We provide ways to test the fit of a parametric copula family for bivariate censored data with or without covariates. The proposed copula family is tested by embedding it in an expanded parametric family of copulas. When parameters in the proposed and the expanded copula models are estimated by maximum likelihood, a likelihood ratio test can be used. However, when they are estimated by two-stage pseudolikelihood estimation, the corresponding test is a pseudolikelihood ratio test. The two-stage procedures offer less computation, which is especially attractive when the marginal lifetime distributions are specified nonparametrically or semiparametrically. It is shown that the likelihood ratio test is consistent even when the expanded model is misspecified. Power comparisons of the likelihood ratio and the pseudolikelihood ratio tests with some other goodness-of-fit tests are performed both when the expanded family is correct and when it is misspecified. They indicate that model expansion provides a convenient, powerful and robust approach. We introduce a semiparametric maximum likelihood estimation method in which the copula parameter is estimated without assumptions on the marginal distributions. This method and the two-stage semiparametric estimation method suggested by Shih and Louis (1995) are generalized to regression models with Cox proportional hazards margins. The two-stage semiparametric estimator of the copula parameter is found to be about as good as the semiparametric maximum likelihood estimator. Semiparametric likelihood ratio and pseudolikelihood ratio tests are considered to provide goodness of fit tests for a copula model without making parametric assumptions for the marginal distributions. Both when the expanded family is correct and when it is misspecified, the semiparametric pseudolikelihood ratio test is almost as powerful as the parametric likelihood ratio and pseudolikelihood ratio tests while achieving robustness to the form of the marginal distributions. The methods are illustrated on applications in medicine and insurance. Sequentially observed survival times are of interest in many studies but there are difficulties in modeling and analyzing such data. First, when the duration of followup is limited and the times for a given individual are not independent, the problem of induced dependent censoring arises for the second and subsequent survival times. Non-identifiability of the marginal survival distributions for second and later times is another issue, since they are observable only if preceding survival times for an individual are uncensored. In addition, in some studies, a significant proportion of individuals may never have the first event. Fully parametric models can deal with these features, but lack of robustness is a concern, and methods of assessing fit are lacking. We introduce an approach to address these issues. We model the joint distribution of the successive survival times by using copula functions, and provide semiparametric estimation procedures in which copula parameters are estimated without parametric assumptions on the marginal distributions. The performance of semiparametric estimation methods is compared with some other estimation methods in simulation studies and shown to be good. The methodology is applied to a motivating example involving relapse and survival following colon cancer treatment.

To combat channel fading, cooperative communication achieves spatial diversity for the transmission between source and destination through the help of relay. However, if the relay behaves abnormally or maliciously and the destination is not aware, the diversity gain of the cooperative system will be significantly reduced, which degrades system performance. In our thesis, we consider an one-relay decode and forward cooperative network, and we assume that the relay may misbehave with a certain probability. If the relay is malicious, it will garble transmission signal, resulting in severe damage to cooperative system. In this work, we discuss three kinds of malicious behavior detection. More specifically, we adopt sequential detection to detect the behavior of relay. If tracing symbols are inserted among the source message, the destination detects malicious after extracting the received tracing symbols. We adopt log-likelihood ratio test to examine these tracing symbols, and then determine the behavior of relay. If the source does not transmit tracing symbols, the destination detects misbehavior according to the received data signal. Furthermore, we employ sequential detection to reduce detection time for a given probabilities of false alarm and miss detection. Through simulation results, for a certain target on probability of errors, our proposed methods can effectively reduce numbers of observations. On the other works, the destination can effectively detect misbehavior of relay, and eliminating the damage causes by malicious relay without requiring large numbers of observations.

Surgeon operation time is a useful and important information for hospital management, which involves operation time estimation for patients under different diagnoses, operation room scheduling, operating room utilization improvements and so on. In this work, we will focus on studying the operation time distributions of thirteen operations performed in the gynecology (GYN) department of one major teaching hospital in southern Taiwan. We firstly investigate what types of distributions are suitable in describing these operation times empirically, where log-normal and mixture log-normal distribution are identified to be acceptable statistically in describing these operation times. Then we compare and characterize the operations into different categories based on the operation time distribution estimates. Later we try to illustrate the possible reason why distributions for some operations with large data set turn out to be mixture of certain log-normal distributions. Finally we end with discussions on possible future work.

