extremefit (version 1.0.2)

Pareto mix: Pareto mixture distribution

Description

Density, distribution function, quantile function and random generation for the Pareto mixture distribution with \(a\) equal to the shape of the first Pareto Distribution, \(b\) equal to the shape of the second Pareto Distribution and \(c\) is the mixture proportion. The locations and the scales parameters are equals to \(0\) and \(1\).

Usage

pparetomix(q, a = 1, b = 2, c = 0.75)
dparetomix(x, a = 1, b = 2, c = 0.75)
qparetomix(p, a = 1, b = 2, c = 0.75, precision = 10^(-10),
  initvalue = 0.5, Nmax = 1000)
rparetomix(n, a = 1, b = 2, c = 0.75, precision = 10^(-10))

Arguments

q

a vector of quantiles.

a

the shape parameter of the first Pareto Distribution.

b

the shape parameter of the second Pareto Distribution.

c

the value of the mixture proportion.

x

a vector of quantiles.

p

a vector of probabilities.

precision

the precision of the Newton method.

initvalue

the initial value of the Newton method.

Nmax

the maximum of iteration done for the Newton method.

n

the number of observations. If length(n) > 1, the length is taken to be the number required.

Value

dparetomix gives the density, pparetomix gives the distribution function, qparetomix gives the quantile function, and rparetomix generates random deviates.

The length of the result is determined by n for rparetomix, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Details

If the \(a\), \(b\) and \(c\) are not specified, they respectively take the default values \(1\), \(2\) and \(0.75\).

The cumulative Pareto mixture distribution is $$ F(x) = c (1- x ^ {-a}) + ( 1 - c ) (1 - x ^ {-b}), x \ge 1, a >0, b > 0, 0 \le c \le 1 $$ where \(a\) and \(b\) are the shapes of the distribution and \(c\) is the mixture proportion.

Examples

# NOT RUN {
par(mfrow = c(3,1))
plot(function(x) dparetomix(x), 0, 5,ylab="density",
     main = " Pareto mixture density ")
mtext("dparetomix(x)", adj = 0)

plot(function(x) pparetomix(x), 0, 5,ylab="distribution function",
     main = " Pareto mixture Cumulative ")
mtext("pparetomix(x)", adj = 0)

plot(function(x) qparetomix(x), 0, 1,ylim=c(0,5),ylab="quantiles",
     main = " Pareto mixture Quantile ")
mtext("qparetomix(x)", adj = 0)

#generate a sample of the Pareto mix distribution of size n
n <- 100
x <- rparetomix(n)


# }