An important problem in statistical inference is to check the adequacy of models upon which inferences are based. Some valuable tools are available for examining a model's suitability of which the most widely used is the goodness-of-fit test. The pioneering work in this area is by Karl Pearson (1900). Since then, a considerable amount of work has been done so far and investigation is still going on in this field due to its importance in the hypothesis testing problem.This thesis contains an expository discussion of the goodness-of-fit tests, intended for the users of the statistical theory. An attempt is made here to give a complete coverage of the historical development, present status and other current problems related to this topic. Numerical examples are provided to best explain the test procedures. The contents, taken as a whole, constitute a unified presentation of some of the most important aspects of goodness-of-fit tests.Ball State UniversityMuncie, IN 57406

The Empirical Likelihood method (ELM) was introduced by A. B. Owen to test hypotheses in the early 1990s. It's a nonparametric method and uses the data directly to do statistical tests and to compute confidence intervals/regions. Because of its distribution free property and generality, it has been studied extensively and employed widely in statistical topics. There are many classical test statistics such as the Cramer-von Mises (CM) test statistic, the Anderson-Darling test statistic, and the Watson test statistic, to name a few. However, none is universally most powerful. This thesis is dedicated to extending the ELM to several interesting statistical topics in hypothesis tests. First of all, we focus on testing the fit of distributions. Based on the CM test, we propose a novel Jackknife Empirical Likelihood test via estimating equations in testing the goodness-of-fit. The proposed new test allows one to add more relevant constraints so as to improve the power. Also, this idea can be generalized to other classical test statistics. Second, when aiming at testing the error distributions generated from a statistical model (e.g., the regression model), we introduce the Jackknife Empirical Likelihood idea to the regression model, and further compute the confidence regions with the merits of distribution free limiting chi-square property. Third, the ELM based on some weighted score equations are proposed for constructing confidence intervals for the coefficient in the simple bilinear model. The effectiveness of all presented methods are demonstrated by some extensive simulation studies.

Cluster analysis is one of the typical tasks in Data Mining, and it groups data objects based only on information found in the data that describes the objects and their relationships. The purpose of this thesis is to verify a modified K-means algorithm in asymmetrical cases, which can be regarded as an extension to the research of Vladislav Valkovsky and Mikael Karlsson in Department of Informatics and Media. In this thesis an experiment is designed and implemented to identify clusters with the modified algorithm in asymmetrical cases. In the experiment the developed Java application is based on knowledge established from previous research. The development procedures are also described and input parameters are mentioned along with the analysis. This experiment consists of several test suites, each of which simulates the situation existing in real world, and test results are displayed graphically. The findings mainly emphasize the limitations of the algorithm, and future work for digging more essences of the algorithm is also suggested.

We are concerned with testing for differential expression and consider three different aspects of such testing procedures. First, we develop an exact ANOVA type model for discrete gene expression data, produced by technologies such as a Massively Parallel Signature Sequencing (MPSS), Serial Analysis of Gene Expression (SAGE) or other next generation sequencing technologies. We adopt two Bayesian hierarchical models—one parametric and the other semiparametric with a Dirichlet process prior that has the ability to borrow strength across related signatures, where a signature is a specific arrangement of the nucleotides. We utilize the discreteness of the Dirichlet process prior to cluster signatures that exhibit similar differential expression profiles. Tests for differential expression are carried out using non-parametric approaches, while controlling the false discovery rate. Next, we consider ways to combine expression data from different studies, possibly produced by different technologies resulting in mixed type responses, such as Microarrays and MPSS. Depending on the technology, the expression data can be continuous or discrete and can have different technology dependent noise characteristics. Adding to the difficulty, genes can have an arbitrary correlation structure both within and across studies. Performing several hypothesis tests for differential expression could also lead to false discoveries. We propose to address all the above challenges using a Hierarchical Dirichlet process with a spike-and-slab base prior on the random effects, while smoothing splines model the unknown link functions that map different technology dependent manifestations to latent processes upon which inference is based. Finally, we propose an algorithm for controlling different error measures in a Bayesian multiple testing under generic loss functions, including the widely used uniform loss function. We do not make any specific assumptions about the underlying probability model but require that indicator variables for the individual hypotheses are available as a component of the inference. Given this information, we recast multiple hypothesis testing as a combinatorial optimization problem and in particular, the 0-1 knapsack problem which can be solved efficiently using a variety of algorithms, both approximate and exact in nature.

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Energy or DFT detector using a fixed window size is very efficient when signal start time and duration is matched with that of the window&#039 / s. However, in the case of unknown signal duration, the performance of this detector decreases. For this scenario, a detector system composed of multiple windows may be preferred. Window sizes of such a system will also be fixed beforehand but they will be different from each other. Therefore, one of the windows will better match the signal duration, giving better detection results. In this study, multiple window detectors are analyzed. Their false alarm and detection probability relations are investigated. Some exact and approximate values are derived for these probabilities. A rule of thumb for the choice of window lengths is suggested for the case of fixed number of windows. Detectors with overlapping window structure are considered for the signals with unknown delay. Simulation results are added for these types of detectors.

Knowledge of fire risk is crucial for manufacturers and regulators to make correct choices in prescribing fire protection systems, especially flame retardants. Methods of determining fire risk are bogged down by a multitude of confounding factors, such as population demographics and overlapping fire protection systems. Teasing out the impacts of one particular choice or regulatory change in such an environment is crucial. Teasing out such detail requires statistical techniques, and knowledge of the field is important for verifying potential methods. Comparing the fire problems between two states might be one way to identify successful approaches to fire safety. California, a state with progressive fire prevention policies, is compared to Texas using logistic regression modeling to account for various common factors such as percentage of rural population and percentage of population in ‘risky’ age brackets. Results indicate that living room fires, fires in which the first item ignited is a flammable liquid, piping, or filter, and fires started by cigarettes, pipes, and cigars have significantly higher odds of resulting in a casualty or fatality than fires started by other areas of origin, items first ignited, or heat sources. Additionally, fires in Texas have roughly 1.5 times higher odds of resulting in casualties than fires in California for certain areas of origin, items first ignited, and heat sources. Methods of estimating fire losses are also examined. The potential of using Ramachandran’s power-law relationship to estimate fire losses in residential home fires in Texas is examined, and determined to be viable but not discriminating. CFAST is likewise explored as a means to model fire losses. Initial results are inconclusive, but Monte Carlo simulation of home geometries might render the approach viable. / text

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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.