ExtDist (version 0.3.3)

Weibull: The Weibull Distribution.

Description

Density, distribution function, quantile function, random generation function and parameter estimation function (based on weighted or unweighted i.i.d. sample) for the Weibull distribution

Usage

dWeibull(x, shape = 2, scale = 2, params = list(shape = 2, scale = 2))
pWeibull(q, shape = 2, scale = 2, params = list(shape = 2, scale = 2))
qWeibull(p, shape = 2, scale = 2, params = list(shape = 2, scale = 2))
rWeibull(n, shape = 2, scale = 2, params = list(shape = 2, scale = 2))
eWeibull(X, w, method = "numerical.MLE")
lWeibull(X, w, shape = 2, scale = 2, params = list(shape = 2, scale = 2),
  logL = TRUE)

Arguments

x,q

vector of quantiles.

shape

shape parameter.

scale

scale parameter.

params

a list includes all parameters

p

vector of probabilities.

n

number of observations.

X

sample observations.

w

weights of sample.

method

parameter estimation method.

logL

logical; if TRUE, lWeibull gives log likelihood.

...

other parameters

Value

• dWeibull gives the density; pWeibull gives the distribution function; qWeibull gives the quantile function; rWeibull generates random variables; eWeibull estimate the parameters

Details

Weibull Distribution

See ../doc/Distributions-Weibull.html{Distributions-Weibull}

Examples

# Parameter estimation
n <- 500
shape <- 1.5
scale <- 0.5
X <- rWeibull(n, shape, scale)
(est.par <- eWeibull(X))

# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dWeibull(den.x,shape=est.par$shape,scale=est.par$scale)
hist(X, breaks=10, col="red", probability=TRUE, ylim = c(0,1.1*max(den.y)))
lines(den.x, den.y, col="blue", lwd=2)

# Q-Q plot and P-P plot
plot(qWeibull((1:n-0.5)/n, params=est.par), sort(X), main="Q-Q Plot",
xlab="Theoretical Quantiles", ylab="Sample Quantiles", xlim = c(0,5), ylim = c(0,5))
abline(0,1)

plot((1:n-0.5)/n, pWeibull(sort(X), params=est.par), main="P-P Plot",
xlab="Theoretical Percentile", ylab="Sample Percentile", xlim = c(0,1), ylim = c(0,1))
abline(0,1)

# A weighted parameter estimation example
n <- 10
par <- list(shape=1, scale=2)
X <- rWeibull(n, params=par)
w <- c(0.13, 0.06, 0.16, 0.07, 0.2, 0.01, 0.06, 0.09, 0.1, 0.12)
eWeibull(X,w) # estimated parameters of weighted sample
eWeibull(X) # estimated parameters of unweighted sample

# Extracting shape or scale parameters
est.par[attributes(est.par)$par.type=="shape"]
est.par[attributes(est.par)$par.type=="scale"]

# evaluate the performance of the parameter estimation function by simulation
eval.estimation(rdist=rWeibull,edist=eWeibull,n = 1000, rep.num = 1e3,
   params = list(shape=1, scale=2))

# evaluate the precision of estimation by Hessian matrix
X <- rWeibull(1000, shape, scale)
(est.par <- eWeibull(X))
H <- attributes(eWeibull(X, method = "numerical.MLE"))$nll.hessian
fisher_info <- solve(H)
sqrt(diag(fisher_info))

# log-likelihood, score vector and observed information matrix
lWeibull(X,param = est.par)
lWeibull(X,param = est.par, logL=FALSE)