cdfwei: Cumulative Distribution Function of the Weibull Distribution

Description

This function computes the cumulative probability or nonexceedance probability of the Weibull distribution given parameters ($\zeta$, $\beta$, and $\delta$) of the distribution computed by parwei. The cumulative distribution function is $$F(x) = 1 - \exp(Y^\delta) \mbox{,}$$ where $Y$ is $$Y = -\frac{x+\zeta}{\beta}\mbox{,}$$ where $F(x)$ is the nonexceedance probability for quantile $x$, $\zeta$ is a location parameter, $\beta$ 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 relations between the Generalized Extreme Value parameters ($\xi$, $\alpha$, and $\kappa$) are $$\kappa = 1/\delta \mbox{,}$$ $$\alpha = \beta/\delta \mbox{, and}$$ $$\xi = \zeta - \beta \mbox{,}$$ which are taken from Hosking and Wallis (1997).

In R, the cumulative 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 1-cdfgev(-x,para).

Usage

cdfwei(x, para)

Value

Nonexceedance probability ($F$) for $x$.

Arguments

x: A real value vector.
para: The parameters from parwei or vec2par.

Author

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.

Examples

Run this code

  # Evaluate Weibull deployed here and within R (pweibull)
  lmr <- lmoms(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 overlaps previous

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