Page 155 - Textos de Matemática Vol. 47

PARAMETER ESTIMATION OF BILINEAR PROCESSES USING ABC 145
aˆ0 b0
aˆ
-0.6 0.1796147
-0.5 0.1796168 -0.4 0.1796255 -0.3 0.1796203 -0.2 0.1796239 -0.1 0.1796172 0.1 0.179616 0.2 0.1796172 0.3 0.1796151 0.4 0.1796115 0.5 0.1796226 0.6 0.1796132 0.7 0.179616
ˆb 0.3224413 0.3224415 0.3224426 0.3224419 0.3224424 0.3224416 0.3224413 0.322442 0.3224414 0.3224408 0.3224421 0.3224436 0.3224434
ˆ2 1.059738 1.059738 1.059738 1.059738 1.059738 1.059738 1.059738 1.059738 1.059738 1.059738 1.059738 1.059738 1.059738
Num. Iterations 28
11
9
8
7
7
5
5
4
5
8
12
10
0.2214939
and
a⇠Unif[ 0.9,0.9], b⇠Unif([ 0.9, 0.1][[0.1,0.9]) 2 |a,b⇠Unif[0.1,(1 a2)/b2],
Table 1. CML estimates - Model I (n = 150) (The model did not converge for b0 = 0.7).
As mentioned before, the bilinear models have likelihood functions di cult to handle, which makes the ABC methodology a especially attractive alternative for parameter estimation (Turkman et al. [38]).
Let D be the fixed random sample of size n simulated from the model (2.2) with some a, b and 2. The prior distributions for (a,b, 2) are:
where Unif[c,d] stands for the uniform distribution in the interval [c,d].
The summary statistics to be considered are meant to reflect the similarities of location, dispersion, autocorrelation, tail-weight and the clustering of large values (not used in this case because the sample size is relatively small) between the fixed sample D and a very large number of samples (N = 107) simulated from the bilinear model presented in (2.2), conditional to the values a, b and 2. By considering the euclidean distance and the 0.90 percentile of the N = 107 distances calculated, a final sample of size kN = 1000 was obtained. The conditions that guarantee stationarity and invertibility were verified at each
iteration.
The values of the approximate posterior distribution, representing the best
1000 values of a, b and 2, corresponding to the sample of size n = 150 previ- ously simulated from the model
Model I: Xt = 0.2Xt 1 + 0.4Xt 1✏t 1 + ✏t,