marma | R Documentation |
Simulation of MARMA(p,q) processes.
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, ...)
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
|
theta |
A vector of non-negative parameters, of length
|
init |
A vector of non-negative starting values, of
length |
n.start |
A non-negative value denoting the length of the
burn-in period. If |
rand.gen |
A simulation function to generate the innovations. |
... |
Additional arguments for |
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}\}.
A numeric vector of length n
.
Davis, R. A. and Resnick, S. I. (1989) Basic properties and prediction of max-arma processes. Adv. Appl. Prob., 21, 781–803.
evmc
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))
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