ExtDist (version 0.3.3)

Beta_ab: The four-Parameter beta 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 4 Parameter beta distribution

Usage

dBeta_ab(x, shape1 = 2, shape2 = 3, a = 0, b = 1,
  params = list(shape1, shape2, a, b))
pBeta_ab(q, shape1 = 2, shape2 = 3, a = 0, b = 1, params = list(shape1
  = 2, shape2 = 5, a = 0, b = 1))
qBeta_ab(p, shape1 = 2, shape2 = 3, a = 0, b = 1, params = list(shape1
  = 2, shape2 = 5, a = 0, b = 1))
rBeta_ab(n, shape1 = 2, shape2 = 3, a = 0, b = 1,
  params = list(shape1, shape2, a, b))
eBeta_ab(X, w, method = "numerical.MLE")
lBeta_ab(X, w, shape1 = 2, shape2 = 3, a = 0, b = 1,
  params = list(shape1, shape2, a, b), logL = TRUE)
sBeta_ab(X, w, shape1 = 2, shape2 = 3, a = 0, b = 1,
  params = list(shape1, shape2, a, b))

Arguments

x,q

vector of quantiles.

shape1,shape2

shape parameters.

a,b

boundary 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, lBeta_ab gives log likelihood.

...

other parameters

Value

• dBeta_ab gives the density; pBeta_ab gives the distribution function; qBeta_ab gives the quantile function; rBeta_ab generates random variables; eBeta_ab estimate the parameters; sBeta_ab gives observed scorn function

Details

four-Parameter beta Distribution

See ../doc/Distributions-Four-Parameter-Beta.html{Distributions-Four-Parameter-Beta}

Examples

# Parameter estimation
n <- 500
a <- 1
b <- 2
shape1 <- 2
shape2 <- 5
X <- rBeta_ab(n, shape1, shape2, a, b)
(est.par <- eBeta_ab(X))

# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dBeta_ab(den.x,params = est.par)
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(qBeta_ab((1:n-0.5)/n, params=est.par), sort(X), main="Q-Q Plot",
xlab="Theoretical Quantiles", ylab="Sample Quantiles", xlim = c(a,b), ylim = c(a,b))
abline(0,1)

plot((1:n-0.5)/n, pBeta_ab(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(shape1=2, shape2=5, a= 1, b=2)
X <- rBeta_ab(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)
eBeta_ab(X,w) # estimated parameters of weighted sample
eBeta_ab(X) # estimated parameters of unweighted sample

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

# evaluate the performance of the parameter estimation function by simulation
eval.estimation(rdist=rBeta_ab,edist=eBeta_ab,n = 1000, rep.num = 1e3,
params = list(shape1=2, shape2=5, a=0, b=1), method ="numerical.MLE")

# evaluate the precision of estimation by Hessian matrix
X <- rBeta_ab(1000, shape1, shape2, a, b)
(est.par <- eBeta_ab(X))
H <- attributes(eBeta_ab(X, method = "numerical.MLE"))$nll.hessian
fisher_info <- solve(H)
sqrt(diag(fisher_info))

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