ExtDist (version 0.3.3)

Gamma: The Gamma 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 Gamma distribution

Usage

dGamma(x, shape = 2, scale = 2, params = list(shape = 2, scale = 2))
pGamma(q, shape = 2, scale = 2, params = list(shape = 2, scale = 2))
qGamma(p, shape = 2, scale = 2, params = list(shape = 2, scale = 2))
rGamma(n, shape = 2, scale = 2, params = list(shape = 2, scale = 2))
eGamma(X, w, method = "numerical.MLE")
lGamma(X, w, shape = 2, scale = 2, params = list(shape = 2, scale = 2),
  logL = TRUE)

Arguments

x,q

vector of quantiles.

shape

shape 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, lGamma gives log likelihood.

...

other parameters

Value

• dGamma gives the density; pGamma gives the distribution function; qGamma gives the quantile function; rGamma generates random variables; eGamma estimate the parameters

Details

Gamma Distribution

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

Examples

# Parameter estimation
n <- 500
shape <- 1.5
scale <- 0.5
X <- rGamma(n, shape, scale)
(est.par <- eGamma(X))

# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dGamma(den.x,shape=est.par$shape,scale=est.par$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(qGamma((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, pGamma(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(shape=1, scale=2)
X <- rGamma(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)
eGamma(X,w) # estimated parameters of weighted sample
eGamma(X) # estimated parameters of unweighted sample

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

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

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

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