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研究生： 邱楷涵
Chiu, Kai-Han
論文名稱： 使用絕對值的自回歸模型的模型選擇
Information Criterion for Infinite Variance Autoregressive Models with Least Absolute Deviation Estimation
指導教授： 銀慶剛
Ing, Ching-Kang
口試委員: 冼芻蕘
Sin, Chor-Yiu
俞淑惠
Yu, Shu-Hui
學位類別： 碩士
Master
系所名稱： 理學院 - 統計學研究所
Institute of Statistics
論文出版年： 2022
畢業學年度： 110
語文別： 英文
論文頁數： 22
中文關鍵詞： 絕對值 、自回歸模型 、模型選擇
外文關鍵詞： Least deviation estimation, Stable distribution
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    • 假設Xt 是一個p 維度的自回歸模型，而他們的隨機項則滿足穩定分布的條件。
    我們的目的是透過訊息量準則選擇p，不同於傳統方差的方法，我們這邊選擇用絕對值得方法來選取^p，最後我們證明當隨機項滿足他的穩定分布的係數α介於1到2之間的時候，我們的估計值^p會強收斂到真實的p值。

    Suppose that Xt is a p-th order autoregressive process whose innovation follows a stable distribution. Our purpose is to choose p via an information criterion. Instead of the conventional least squares estimate, our estimate, ^p, of p is based on the least absolute
    deviation. We prove that ^p is strongly consistent when the innovation is stable, with index 1 < α < 2.

    Contents 摘要i Abstract ii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .. . . .1 2 Background Knowledge . .. . . . . . . . . . . . . . . . .. .. . . .2 2.1 Stable distribution . . . .. . . . . . . . . . . . . . . . . . . 2 2.2 LAD estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Theoretical results. . . . . . . . . . . . . . . . . . . . . . . . 4 4 Simulation . . . . . . . . . . . . . . . . . . . . . .. . . .. . . 9 5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .. . .21

    References
    [1] Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (B. Petrov and F. Csaki, eds.) 267-281. Akedemiai Kiado,16 Budapest.
    [2] An, H. Z. and Chen, Z. G. (1982) On convergence of LAD estimates in autoregression with infinite variance.
    J. Multiv. Anal., 12, 335-345.
    [3] Knight, K. (1989) Consistency of Akaike's information criterion for infinite variance autoregressive processes. The Annals of Statistics. 17, 824-840.
    [4] Davis R. A., Knight, K. and Liu, J. (1992) M-estimation for autoregressions with infinite variance. Stoch. Processes Appl., 40, 145-180.
    [5] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, Second ed., Wiley, New York.
    [6] Gross, S. and Steiger, W. L. (1979) Least absolute deviation estimates in autoregression with infinite variance. J. Appl. Probab., 16, 104-116.
    [7] Cline, D. B. H. (1983) Estimation and linear prediction for regression, autoregression and ARMA with infinite variance data. Ph.D. Dissertation. Department of Statistics, Colorado State University, Fort Collins, Colorado.
    [8] Deheuvels, P., Haeusler, E., and Mason, D. (1988) Almost sure convergence of the Hill estimator. Math. Proc. Cambridge Philos. Soc., 104, 371-381.
    [9] Nelder, John A.; R. Mead (1965). A simplex method for function minimization. Computer Journal. 7, 308-313.

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