evd (version 2.3-3)

marma: Simulate MARMA(p,q) Processes

Description

Simulation of MARMA(p,q) processes.

Usage

marma(n, p = 0, q = 0, psi, theta, init = rep(0, p), n.start = p,
    rand.gen = rfrechet, …)
mar(n, p = 1, psi, init = rep(0, p), n.start = p, rand.gen =
    rfrechet, …)
mma(n, q = 1, theta, rand.gen = rfrechet, …)

Arguments

n

The number of observations.

p

The AR order of the MARMA process.

q

The MA order of the MARMA process.

psi

A vector of non-negative parameters, of length p. Can be omitted if p is zero.

theta

A vector of non-negative parameters, of length q. Can be omitted if q is zero.

init

A vector of non-negative starting values, of length p.

n.start

A non-negative value denoting the length of the burn-in period. If n.start is less than p, then p minus n.start starting values will be included in the output series.

rand.gen

A simulation function to generate the innovations.

…

Additional arguments for rand.gen. Most usefully, the scale and shape parameters of the innovations generated by rfrechet can be specified by scale and shape respectively.

Value

A numeric vector of length n.

Details

A max autoregressive moving average process \(\{X_k\}\), denoted by MARMA(p,q), is defined in Davis and Resnick (1989) as satisfying $$X_k = \max\{\phi_1 X_{k-1}, \ldots, \phi_p X_{k-p}, \epsilon_k, \theta_1 \epsilon_{k-1}, \ldots, \theta_q \epsilon_{k-q}\}$$ where \(\code{phi} = (\phi_1, \ldots, \phi_p)\) and \(\code{theta} = (\theta_1, \ldots, \theta_q)\) are non-negative vectors of parameters, and where \(\{\epsilon_k\}\) is a series of iid random variables with a common distribution defined by rand.gen.

The functions mar and mma generate MAR(p) and MMA(q) processes respectively. A MAR(p) process \(\{X_k\}\) is equivalent to a MARMA(p, 0) process, so that $$X_k = \max\{\phi_1 X_{k-1}, \ldots, \phi_p X_{k-p}, \epsilon_k\}.$$ A MMA(q) process \(\{X_k\}\) is equivalent to a MARMA(0, q) process, so that $$X_k = \max\{\epsilon_k, \theta_1 \epsilon_{k-1}, \ldots, \theta_q \epsilon_{k-q}\}.$$

References

Davis, R. A. and Resnick, S. I. (1989) Basic properties and prediction of max-arma processes. Adv. Appl. Prob., 21, 781--803.

See Also

evmc

Examples

# NOT RUN {
marma(100, p = 1, q = 1, psi = 0.75, theta = 0.65)
mar(100, psi = 0.85, n.start = 20)
mma(100, q = 2, theta = c(0.75, 0.8))
# }