ExtDist (version 0.3.7)

Beta: The standard 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 Beta distribution

Usage

dBeta(x, shape1 = 2, shape2 = 3, params = list(shape1, shape2))
pBeta(q, shape1 = 2, shape2 = 3, params = list(shape1, shape2))
qBeta(p, shape1 = 2, shape2 = 3, params = list(shape1, shape2))
rBeta(n, shape1 = 2, shape2 = 3, params = list(shape1, shape2))
eBeta(X, w, method = "numerical.MLE")
lBeta(X, w, shape1 = 2, shape2 = 3, params = list(shape1, shape2),
  logL = TRUE)
sBeta(X, w, shape1 = 2, shape2 = 3, params = list(shape1, shape2))
iBeta(X, w, shape1 = 2, shape2 = 3, params = list(shape1, shape2))

Arguments

x,q

vector of quantiles.

shape1,shape2

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, lBeta gives log likelihood.

...

other parameters

Value

• dBeta gives the density; pBeta gives the distribution function; qBeta gives the quantile function; rBeta generates random variables; eBeta estimate the parameters; sBeta gives observed scorn function

Details

standard Beta Distribution

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

Examples

# Parameter estimation
n <- 500
shape1 <- 1
shape2 <- 2
X <- rBeta(n, shape1, shape2)
(est.par <- eBeta(X))

# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dBeta(den.x,shape1=est.par$shape1,shape2=est.par$shape2)
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((1:n-0.5)/n, params=est.par), sort(X), main="Q-Q Plot", xlab="Theoretical Quantiles",
ylab="Sample Quantiles", xlim = c(0,1), ylim = c(0,1))
abline(0,1)

plot((1:n-0.5)/n, pBeta(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=1, shape2=2)
X <- rBeta(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(X,w) # estimated parameters of weighted sample
eBeta(X) # estimated parameters of unweighted sample

# Alternative parameter estimation methods
(est.par <- eBeta(X, method = "numerical.MLE"))

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

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

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

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