ExtDist (version 0.3.3)

Normal: The Normal 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 Normal distribution

Usage

dNormal(x, mean = 0, sd = 1, params = list(mean, sd))
pNormal(q, mean = 0, sd = 1, params = list(mean, sd))
qNormal(p, mean = 0, sd = 1, params = list(mean, sd))
rNormal(n, mean = 0, sd = 1, params = list(mean, sd))
eNormal(X, w, method = "numerical.MLE")
lNormal(X, w, mean = 0, sd = 1, params = list(mean, sd), logL = TRUE)
sNormal(X, w, mean = 0, sd = 1, params = list(mean, sd))
iNormal(X, w, mean = 0, sd = 1, params = list(mean, sd))

Arguments

params

a list includes all parameters

x,q

vector of quantiles.

w

weights of sample.

p

vector of probabilities.

n

number of observations.

X

sample observations.

mean

location parameter.

sd

scale parameter.

method

parameter estimation method.

logL

logical; if TRUE, lNormal gives log likelihood.

...

other parameters

Value

• dNormal gives the density; pNormal gives the distribution function; qNormal gives the quantile function; rNormal generates random variables; eNormal estimate the parameters; sNormal gives observed scorn function

Details

Normal Distribution

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

Examples

# Parameter estimation
n <- 500
mean <- 1
sd <- 2
X <- rNormal(n, mean, sd)
(est.par <- eNormal(X))

# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dNormal(den.x,mean=est.par$mean,sd=est.par$sd)
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(qNormal((1:n-0.5)/n, params=est.par), sort(X), main="Q-Q Plot",
xlab="Theoretical Quantiles", ylab="Sample Quantiles", xlim = c(-5,5), ylim = c(-5,5))
abline(0,1)

plot((1:n-0.5)/n, pNormal(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(mean=1, sd=2)
X <- rNormal(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)
eNormal(X,w) # estimated parameters of weighted sample
eNormal(X) # estimated parameters of unweighted sample

# Alternative parameter estimation methods
eNormal(X, method = "numerical.MLE")
eNormal(X, method = "bias.adjusted.MLE")
mean(X); sd(X); sd(X)*sqrt((n-1)/n)

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

# evaluate the performance of the parameter estimation function by simulation
eval.estimation(rdist=rNormal,edist=eNormal,n = 1000, rep.num = 1e3, params = list(mean=1, sd=2))
eval.estimation(rdist=rNormal,edist=eNormal,n = 1000, rep.num = 1e3, params = list(mean=1, sd=2),
method ="analytical.MLE")

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

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