Page 103 - Textos de Matemática Vol. 47

PERIODIC THRESHOLD PROCESSES 93
In order to generate the h-step ahead prediction the mean, median or mode of the predictive distribution of Xi+NT+h|Xi+NT can be employed as a point fore- cast. Note that the median and mode are considered as coherent (i.e., integer- valued) predictions, whereas the mean is not. In order to evaluate the prediction performance given by the mean, median or mode of the predictive distribution we can use the square root of the mean squared error (RMSE), the mean absolute error (MAE) or the loss function everything or nothing (LFEN), re- spectively. Note that the h-step-ahead point predictor that minimizes the mean square error (MSE) is given by
with
Xˆi+NT+h
= E [Xi+NT+h|Xi+NT ] X⇥ ` ⇤
= E i+j,j T,T Xi+NT |Xi+NT
⇣⌘
j 1
+ i+j,m (1) p(1) + (2) p(2)
min(x,y) 2 ⇣ ⌘ = P P Cx ↵(k)
m=0 k=1 m j+1 j+1 j+NT+1
m
1 ↵(k) j+1
x m e (k) j+1
(k) (y m) j+1
I(k) , j+1
i+j m i+j m i+j m i+j m
⇣⌘
(1) p(1) + (2) p(2) i m i m i m i m
p(1) := P (Xi+j m+NT 1  ri+j m); i+j m
m=0
` 1 T 1
+ X X i+j,m+j+T w w=0 m=0
p(2) = 1 p(1) , i + j > m;
i+j m i+j m

p(1):=PX
i m i+(N +` w)T m 1
p(2) =1 p(1) , i>m. i m i m
r ; i m
Turning now to the particular case h = 1, the one-step-ahead predictive func- tion is given by
P(Xj+NT+1 = y|Xj+NT = x)
⇣
⌘
⇣
⌘
(y m)!
with (k) = E[Z(k) ], k 2 {1,2}. Finally, from (4.2), the most commonly
used one-step-ahead predictor of Xj+NT+1, takes the form Xˆj+NT+1 = ⇣↵(1) Xj+NT + (1) ⌘P(Xi+NT ri+1)
+ ⇣↵(2) Xj+NT + (2) ⌘P(Xi+NT >ri+1). j+1 j+1
j+1 j+1