In this article, we propose a generalized Akaike's information criterion (AIC) (GAIC), which includes the usual AIC as a special case, for general class of stochastic models (i.e. i.i.d., non-i.i.d., time series models etc.). Then we derive the asymptotic distribution of selected orderp by GAIC, and show thatp is inconsistent, i.e.p 6 ! p 0 (true order). This is the problem of selection by completely specified models. In practice, it is natural to suppose that the true model g would be incompletely specified by uncertain prior information, and be contiguous to a fundamental parametric model f h 0 with dim h 0 ¼ p 0 . One plausible parametric description for g is f ðh0;h= ffiffi n p Þ , h ¼ (h 1 , . . . ,h K À p 0 ) 0 where n is the sample size, and the true order is K. Under this setting, we derive the asymptotic distribution ofp. Then it is shown that GAIC has admissible properties for perturbation of models with order of Oðkhk= ffiffiffi n p Þ, where the length ||h|| is large. This observation seems important. Also numerical studies will be given to confirm the results.
Time series analysis under stationary assumption has been well established. However, stationary time series models are not plausible to describe the real world. Indeed, relatively long stretches of time series data should contain either slow or rapid changes in the spectra. To develop a general non-stationary theory, we have to pay careful attention to constituting a suitable model, otherwise the observations obtained in the future give no information about the present structure. Dahlhaus [1–4] has introduced an important class of non-stationary processes, called locally stationary processes which have the time varying spectral densities. In this paper, for a clustering problem of stock returns in Tokyo Stock Exchanges, we propose nonparametric approach based on generalized integral functional measures of the time varying spectral densities. The generalized measures include Gaussian Kullback–Leibler information and Chernoff information measures. The clustering results well extract the features of the relationship among the companies.
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