# cdfwei: Cumulative Distribution Function of the Weibull Distribution

### Description

This function computes the cumulative probability or nonexceedance probability of the Weibull distribution given parameters (ζ, β, and δ) of the distribution computed by parwei. The cumulative distribution function is

F(x) = 1 - \exp(Y^δ) \mbox{,}

where Y is

Y = -\frac{x+ζ}{β}\mbox{,}

where F(x) is the nonexceedance probability for quantile x, ζ is a location parameter, β is a scale parameter, and δ 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 (ξ, α, and κ) are

κ = 1/δ \mbox{,}

α = β/δ \mbox{, and}

ξ = ζ - β \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$para[2]). For the current implementation for this package, the reversed Generalized Extreme Value distribution is used 1-cdfgev(-x,para). ### Usage  1 cdfwei(x, para)  ### Arguments  x A real value vector. para The parameters from parwei or vec2par. ### Value Nonexceedance probability (F) for x. ### Author(s) W.H. Asquith ### References Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press. ### See Also pdfwei, quawei, lmomwei, parwei ### Examples   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17  # Evaluate Weibull deployed here and within R (pweibull) lmr <- lmoms(c(123,34,4,654,37,78)) WEI <- parwei(lmr) F1 <- cdfwei(50,WEI) F2 <- pweibull(50+WEI$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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