pdfwei {lmomco}R Documentation

Probability Density Function of the Weibull Distribution

Description

This function computes the probability density of the Weibull distribution given parameters (zeta, β, and delta) of the distribution computed by parwei. The probability density function of the distribution is

f(x) =

where f(x) is the probability density for quantile x, zeta is a location parameter, β is a scale parameter, and delta is a shape parameter.

The Weibull distribution is a reverse Generalized Extreme Value distribution. As result, the Generalized Extreme Value algorithms are used for implementation of the Weibull in this package. The relation between the Generalized Extreme Value parameters (xi, α, and kappa) is

kappa = 1/delta mbox{,}

α = β/delta mbox{, and}

xi = zeta - β mbox{.}

These relations are taken from Hosking and Wallis (1997).

In R the probability distribution function of the Weibull distribution is pweibull. Given a Weibull parameter object para, the R syntax is pweibull(x+para$para[1], para$para[3], scale=para$para[2]). For the current implementation for this package, the reversed Generalized Extreme Value distribution is used pdfgev(-x,para).

Usage

pdfwei(x, para)

Arguments

x A real value.
para The parameters from parwei or similar.

Value

Probability density (f) for x.

Author(s)

W.H. Asquith

References

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

quawei, parwei

Examples

  # Evaluate Weibull deployed here and within R (pweibull)
  lmr <- lmom.ub(c(123,34,4,654,37,78))
  WEI <- parwei(lmr)
  F1  <- cdfwei(50,WEI)
  F2  <- pweibull(50+WEI$para[1],shape=WEI$para[3],scale=WEI$para[2])
  if(F1 == F2) EQUAL <- TRUE

  # The Weibull is a reversed generalized extreme value
  Q <- sort(rlmomco(34,WEI)) # generate Weibull sample
  lm1 <- lmoms(Q)    # regular L-moments
  lm2 <- lmoms(-Q)   # L-moment of negated (reversed) data
  WEI <- parwei(lm1) # parameters of Weibull
  GEV <- pargev(lm2) # parameters of GEV
  F <- nonexceeds()  # Get a vector of nonexceedance probs
  plot(pp(Q),Q) 
  lines(cdfwei(Q,WEI),Q,lwd=5,col=8)
  lines(1-cdfgev(-Q,GEV),Q,col=2) # line over laps previous

[Package lmomco version 0.96.3 Index]