在日常生活中，許多連續型資料由於記錄機制的精確性，記錄值常被捨入到小數位。而這些捨入誤差將影響測量和估計的準確性。本研究介紹三種對於捨入型資料參數估計的方法，分別是 A-K 調整估計法、近似最大概似估計值 和 SOS 估計法，並且比較三種方法估計值的表現。為了進行比較，我們先推導在 MA(1) 模型的 A-K 調整估計值。另一方面，為了提高估計上的效率，利用上述三種估計值作線性組合，提出了兩種變異數降低法，得到一個新的不偏估計值。模擬結果顯示，新提出的降低變異數的估計值顯著提升估計的效率性。

Most recorded data are rounded to the nearest decimal place due to the precision of the recording mechanism. This rounding entails errors in estimation and measurement. In this paper, we compare the performances of three types of estimators based on rounded data from time series models, namely A-K corrected estimator, approximate MLE and the SOS estimator. In order to perform the comparison, the A-K corrected estimators for the MA(1) model are derived theoretically. To improve the efficiency of the estimation, two types of variance-reduction estimators are further proposed, which are based on linear combinations of aforementioned three estimators. Simulation results show the proposed variance reduction estimators significantly improve the estimation efficiency.

論文審定書 i
謝誌 ii
摘要 iii
Abstract iv
1 Introduction 1
2 Literature Review 3
2.1 A-K correction method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Approximate MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 AMLE: AR(1) model . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 AMLE: MA(p) model . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 SOS method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Modified SOS method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Newly proposed estimators 13
3.1 A-K correction of MA(1) model . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Variance reduction techniques . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Simulation study 17
4.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 AR(1) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 MA(1) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 AR(2) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.5 ARMA(1,1) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Conclusion 23
A Appendix 24
A.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.1.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.1.2 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A.1.3 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.1.4 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
A.1.5 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
A.2 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
A.3 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
References 50

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