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)