人類巨噬細胞病毒(CMV)的傳播範圍遍佈全球，它會侵入特定的細胞組織並加以控制，進而達到生存繁殖的目的。在巨噬細胞病毒的DNA基因組中，「複製源」是一段記錄複製病毒的基因序列。本文中我們推導出一些互補迴文檢定統計方法，以縮短找尋複製源位置所需要的時間。
定義X_(2k)為DNA序列中長度2k的互補迴文個數，Y_(2k)則為長度2k且剔除覆蓋情況的互補迴文個數。以DNA序列中四種核甘酸出現機率相同與否，定為虛無假設及對立假設，並利用Y_(2k)條件分佈所組成的聯合機率分佈進行概似比檢定。在虛無假設下，檢定統計量近似一個乘上係數的卡方分配，其係數與自由度則用動差法加以估計。利用X_(2k)的邊際分佈，我們也可以做Pearson卡方檢定，其虛無假設下的檢定統計量也近似一個乘上係數的卡方分配。此外，X_(2k)/X_(2(k+1))和Y_(2k)/Y_(2(k+1))的比值在虛無假設下會近似特殊值。本文最後，再以模擬研究方式，對其理論加以驗證。

The human cytomegalovirus (CMV) is one of the viruses which extensively infect in the world. In order to grow and reproduce, the CMV invades designated cellular lives and influences their behavior. The origin of replication (also called the replication origin) is a particular sequence in the CMV DNA genome at which replication is initiated. In this study, we develop some statistical tests of complementary palindromes, which can be applied to narrow the search for replication origin of the CMV DNA sequence.
Let X_(2k) be the number of complementary palindromes with length 2k and Y_(2k) be the number of non-covered complementary palindromes with length 2k inside a given DNA sequence. Consider the null hypothesis that the marginal probabilities of the four nucleotides remain the same (1/4) over the given sequence versus the alternative hypothesis that the marginal probabilities are different. The likelihood ratio test based on the joint distributions of Y_(18) and Y_(2k) | (Y_(2(k+1)), ...,Y_(18)), where k=1, ..., 8, under the null and the alternative hypotheses are derived. The null distribution of the test statistic is approximated by a scaled chi-squared distribution. The scale parameter and the degree of freedom are estimated by the method of moments. The Pearson's chi-squared test based on the marginal distributions of X_(2k), where k=1, ..., 9. The null distribution of the test statistic is also approximated by a scaled chi-squared distribution. There is an another focus about ratios statistics X_(2k)/X_(2(k+1)) and Y_(2k)/Y_(2(k+1)), which approximate a specific value under the null hypotheses. Simulation studies are performed to confirm the theoretical findings.

1 Introduction . . . . . . 1
1.1 CMV . . . . . . 1
1.2 Research Issues . . . . . . 3
2 Data Description . . . . . . 7
2.1 CMV DNA . . . . . . 7
2.2 Searching Algorithm of Complementary Palindrome . . . . . . 9
3 Distributions of the complementary palindrome Numbers . . . . . . 12
3.1 The Complementary Palindrome Number X_(2k) . . . . . . 12
3.1.1 A, T, G, C with Equal Probability . . . . . . 12
3.1.2 A, T, G, C with Unequal Probabilities . . . . . . 13
3.2 The Complementary Palindrome Number Y_(2k) . . . . . . 13
3.2.1 A, T, G, C with Equal Probability . . . . . . 13
3.2.2 A, T, G, C with Unequal Probabilities . . . . . . 15
3.3 Likelihood Ratio Test . . . . . . 15
3.4 Pearson's Chi-square Test . . . . . . 17
3.4.1 A, T, G, C with Equal Probability . . . . . . 18
3.4.2 A, T, G, C with Unequal Probabilities . . . . . . 19
3.5 Ratio Statistics X_(2k)/X_(2(k+1)) and Y_(2k)/Y_(2(k+1)) . . . . . . 20
3.5.1 A, T, G, C with Equal Probability . . . . . . 20
3.5.2 A, T, G, C with Unequal Probabilities . . . . . . 20
4 Simulation Study . . . . . . 22
4.1 Simulation Algorithm . . . . . . 22
4.2 Tests of the CMV Complementary Palindrome Numbers . . . . . . 24
4.3 Simulating the Complementary Palindrome Number Scenario of the CMV . . . . . . 25
4.3.1 A, T, G, C with Equal Probability . . . . . . 25
4.3.2 A, T, G, C with Unequal Probabilities . . . . . . 26
5 Conclusions and Future Work . . . . . . 27
6 Appendix . . . . . . 28
6.1 Figures . . . . . . 28
6.2 Tables . . . . . . 39
References . . . . . . 50

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