本篇論文著重於探討ARMA模型的選模準則，過去較為著名的AIC、BIC等選模準則中，若總參數個數相同，模型選擇便簡化為比較各模型的概似函數在MLE下的值，故本研究將假設檢定定義為檢定總參數個數；截至目前為止，選模準則在使用上以AIC及BIC較為普遍，此兩種選模準則從本研究所定義的假設檢定的觀點來看，AIC犯型一誤差機率高，同時檢定力也高；BIC犯型一誤差的機率極低，同時檢定力也相對不高，本研究從此觀點提出一個選模準則方法，嘗試將上述兩種方法折衷，將型一誤差控制在5%，且檢定力略高於BIC。模擬的結果在理想的情形下皆符合預期，但在真實情形本研究方法涉及第一階段的模型選取，本研究提供兩種第一階段的模型選取方法，模擬的結果顯示，方法一型一誤差略為膨脹，檢定力增幅顯著；方法二型一誤差控制精準，但檢定力表現較差。本研究所提出的方法計算時間較為冗長，但若想將 AIC 及 BIC 方法折衷，可考慮嘗試本研究方法。 / This thesis focuses on model selection criteria for ARMA models. For information-based criteria such as AIC and BIC, the task of model selection is reduced to the comparison among likelihood values at maximum likelihood estimates if the numbers of parameters in candidate models are all the same. Thus the key step in model selection is the determination of the total number of parameters. The determination of number of parameters can be addressed using a hypothesis testing approach, where the null hypothesis is that the total number of model parameters is equal to a given number k and the alternative hypothesis is that the total number of parameters is equal to k+1. In this thesis, an information-based model selection method is proposed, where the number of parameters is determined using a two-stage testing procedure, which is constructed with the attempt to control the average type I error probability to be 5%. When using BIC in the above testing problem, simulation results indicate that the average type I error probability for BIC is lower than 0.05, so it is expected the proposed test is more powerful than BIC. The first stage of the proposed test involves selecting the most likely models under the null and the alternative hypothesis respectively. Two methods are considered for the first-stage selection. For the first method, the type I error probability can be larger than 0.05, but the power is significantly larger than BIC. For the second method, the type I error probability is under control, but its power increment is comparatively low. The computing time for the proposed test is rather long. However, for those who need an eclectic method between AIC and BIC, the proposed test can serve as a reasonable choice.

Hypothesis testing is foundational to the discipline of statistics. Procedures exist which control for individual Type I error rates and more global or family-wise error rates for a series of hypothesis tests. However, the ability of scientists to produce very large data sets with increasing ease has led to a rapid rise in the number of statistical tests performed, often with small sample sizes. This is seen particularly in the area of biotechnology and the analysis of microarray data. This thesis considers this high-dimensional context with particular focus on the effects of dependence on existing multiple hypothesis testing procedures. / While dependence is often ignored, there are many existing techniques employed currently to deal with this context but these are typically highly conservative or require difficult estimation of large correlation matrices. This thesis demonstrates that, in this high-dimensional context when the distribution of the test statistics is light-tailed, dependence is not as much of a concern as in the classical contexts. This is achieved with the use of a moving average model. One important implication of this is that, when this is satisfied, procedures designed for independent test statistics can be used confidently on dependent test statistics. / This is not the case however for heavy-tailed distributions, where we expect an asymptotic Poisson cluster process of false discoveries. In these cases, we estimate the parameters of this process along with the tail-weight from the observed exceedences and attempt to adjust procedures. We consider both conservative error rates such as the family-wise error rate and more popular methods such as the false discovery rate. We are able to demonstrate that, in the context of DNA microarrays, it is rare to find heavy-tailed distributions because most test statistics are averages.

Thesis (Ph. D.)--University of Missouri-Columbia, 2004. / Typescript. Vita. Includes bibliographical references (leaves 159-165). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2004. / Typescript. Vita. Includes bibliographical references (leaves 159-165). Also available on the Internet.

Thesis (Ph. D.)--University of California, San Diego, 2009. / Title from first page of PDF file (viewed August 11, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 134-136).

Thesis (M.S.)--Brigham Young University. Dept. of Statistics, 2006. / Includes bibliographical references (p. 83-85).

Thesis (Ph.D.)--Ohio University, June, 2010. / Title from PDF t.p. Includes bibliographical references.

Thesis (M. A.)--University of Notre Dame, 2005. / Thesis directed by Ke-Hai Yuan for the Department of Psychology. "October 2005." Includes bibliographical references (leaves 60-63).

Thesis (Ph. D.)--University of Notre Dame, 2006. / Thesis directed by George S. Howard for the Department of Psychology. "December 2006." Includes bibliographical references (leaves 49-52).

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Includes bibliographical references (p. 263-266).