ExtDist (version 0.3.3)

JohnsonSB: The Johnson SB 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

dJohnsonSB(x, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
  params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2))
pJohnsonSB(q, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
  params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2))
qJohnsonSB(p, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
  params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2))
rJohnsonSB(n, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
  params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2))
eJohnsonSB(X, w, method = "numerical.MLE")
lJohnsonSB(X, w, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
  params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2),
  logL = TRUE)

Arguments

x,q

vector of quantiles.

gamma,delta

shape parameters.

xi,lambda

location-scale 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, lJohnsonSB gives log likelihood.

...

other parameters

Value

• dJohnsonSB gives the density; pJohnsonSB gives the distribution function; qJohnsonSB gives the quantile function; rJohnsonSB generates random variables; eJohnsonSB estimate the parameters

Details

Johnson SB Distribution

Johnson SB is bounded in [xi,xi+lambda]. See ../doc/Distributions-Johnson-SB.html{Distributions-Johnson-SB}

Examples

# Parameter estimation
n <- 500
gamma = -0.5; delta = 2; xi = -0.5; lambda = 2
X <- rJohnsonSB(n, gamma, delta, xi, lambda)
(est.par <- eJohnsonSB(X))
# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dJohnsonSB(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(qJohnsonSB((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, pJohnsonSB(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(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2)
X <- rJohnsonSB(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)
eJohnsonSB(X,w) # estimated parameters of weighted sample
eJohnsonSB(X) # estimated parameters of unweighted sample

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

# evaluate the performance of the parameter estimation function by simulation
eval.estimation(rdist=rJohnsonSB,edist=eJohnsonSB,n = 1000, rep.num = 1e3,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2), method ="numerical.MLE")

# evaluate the precision of estimation by Hessian matrix
X <- rJohnsonSB(1000, gamma, delta, xi, lambda)
(est.par <- eJohnsonSB(X))
H <- attributes(eJohnsonSB(X, method = "numerical.MLE"))$nll.hessian
fisher_info <- solve(H)
sqrt(diag(fisher_info))

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