本論文考慮在生物資訊、經濟以及無線通訊領域中的統計推論。在生物資訊方面，我們考慮有致命危險的巨細胞病毒當中的互補迴文模式。為了幫助找到巨細胞病毒的複製起點，我們以互補迴文個數為基礎，提供了四種統計量。推導出這四種統計量的分佈及漸進分佈。除了以模擬研究確認理論的正確性，也探討巨細胞病毒的互補迴文表現。

關於經濟方面，我們提出一個關於鞅差過程的檢定統計量。利用 V 統計量的一些理論，得到了所提出的統計量之近似分佈。以模擬研究驗證了該統計量的近似常態性，及其在有限樣本下的檢定水準與檢定力。

關於無線通訊方面的應用，考慮一個立基於移動發射器、基地台接收天線以及頻道散射體間幾何關係的統計模型。我們嚴格地推導出，在接受探測器陣列間，上行線路接受訊號的四階空間相關係數。散射體的空間位置為卜瓦松分佈，且具有二維高斯強度。最終得到的公式僅由幾何模型中少數參數決定。也利用了蒙地卡羅法驗證理論結果的正確性。

最後，為了協助科學家找到具有相似基因表現及改變模式的基因群，計算了基因表現資料的傅立葉係數。無論基因表現資料相關與否，我們都推導出這些傅立葉係數的聯合分佈。未來可利用這些分佈探討立基於模型的聚類分析方法。

In this thesis, we consider statistical inferences for Bioinformatics, Economics and Wireless Communication models. For application of Bioinformatics, we consider the complementary palindrome (CP) pattern for the life-threatening disease virus called cytomegalovirus (CMV). To help search the replication origin of the CMV, four kinds of statistics based on the number of CP's are proposed. The distributions of the four statistics and the asymptotic distributions of some statistics are also derived. A simulation study is performed to confirm the theoretical findings. Some empirical results of CMV DNA sequence are also considered.

For Economic models, we propose a novel test statistic Tn for martingale difference problem. The asymptotic distribution of the proposed Tn is derived by using the theory of V statistic. A simulation study is performed to validate the asymptotic normality of Tn. The simulated sizes and powers of Tn statistic are also presented.

For Wireless Communication application, we consider a model based on idealized geometric relationships among the mobile transmitter, the base-station's receiving antennas, and the scatterers of the channel. The uplink received signal's fourth-order spatial-correlation coefficient across a receiving sensor-array's aperture is rigorously derived. The scatterers' spatial locations are modeled as Poisson distributed, with a Gaussian intensity over a two-dimensional space. The final formula is explicitly in terms of the simple geometric model's few independent parameters. Monte Carlo verification is carried out to support the theoretical results.

Finally, for the purpose of finding similar gene expression and change patterns, the Fourier coefficients (FC's) of gene expression data are calculated. For correlated and uncorrelated gene expression data, the joint distributions of these FC's are derived. With these distributions, the model-based clustering analysis can be considered in the future.

論文審定書 i

誌謝 ii

中文摘要 iv

Abstract v

1 Introduction 1
Reference 3

2 Complementary Palindrome in CMV Data 7
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Notations and De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Four Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 I2k;t Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Test Statistic I 2k,t . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Quadratic Test Statistic T . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Ratio Statistic Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Martingale Di erence Test 31
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 The Proposed Test Statistic vs the V Statistic . . . . . . . . . . . . . . . . 33
3 Simulation Study and Conclusion . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Higher-Order Correlations Across the Uplink Receiver's Spatial Aper-
ture 51
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2 The Proposed Geometric Model" . . . . . . . . . . . . . . . . . . . . . . . 54
3 Third-Order Spatial Correlation-Coe cient Functions . . . . . . . . . . . . 58
3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Su cient Conditions for Zero Third-Order Spatial Correlation-Coe cient
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Fourth-Order Correlation-Coe cient Function . . . . . . . . . . . . . . . . 61
4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 The Special Case of a Uniform Linear Array of Identical Isotropic
Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 Monte Carlo Veri cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.2 Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Fourier Coe cients in Gene Expression Data 79
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 The Joint Distribution of the Fourier Coe cients . . . . . . . . . . . . . . 81
3.1 Independent Noise Sequence . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Dependent Noise Sequence . . . . . . . . . . . . . . . . . . . . . . . 83
4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1 Proof of Lemma 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Proof of Lemma 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 Proof of Lemma 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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Banfeild, J. D. and Raftery, A. E. (1993). Model-based gaussian and non-gaussian clustering. Biometrics, 49:803-821.
Kim, J. (2011). Clustering change patterns using fourier transformation with time-course gene expression data. Methods in molecular biology (Clifton, N.J.), 734:201-220.
Kim, J. and Kim, H. (2008). Clustering of change patterns using fourier coe cients. Bioinformatics, 24:184-191.