Selected Papers of Hirotugu Akaike pp 215-222 | Cite as

# A New Look at the Statistical Model Identification

## Abstract

*The* history of the development of statistical hypothesis testing in time series analysis is reviewed briefly and it is pointed out that the hypothesis testing procedure is not adequately defined as the procedure for statistical model identification. The classical maximum likelihood estimation procedure is reviewed and a new estimate minimum information theoretical criterion (AIC) estimate (MAICE) which is designed for the purpose of statistical identification is introduced. When there are several competing models the MAICE is defined by the model and the maximum likelihood estimates of the parameters which give the minimum of AIC defined by AIC = (−2)log^{-} (maximum likelihood) + 2(number of independently adjusted parameters within the model). MAICE provides a versatile procedure for statistical model identification which is free from the ambiguities inherent in the application of conventional hypothesis testing procedure. The practical utility of MAICE in time series analysis is demonstrated with some numerical examples.

## Keywords

Time Series Analysis Hankel Matrix Classical Maximum Likelihood Gaussian Process Model Final Prediction Error## Preview

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## References

- [1]H. Akaike, “Stochastic theory of minimal realization,” this issue, pp. 667–674.Google Scholar
- [2]E. L. Lehman,
*Testing Statistical Hypothesis*. New York: Wiley, 1959.Google Scholar - [3]H. Akaike, “Information theory and an extension of the maximum likelihood principle,” in Proc.
*2nd Int. Symp. Information Theory*,*Supp. to Problems of Control and Information Theory*, 1972, pp. 267–281.Google Scholar - [4]M. H. Quenouille, “A large-sample test for the goodness of fit of autoregressive schemes,”
*J. Roy. Statist. Soc*., vol. 110, pp. 123–129, 1947.MathSciNetCrossRefzbMATHGoogle Scholar - [5]H. Weld, “A large-sample test for moving averages,”
*J. Roy. Statist. Soc*.,*B*, vol. 11, pp. 297–305, 1949.Google Scholar - [6]M. S. Barlett and P. H. Diananda, “Extensions of Quenouille’s test for autoregressive scheme,”
*J. Roy. Statist. Soc*.,*B*, vol. 12, pp. 108–115, 1950.Google Scholar - [7]M. S. Bartlett and D. V. Rajalakshman, “Goodness of fit test for simultaneous autoregressive series,”
*J. Roy. Statist. Soc*.,*B*, vol. 15, pp. 107–124, 1953.MathSciNetzbMATHGoogle Scholar - [8]A. M. Walker, “Note on a generalization of the large sample goodness of fit test for linear autoregressive schemes,”
*J. Roy. Statist. Soc*.,*B*, vol. 12, pp. 102–107, 1950.zbMATHGoogle Scholar - [9]A. M. Walker,, “The existence of Bartlett-Rajalakshman goodness of fit G-tests for multivariate autoregressive processes with finitely dependent residuals,”
*Proc. Cambridge Phil. Soc*., vol. 54, pp. 225–232, 1957.CrossRefGoogle Scholar - [10]P. Whittle,
*Hypothesis Testing in Time-Series Analysis*. Uppsala, Sweden: Almqvist and Wiksell, 1951.Google Scholar - [11]Some recent contributions to the theory of stationary processes,“ A
*Study in the Analysis of Stationary Time Series*. Uppsala, Sweden: Almqvist and Wiksell, 1954, appendix 2.Google Scholar - [12]G. E. P. Box and G. M. Jenkins,
*Time Series*,*Forecasting and Control*. San Francisco, Calif.: Holden-Day, 1970.Google Scholar - [13]I. Gustaysson, “Comparison of different methods for identifies-t197ion2. of industrial processes,”
*Automatics*,vol. 8, pp. 127–142Google Scholar - [14]R. K. Mehra, “On the identification of variances and adaptive Kalman filtering,”
*IEEE Trans. Automat. Contr*., vol. AC-15, pp. 175–184, Apr. 1970.Google Scholar - [15]R. K. Mehra, “On-line identification of linear dynamic systems with applications to Kalman filtering,”
*IEEE Trans. Automat. Contr*., vol. AC-16, pp. 12–21, Feb. 1971.Google Scholar - [16]E. J. Hannan,
*Time Series Analysis*. London, England: Methuen, 1960.zbMATHGoogle Scholar - [17]T. W. Anderson, “Determination of the order of dependence in normally distributed time series,” in
*Time Series Analysis*, M. Rosenblatt, Ed. New York: Wiley, 1963, pp. 425–446.Google Scholar - [18]C. L. Mallows, “Some comments on Cr,”
