ExtDist (version 0.3.3)

Gumbel: The Gumbel 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 Gumbel distribution

Usage

dGumbel(x, location = 0, scale = 1, params = list(location = 0, scale =
  1))
pGumbel(q, location = 0, scale = 1, params = list(location = 0, scale =
  1))
qGumbel(p, location = 0, scale = 1, params = list(location = 0, scale =
  1))
rGumbel(n, location = 0, scale = 1, params = list(location = 0, scale =
  1))
eGumbel(X, w, method = "numerical.MLE")
lGumbel(X, w, location = 0, scale = 1, params = list(location = 0, scale =
  1), logL = TRUE)

Arguments

x,q

vector of quantiles.

location

location 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, lGumbel gives log likelihood.

...

other parameters

Value

• dGumbel gives the density; pGumbel gives the distribution function; qGumbel gives the quantile function; rGumbel generates random variables; eGumbel estimate the parameters

Details

Gumbel Distribution

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

Examples

# Parameter estimation
n <- 500
location <- 1.5
scale <- 0.5
X <- rGumbel(n, location, scale)
(est.par <- eGumbel(X))

# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dGumbel(den.x,location=est.par$location,scale=est.par$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(qGumbel((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, pGumbel(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(location=1, scale=2)
X <- rGumbel(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)
eGumbel(X,w) # estimated parameters of weighted sample
eGumbel(X) # estimated parameters of unweighted sample

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

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

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

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