Pareto mix {extremefit} R Documentation

## 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.

### 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 ≥ 1, a >0, b > 0, 0 ≤ c ≤ 1

where a and b are the shapes of the distribution and c is the mixture proportion.

### 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.

### Examples

```par(mfrow = c(3,1))
plot(function(x) dparetomix(x), 0, 5,ylab="density",
main = " Pareto mixture density ")

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