臺灣博碩士論文加值系統

本論文中,探討非常態誤差項的自我迴歸模型,且令誤差項服從 Azzalini (1985) 的偏斜常態分配及 Bondon (2009) 的微偏斜常態分配兩種情形。分別推導了其動差估計量,最小平方估計量與最大概似估計量,然後推論其漸近特性。並且用蒙地卡羅法模擬模型參數的估計表現,最後利用此模型到真實的時間序列資料上,並觀察不同模型與真實資料的配適狀況。

In the modelling of non-Gaussian time series, one strategy is to retain the general autoregressive moving average (ARMA) framework and allow the white noise to be non-Gaussian distribution. In this work, we are interested in correlated data exhibiting asymmetry by adopting a non-Gaussian autoregressive model with Azzalini's (1985) skew-normal distribution and Bondon's (2009) epsilon-skew-normal innovations. The moments estimate, least squares estimate and conditional maximum likelihood estimates of the parameters are derived, and their limit distributions are proved. For small sample behavior, we assess the performance of proposed methods through Monte Carlo simulations. Finally, the flexibility of this model is illustrated by fitting it to a real time series.

Contents
1 Introduction
2 Parameter estimation
2.1 Method of moments estimate (MME)
2.2 Method of least squares estimate (LSE)
2.3 Conditional maximum likelihood method (MLE)
3 Simulation study
3.1 Simulation study
3.2 A real time series example
4 Conclusion
Appendix
A. Some statistical Theorems and Lemmas
B. Proofs
Reference

List of Tables
3.1 The true and estimate values of AR(1) sample with phi = 0.2 and Zt ~ SN(10; 1; 2)
3.2 The true and estimate values of AR(1) sample with phi = 0.2 and Zt ~ SN(10; 1; 1)
3.3 The true and estimate values of AR(1) sample with phi = 0.2 and Zt ~ SN(10; 1; 0)
3.4 The true and estimate values of AR(1) sample with phi = 0.2 and Zt ~ ESN(10; 1; 0.9)
3.5 The true and estimate values of AR(1) sample with phi = 0.2 and Zt ~ ESN(10; 1; 0.5)
3.6 The true and estimate values of AR(1) sample with phi = 0.2 and Zt ~ ESN(10; 1; 0)
3.7 The MLE of AR(1) model tted to the dierenced series of the Dow Jones Utilities Index

List of Figures
3.1 The MSE of eta, when phi = 0.2 and Zt ~ SN(10; 1; alpha)
3.2 The MSE of eta, when phi = 0.2 and Zt ~ ESN(10; 1; alpha)
3.3 Basic information for dierenced of Dow JonesUtilities Index, denoted as Xt
3.4 MSE of AR(1) model with SN, ESN and Normal noise are estimated by the estimators of Table 3.7.
3.5 The MSE of AR(1) model with SN, ESN and Normal, noise respectively

Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12(2):171-178.

Bondon, P. (2009). Estimation of autoregressive models with epsilon-skewnormal innovations. Journal of Multivariate Analysis, 100(8):1761{1776.

Brockwell, P. J. and Davis, R. A. (2009). Time series: theory and methods. Springer Science & Business Media.

Genton, M. G. (2004). Skew-elliptical distributions and their applications: a journey beyond normality. CRC Press.

Hutson, A. (2004). Utilizing the fexibility of the epsilon-skew-normal distribution for common regression problems. Journal of Applied Statistics, 31(6):673-683.

Lo, A. W. and Newey, W. K. (1985). A large-sample chow test for the linear simultaneous equation. Economics Letters, 18(4):351-353.

Mudholkar, G. S. and Hutson, A. D. (2000). The epsilon-skew-normal distribution for analyzing near-normal data. Journal of Statistical Planning and Inference, 83(2):291-309.

Stout, W. F. (1974). Almost sure convergence, volume 154. Academic press New York.