Geodesic
Improvement of prediction for a larger number of steps in discrete stationary processes
Applications of Mathematics, Tome 27 (1982) no. 2, pp. 118-127
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Let $\{W_t\}=\{(X'_{t'}, Y'_t)'\}$ be vector ARMA $(m,n)$ processes. Denote by $\hat{X}_t(a)$ the predictor of $X_t$ based on $X_{t-a}, X_{t-a-1}, \ldots$ and by $\hat{X}_t(a,b)$ the predictor of $X_t$ based on $X_{t-a}, X_{t-a-1}, \ldots, Y_{t-b},Y_{t-b-1}, \ldots$. The accuracy of the predictors is measured by $\Delta_X(a)=\text{E}[X_t-\hat{X}_t(a)][X_t-\hat{X}_t(a)]'$ and $\Delta_X(a,b)=\text{E}[X_t-\hat{X}_t(a,b)][X_t-\hat{X}_t(a,b)]'$. A general sufficient condition for the equality $\Delta_X(a)=\Delta_X(a,a)]$ is given in the paper and it is shown that the equality $\Delta_X(1)=\Delta_X(1,1)]$ implies $\Delta_X(a)=\Delta_X(a,a)]$ for all natural numbers $a$.
Let $\{W_t\}=\{(X'_{t'}, Y'_t)'\}$ be vector ARMA $(m,n)$ processes. Denote by $\hat{X}_t(a)$ the predictor of $X_t$ based on $X_{t-a}, X_{t-a-1}, \ldots$ and by $\hat{X}_t(a,b)$ the predictor of $X_t$ based on $X_{t-a}, X_{t-a-1}, \ldots, Y_{t-b},Y_{t-b-1}, \ldots$. The accuracy of the predictors is measured by $\Delta_X(a)=\text{E}[X_t-\hat{X}_t(a)][X_t-\hat{X}_t(a)]'$ and $\Delta_X(a,b)=\text{E}[X_t-\hat{X}_t(a,b)][X_t-\hat{X}_t(a,b)]'$. A general sufficient condition for the equality $\Delta_X(a)=\Delta_X(a,a)]$ is given in the paper and it is shown that the equality $\Delta_X(1)=\Delta_X(1,1)]$ implies $\Delta_X(a)=\Delta_X(a,a)]$ for all natural numbers $a$.
DOI : 10.21136/AM.1982.103952
Classification : 60G10, 60G25, 62M20
Keywords: improvement of prediction; discrete stationary process 
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Cipra, Tomáš. Improvement of prediction for a larger number of steps in discrete stationary processes. Applications of Mathematics, Tome 27 (1982) no. 2, pp. 118-127. doi: 10.21136/AM.1982.103952

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