Page 153 - Textos de Matemática Vol. 47

PARAMETER ESTIMATION OF BILINEAR PROCESSES USING ABC 143
3.2. The ABC algorithm for the simple bilinear process. For bilinear processes, no su cient statistics, other than the order statistics are known. Therefore, a set of summary statistics capturing several aspects of the model has to be chosen. The following seven summary statistics are selected in the case of the simple bilinear model that is being studied:
(1) the median and the inter-quartile range;
(2) Y =
h=1
1⇢2 2(h) where ⇢Y (h) and ⇢Y2(h) are hYYhY
PP 1⇢2 (h) and 2 =
10
10
h=1
the values of the autocorrelation functions (ACF) of the time series
{Yt} and of the squared values, {Yt2}, computed at lag h, respectively. The weight 1/h is used to give more emphasis to correlations associated to low lags.
(3) The left and right tail indexes, L and R, using the moments estimator defined by Dekkers et al. [10] as
where
Mn 1 kX 1
ˆ=M(1)+1 1 1 (Mn ) n 2 (2)
M(j) =
n k
(logY
n i:n
logY
n k:n
)j,
" (1) 2# 1
i=0
where j = 1, 2. In this case, (Y1:n, Y2:n, . . . , Yn:n) represents an ascend- ing ordered sample of size n. This estimator can take any real value and as such is able to reflect heavy or light-tailed distributions, as well as exponential tails.
(4) The right extremal index, ✓⇤ , where 0 < ✓⇤  1. The extremal index measures how strongly the largest values in the sample cluster. The smaller the value of the extremal index, the stronger the clustering of extreme observations is. If the largest values are independent then ✓⇤ = 1. However, the converse is not true (see, e.g., Coles [8] for details and examples). The usual estimate of the extremal index is ✓ˆ⇤ = nc/nu, where nu represents the number of exceedances over the threshold u and nc the number of clusters above u.
The reason for choosing this set of statistics is to control the similarities between the first and second order properties, the degree of nonlinearity, as well as the tail behavior of the observed and the simulated time series, and the (possible) clustering of large values.