ExtDist (version 0.3.3)

Burr: The Burr's 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 Burr distribution

Usage

dBurr(x, b = 1, g = 2, s = 2, params = list(b = 1, g = 2, s = 2))
pBurr(q, b = 1, g = 2, s = 2, params = list(b = 1, g = 2, s = 2))
qBurr(p, b = 1, g = 2, s = 2, params = list(b = 1, g = 2, s = 2))
rBurr(n, b = 1, g = 2, s = 2, params = list(b = 1, g = 2, s = 2))
eBurr(X, w, method = "numerical.MLE")
lBurr(X, w, b = 1, g = 2, s = 2, params = list(b = 1, g = 2, s = 2),
  logL = TRUE)

Arguments

x,q

vector of quantiles.

b

scale parameters.

g,s

shape parameters.

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, lBurr gives log likelihood.

...

other parameters

Value

• dBurr gives the density; pBurr gives the distribution function; qBurr gives the quantile function; rBurr generates random variables; eBurr estimate the parameters

Details

Burr's Distribution

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

Examples

# Parameter estimation
n <- 500
b = 1; g = 2; s = 2
X <- rBurr(n, b = 1, g = 2, s = 2)
(est.par <- eBurr(X))

# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dBurr(den.x, b=est.par$b, g=est.par$g, s=est.par$s)
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(qBurr((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, pBurr(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(b=1, g=2, s =2)
X <- rBurr(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)
eBurr(X,w) # estimated parameters of weighted sample
eBurr(X) # estimated parameters of unweighted sample

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

# evaluate the performance of the parameter estimation function by simulation
eval.estimation(rdist=rBurr,edist=eBurr,n = 1000, rep.num = 1e3, params = list(b=1, g=2, s =2))

# evaluate the precision of estimation by Hessian matrix
X <- rBurr(1000, b = 1, g = 2, s = 2)
(est.par <- eBurr(X))
H <- attributes(eBurr(X, method = "numerical.MLE"))$nll.hessian
fisher_info <- solve(H)
sqrt(diag(fisher_info))

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