A new look at the statistical model identification

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.

[1]  M. H. Quenouille.  A Large‐Sample Test for the Goodness of Fit of Autoregressive Schemes , 1947 .

[2]  H. Wold,et al.  A Large-Sample Test for Moving Averages , 1949 .

[3]  A. M. Walker.  Note on a Generalization of the Large Sample Goodness of Fit Test for Linear Autoregressive Schemes , 1950 .

[4]  M. S. Bartlett,et al.  Extensions of Quenouille's Test for Autoregressive Schemes , 1950 .

[5]  Peter Whittle,et al.  Hypothesis Testing in Time Series Analysis. , 1951 .

[6]  M. Bartlett,et al.  The statistical approach to the analysis of time-series , 1953, Trans. IRE Prof. Group Inf. Theory.

[7]  M. Bartlett,et al.  Goodness of Fit Tests for Simultaneous Autoregressive Series , 1953 .

[8]  M. Kendall,et al.  A Study in the Analysis of Stationary Time-Series. , 1955 .

[9]  D. Lindley,et al.  The existence of Bartlett-Rajalakshman goodness of fit G -tests for multivariate autoregressive processes with finitely dependent residuals , 1958 .

[10]  Solomon Kullback,et al.  Information Theory and Statistics , 1960 .

[11]  E. Lehmann,et al.  Testing Statistical Hypothesis. , 1960 .

[12]  D. Cox.  Tests of Separate Families of Hypotheses , 1961 .

[13]  F. Downton,et al.  Time Series Analysis , 1961, Mathematical Gazette.

[14]  T. W. Anderson.  DETERMINATION OF THE ORDER OF DEPENDENCE IN NORMALLY DISTRIBUTED TIME SERIES , 1962 .

[15]  David R. Cox,et al.  Further Results on Tests of Separate Families of Hypotheses , 1962 .

[16]  Lee D. Davisson,et al.  The prediction error of stationary Gaussian time series of unknown covariance , 1965, IEEE Trans. Inf. Theory.

[17]  P. Whittle,et al.  Prediction and Regulation. , 1965 .

[18]  Lee D. Davisson.  A theory of adaptive filtering , 1966, IEEE Trans. Inf. Theory.

[19]  Robert B. Ash,et al.  Information Theory , 2020, The SAGE International Encyclopedia of Mass Media and Society.

[20]  P. J. Huber.  The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[21]  A. M. Walker.  Some tests of separate families of hypotheses in time series analysis. , 1967, Biometrika.

[22]  D. G. Watts,et al.  Spectral analysis and its applications , 1968 .

[23]  H. Akaike.  Fitting autoregressive models for prediction , 1969 .

[24]  H. Akaike.  Statistical predictor identification , 1970 .

[25]  R. Mehra.  On the identification of variances and adaptive Kalman filtering , 1970 .

[26]  W. Kennedy,et al.  Model Building for Prediction in Regression Based Upon Repeated Significance Tests , 1971 .

[27]  R. Mehra.  On-line identification of linear dynamic systems with applications to Kalman filtering , 1971 .

[28]  H. Akaike.  Autoregressive model fitting for control , 1971 .

[29]  I. Gustavsson.  Comparison of different methods for identification of industrial processes , 1972 .

[30]  T. Nakagawa,et al.  Statistical approach to computer control of cement rotary kilns , 1972 .

[31]  R. J. Bhansali,et al.  A Monte Carlo Comparison of the Regression Method and the Spectral Methods of Prediction , 1973 .

[32]  W. Gersch,et al.  Estimation of power spectra with finite-order autoregressive models , 1973 .

[33]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[34]  C. L. Mallows.  Some comments on C_p , 1973 .

[35]  H. Akaike.  Markovian Representation of Stochastic Processes and Its Application to the Analysis of Autoregressive Moving Average Processes , 1974 .

[36]  H. Akaike.  Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes , 1974 .

[37]  H. Akaike.  Stochastic theory of minimal realization , 1974 .

[38]  James Clary,et al.  Stochastic theory of minimal realization , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.