*Technometrics*, vol. 15, pp. 661–675, 1973.zbMATHGoogle Scholar - [19]L. D. Davisson, “The prediction error of stationary Gaussian time series of unknown covariance,”
*IEEE Trans. Inform. Theory*, vol. IT-11, pp. 527–532, Oct. 1965.Google Scholar - [20]A theory of adaptive filtering,“
*IEEE*Trans.*Inform. Theory*,vol. IT-12, pp. 97–102, Apr. 1966.Google Scholar - [21]H. Akaike, “Fitting autoregressive models for prediction,”
*Ann. Inst. Statist. Math*., vol. 21, pp. 243–247, 1969.MathSciNetCrossRefzbMATHGoogle Scholar - [22]Statistical predictor identification,“ Ann.
*Inst. Statist.Math*., vol. 22, pp. 203–217, 1970.Google Scholar - [23]On a semiautomatic power spectrum estimation pro-cedure,“ in
*Proc. 3rd Hawaii Int. Conf. System Sciences*,1970, pp. 974–977.Google Scholar - [24]R. H. Jones, “Autoregressive spectrum estimation,” in
*3rd Conf. Probability*and*Statistics In Atmospheric Sciences*,*Preprints*, Boulder, Colo., June 19–22, 1973.Google Scholar - [25]W. Gersch and D. R. Sharpe, “Estimation of power spectra with finite-order autoregressive models,” IEEE
*Trans. Automat. Contr*., vol. AC-18, pp. 367–379, Aug. 1973.Google Scholar - [26]R. J. Bhansali, “A Monte Carlo comparison of the regression method and the spectral methods of prediction,”
*J. Amer. Statist. Ass*., vol. 68, pp. 621–625, 1973.CrossRefzbMATHGoogle Scholar - [27]H. Akaike, “Autoregressive model fitting for control,”
*Ann. Inst. Statist. Math*., vol. 23, pp. 163–180, 1971.MathSciNetCrossRefzbMATHGoogle Scholar - [28]T. Otomo, T. Nakagawa, and H. Akaike, “Statistical approach to computer control of cement rotary kilns,”
*Automatica*, vol. 8, pp. 3548, 1972.CrossRefGoogle Scholar - [29]H. Cramer,
*Mathematical Methods of Statistics*. Princeton, N. J.: Princeton Univ. Press, 1946.Google Scholar - [30]S. Kullback,
*Information Theory and Statistics*. New York: Wiley, 1959.zbMATHGoogle Scholar - [31]M. S. Bartlett, “The statistical approach to the analysis of time series,” in
*Proc. Symp. Information Theory*, London, England, Ministry of Supply, 1950, pp. 81–101.Google Scholar - [32]P. J. Hither, “The behavior of maximum likelihood estimates under nonstandard conditions,” in
*Proc. 5th Berkeley Symp. Mathematical Statistics and Probability*, vol. 1, pp. 221–233, 1967.Google Scholar - [33]H. Akaike, “Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes,”
*Ann. Inst. Statist. Math*.,to be published.Google Scholar - [34]P. Whittle, “Gaussian estimation in stationary time series, ”
*Bull. Int. Statist. Inst*., vol. 39, pp. 105–129, 1962.zbMATHGoogle Scholar - [35]H. Akaike, “Use of an information theoretic quantity for statistical model identification,”
*in Proc. 5th Hawaii Int. Conf. System Sciences*, pp. 249–250, 1972.Google Scholar - [36]H. Akaike, “Automatic data structure search by the maximum likelihood,” in
*Computer in Biomedicine Suppl. to Proc. 5th Hawaii Int. Conf. on System Sciences*, pp. 99–101, 1972.Google Scholar - [37]T. W. Anderson,
*The Statistical Analysis of Time Series*. New York: Wiley, 1971.zbMATHGoogle Scholar - [38]J. D. Sargon, “An approximate treatment of the properties of the cxrrelogram and periodgram,”
*J. Roy. Statist. Soc. B*, vol. 15, pp. 140–152, 1953.Google Scholar - [39]G. M. Jenkins and D. G. Watts,
*Spectral Analysis and its Applications*. San Francisco, Calif.: Holden-Day, 1968.zbMATHGoogle Scholar - [40]P. Whittle, “The statistical analysis of a seiche record,”
*J. Marine Res*., vol. 13, pp. 76–100, 1954.MathSciNetGoogle Scholar - [41]P. Whittle,
*Prediction and Regulation*. London, England: English Univ. Press, 1963.Google Scholar - [42]D. R. Cox, “Tests of separate families of hypotheses,” in
*Proc. 4th Berkeley Symp. Mathematical Statistics and Probability*,*vol*. 1, 1961, pp. 105–123.Google Scholar - [43]D. R. Cox, “Further results on tests of separate families of hypotheses,”
*J. Roy. Statist. Soc*.,*B*, vol. 24, pp. 406–425, 1962.zbMATHGoogle Scholar - [44]A. M. Walker, “Some tests of separate families of hypotheses in time series analysis,”
*Biometrika*, vol. 54 pp. 39–68, 1987.Google Scholar - [45]W. J. Kennedy and T. A. Bancroft, “Model building for prediction in regression based upon repeated significance testa,” Ann.
*Math. Statist*., vol. 42, pp. 1273–1284, 1971.MathSciNetCrossRefzbMATHGoogle Scholar