shape = 3, rate = 0.25)

# With scale parameter
TVatRWeibull(kap = .2, shape = 3, scale = 4)

# With rate parameter
TVatRWeibull(kap = .2, shape = 3, rate = 0.25)