MixExp2: Mixture of two exponential distributions

In Renext: Renewal Method for Extreme Values Extrapolation

Mixture of two exponential distributions

Description

Probability functions associated to the mixture of two exponential distributions.

Usage

   dmixexp2(x, prob1,
            rate1 = 1.0, rate2 = rate1 + delta, delta,
            log = FALSE)
   pmixexp2(q, prob1,
            rate1 = 1.0, rate2 = rate1 + delta, delta,
            log = FALSE)
   qmixexp2(p, prob1,
            rate1 = 1.0, rate2 = rate1 + delta, delta)
   rmixexp2(n, prob1,
            rate1 = 1.0, rate2 = rate1 + delta, delta)
   hmixexp2(x, prob1,
            rate1 = 1.0, rate2 = rate1 + delta, delta)
   Hmixexp2(x, prob1,
            rate1 = 1.0, rate2 = rate1 + delta, delta)

Arguments

x, q	Vector of quantiles.
p	Vector of probabilities.
n	Number of observations.
log	Logical; if TRUE, the log density is returned.
prob1	Probability weight for the "number 1" exponential density.
rate1	Rate (inverse expectation) for the "number 1" exponential density.
rate2	Rate (inverse expectation) for the "number 2" exponential density. Should in most cases be > rate1. See Details.
delta	Alternative parameterisation delta = rate2 - rate1.

Details

The density function is the mixture of two exponential densities

f(x) = \alpha_1 \lambda_1 \, e^{-\lambda_1 x} + (1-\alpha_1) \lambda_2 \, e^{-\lambda_2x} \qquad x > 0

where \alpha_1 is the probability given in prob1 while \lambda_1 and\lambda_2 are the two rates given in rate1 and rate2.

A 'naive' identifiability constraint is

\lambda_1 < \lambda_2

i.e. rate1 < rate2, corresponding to the simple constraint delta > 0. The parameter delta can be given instead of rate2.

The mixture distribution has a decreasing hazard, increasing Mean Residual Life (MRL) and has a thicker tail than the usual exponential. However the hazard, MRL have a finite non zero limit and the distribution behaves as an exponential for large return levels/periods.

The quantile function is not available in closed form and is computed using a dedicated numerical method.

Value

dmiwexp2, pmiwexp2, qmiwexp2, evaluates the density, the distribution and the quantile functions. dmixexp2 generates a vector of n random draws from the distribution. hmixep2 gives hazard rate and Hmixexp2 gives cumulative hazard.

Examples

rate1 <- 1.0
rate2 <- 4.0
prob1 <- 0.8
qs <- qmixexp2(p = c(0.99, 0.999), prob1 = prob1,
               rate1 = rate1, rate2 = rate2) 
x <- seq(from = 0, to = qs[2], length.out = 200)
F <- pmixexp2(x, prob1 = prob1, rate1 = rate1, rate2 = rate2)
plot(x, F, type = "l", col = "orangered", lwd = 2,
     main = "Mixexp2 distribution and quantile for p = 0.99")
abline(v = qs[1])
abline(h = 0.99)

Renext documentation built on Aug. 30, 2023, 1:06 a.m.