Improvement of prediction for a larger number of steps in discrete stationary processes

Tomáš Cipra

Aplikace matematiky (1982)

  • Volume: 27, Issue: 2, page 118-127
  • ISSN: 0862-7940

Abstract

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Let { W t } = { ( X t ' ' , Y t ' ) ' } be vector ARMA ( m , n ) processes. Denote by X ^ t ( a ) the predictor of X t based on X t - a , X t - a - 1 , ... and by X ^ t ( a , b ) the predictor of X t based on X t - a , X t - a - 1 , ... , Y t - b , Y t - b - 1 , ... . The accuracy of the predictors is measured by Δ X ( a ) = E [ X t - X ^ t ( a ) ] [ X t - X ^ t ( a ) ] ' and Δ X ( a , b ) = E [ X t - X ^ t ( a , b ) ] [ X t - X ^ t ( a , b ) ] ' . A general sufficient condition for the equality Δ X ( a ) = Δ X ( a , a ) ] is given in the paper and it is shown that the equality Δ X ( 1 ) = Δ X ( 1 , 1 ) ] implies Δ X ( a ) = Δ X ( a , a ) ] for all natural numbers a .

How to cite

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Cipra, Tomáš. "Improvement of prediction for a larger number of steps in discrete stationary processes." Aplikace matematiky 27.2 (1982): 118-127. <http://eudml.org/doc/15230>.

@article{Cipra1982,
abstract = {Let $\lbrace W_t\rbrace =\lbrace (X^\{\prime \}_\{t^\{\prime \}\}, Y^\{\prime \}_t)^\{\prime \}\rbrace $ 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 $\Delta _X(a)=\text\{E\}[X_t-\hat\{X\}_t(a)][X_t-\hat\{X\}_t(a)]^\{\prime \}$ and $\Delta _X(a,b)=\text\{E\}[X_t-\hat\{X\}_t(a,b)][X_t-\hat\{X\}_t(a,b)]^\{\prime \}$. 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 $a$.},
author = {Cipra, Tomáš},
journal = {Aplikace matematiky},
keywords = {improvement of prediction; discrete stationary process; improvement of prediction; discrete stationary process},
language = {eng},
number = {2},
pages = {118-127},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Improvement of prediction for a larger number of steps in discrete stationary processes},
url = {http://eudml.org/doc/15230},
volume = {27},
year = {1982},
}

TY - JOUR
AU - Cipra, Tomáš
TI - Improvement of prediction for a larger number of steps in discrete stationary processes
JO - Aplikace matematiky
PY - 1982
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 27
IS - 2
SP - 118
EP - 127
AB - Let $\lbrace W_t\rbrace =\lbrace (X^{\prime }_{t^{\prime }}, Y^{\prime }_t)^{\prime }\rbrace $ 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 $\Delta _X(a)=\text{E}[X_t-\hat{X}_t(a)][X_t-\hat{X}_t(a)]^{\prime }$ and $\Delta _X(a,b)=\text{E}[X_t-\hat{X}_t(a,b)][X_t-\hat{X}_t(a,b)]^{\prime }$. 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 $a$.
LA - eng
KW - improvement of prediction; discrete stationary process; improvement of prediction; discrete stationary process
UR - http://eudml.org/doc/15230
ER -

References

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  4. T. Cipra, Correlation and improvement of prediction in multivariate stationary processes, Ph. D. dissertation (Department of Statistics, Charles University, Prague, 1980) (in Czech). (1980) 
  5. T. Cipra, Improvement of prediction in multivariate stationary processes, Kybernetika 17 (1981), 234-243. (1981) MR0628211
  6. T. Cipra, On improvement of prediction in ARMA processes, Math. Operationsforsch. Statist., Ser. Statistics 12(1981), 567-580. (1981) Zbl0514.62103MR0639253
  7. W. A. Fuller, Introduction to statistical time series, Wiley, New York, 1976. (1976) Zbl0353.62050MR0448509
  8. C. W. J. Granger, 10.2307/1912791, Econometrica 37 (1969), 424-438. (1969) DOI10.2307/1912791
  9. D. A. Pierce L. D. Haugh, 10.1016/0304-4076(77)90039-2, Journal of Econometrics 5 (1977), 265-293. (1977) MR0443261DOI10.1016/0304-4076(77)90039-2
  10. Yu. V. Rozanov, Stationary random processes, Gos. izd., Moskva, 1963 (in Russian). (1963) 
  11. C. A. Sims, Money, income and causality, American Economic Review 62 (1972), 540-552. (1972) 

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