arrob.gm: Robust Fitting of Autoregressive Models Based on GM...

In robts: Robust Time Series Analysis

Description

Robust fit of an autoregressive model to a time series by generalized M estimators as described in Maronna et al. (2006).

Usage

arrob.gm(x, order.max, aic = TRUE, aicpenalty = function(p) 2*p,
    na.action = na.fail, series = deparse(substitute(x)), k1 = 1.37, k2 = 1,
    delta = 1/2, maxit = 10^3, epsilon = 10^(-4))        

Arguments

x	numeric vector of a univariate time series.
aic	logical indicating whether the AR order should be estimated by robust AIC criteria considering orders up to order.max or should be equal to order.max.
order.max	integer value determining the (maximal) order of the AR fit. If missing, this value is chosen to be floor(min(c((n - 1) / 4, 10 * log(n, base = 10)))), where n is the number of observations.
na.action	function to be called to handle missing values. Default is na.fail. Other reasonable possibilities are na.omit (omits all missing values at the beginning and the end of the time series and fails if there are more missing values), na.contiguous (uses longest contiguous stretch of non-missing observations) and na.extremify (experimental, sets missing values to very extreme values in the expectation that they have only little influence on the outcome of robust estimation methods).
series	the time series name.
aicpenalty	function of the model order defining the penalty term of the model selection criteria. The default results in a robust AIC.
k1	numeric value giving the tuning parameter for the Huber weights.
k2	numeric value giving the tuning parameter for the psi function applied to the regressors.
delta	numeric value for delta, see Details.
maxit	integer value giving rhe maximal number of iterations for the iterative weighting procedure of the M estimators
epsilon	numeric value giving the desired accuracy of the iterated solution of the M estimator.

Details

This procedure fits an AR model to a time series. The AR coefficients are determined by robust regression, using the time series as dependent variable and the lagged time series as independent one. So if there are outliers in the time series, they are also in the regressors. That is why generalized M estimators are proposed which downweight also the explanatory variables. To minimize the complexity of the problem, the Durbin-Levinson algorithm is applied, fitting a sequence of AR processes of increasing order, which break the estimation down to maxp regressions with one prediction variable instead of one regression with maxp predictors.

The GM estimator is of Mallows type with a Huber function for the regression residuals and a bisquare function for the regressors. Both can be tuned by k1 respectively k2. The weights vi of the regressors are based on Malahanobis distances with the covariance matrix build of autocovariances derived by the so far estimated AR parameters, see Maronna et al. (2006) chapter 8.5 for details.

The GM estimator is computed as an iteratively weighted sum. Based on an initial estimator, which is here the least trimmed squares estimator ltsReg of the robustbase package, residuals ri and its related Huber wi and bisquare weights ui are calculated. The updated regression estimate is then

β=\frac{∑_{i=1}^n w_iv_i x_iy_i}{∑_{i=1}^n w_iv_i x_i^2}

and the scale estimate

σ^2_{new}=\frac{σ^2_{old}}{n δ}∑_{i=1}^n u_i r_i^2,

where delta determines the breakdown point of the scale estimator. For delta<1/2 it is equivalent to the breakpoint (in fact the explosion breakdown point) and for delta<1/2 the (implosion) breakdown point equals 1-delta.

The iteration stops if either the maximal change of the residuals is smaller than epsilon or the maximal number of iterations maxit is reached. Subsequently the scale estimate is transformed to be consistent under normality.

Value

Object of classes "arrob" and "ar". This is a list which includes all elements of an object of class "ar" (see ar for details) plus the following additional element:

x	the original time series x.

Note that list element aic gives the value of the robust information criterion and not its difference with the lowest information criterion of all considered models, as it is returned by the function ar.

Author(s)

Alexander Dürre, Tobias Liboschik and Jonathan Rathjens

References

Maronna, R. A., Martin, R. D., and Yohai, V. J. (2006): Robust Statistics: Theory and Methods, Wiley, chapter 8, doi: 10.1002/0470010940.

See Also

The wrapper function arrob.

Classical, nonrobust fitting is provided by the function ar.

Examples

set.seed(1066)
tss <- arima.sim(model = list(ar = 0.3, ma = 0.5), n = 100)
arrob(tss, method="gm")

robts documentation built on May 2, 2019, 4:55 p.m.