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.

The main goal in analyzing microarray data is to determine the genes that are differentially expressed across two types of tissue samples or samples obtained under two experimental conditions. In this dissertation we proposed two methods to determine differentially expressed genes. For the penalized normal mixture model (PMMM) to determine genes that are differentially expressed, we penalized both the variance and the mixing proportion parameters simultaneously. The variance parameter was penalized so that the log-likelihood will be bounded, while the mixing proportion parameter was penalized so that its estimates are not on the boundary of its parametric space. The null distribution of the likelihood ratio test statistic (LRTS) was simulated so that we could perform a hypothesis test for the number of components of the penalized normal mixture model. In addition to simulating the null distribution of the LRTS for the penalized normal mixture model, we showed that the maximum likelihood estimates were asymptotically normal, which is a first step that is necessary to prove the asymptotic null distribution of the LRTS. This result is a significant contribution to field of normal mixture model. The modified p-value approach for detecting differentially expressed genes was also discussed in this dissertation. The modified p-value approach was implemented so that a hypothesis test for the number of components can be conducted by using the modified likelihood ratio test. In the modified p-value approach we penalized the mixing proportion so that the estimates of the mixing proportion are not on the boundary of its parametric space. The null distribution of the (LRTS) was simulated so that the number of components of the uniform beta mixture model can be determined. Finally, for both modified methods, the penalized normal mixture model and the modified p-value approach were applied to simulated and real data.

This thesis concerns inference problems in balanced random effects models with a so-called block circular Toeplitz covariance structure. This class of covariance structures describes the dependency of some specific multivariate two-level data when both compound symmetry and circular symmetry appear simultaneously. We derive two covariance structures under two different invariance restrictions. The obtained covariance structures reflect both circularity and exchangeability present in the data. In particular, estimation in the balanced random effects with block circular covariance matrices is considered. The spectral properties of such patterned covariance matrices are provided. Maximum likelihood estimation is performed through the spectral decomposition of the patterned covariance matrices. Existence of the explicit maximum likelihood estimators is discussed and sufficient conditions for obtaining explicit and unique estimators for the variance-covariance components are derived. Different restricted models are discussed and the corresponding maximum likelihood estimators are presented. This thesis also deals with hypothesis testing of block covariance structures, especially block circular Toeplitz covariance matrices. We consider both so-called external tests and internal tests. In the external tests, various hypotheses about testing block covariance structures, as well as mean structures, are considered, and the internal tests are concerned with testing specific covariance parameters given the block circular Toeplitz structure. Likelihood ratio tests are constructed, and the null distributions of the corresponding test statistics are derived.

abstract: This dissertation presents methods for addressing research problems that currently can only adequately be solved using Quality Reliability Engineering (QRE) approaches especially accelerated life testing (ALT) of electronic printed wiring boards with applications to avionics circuit boards. The methods presented in this research are generally applicable to circuit boards, but the data generated and their analysis is for high performance avionics. Avionics equipment typically requires 20 years expected life by aircraft equipment manufacturers and therefore ALT is the only practical way of performing life test estimates. Both thermal and vibration ALT induced failure are performed and analyzed to resolve industry questions relating to the introduction of lead-free solder product and processes into high reliability avionics. In chapter 2, thermal ALT using an industry standard failure machine implementing Interconnect Stress Test (IST) that simulates circuit board life data is compared to real production failure data by likelihood ratio tests to arrive at a mechanical theory. This mechanical theory results in a statistically equivalent energy bound such that failure distributions below a specific energy level are considered to be from the same distribution thus allowing testers to quantify parameter setting in IST prior to life testing. In chapter 3, vibration ALT comparing tin-lead and lead-free circuit board solder designs involves the use of the likelihood ratio (LR) test to assess both complete failure data and S-N curves to present methods for analyzing data. Failure data is analyzed using Regression and two-way analysis of variance (ANOVA) and reconciled with the LR test results that indicating that a costly aging pre-process may be eliminated in certain cases. In chapter 4, vibration ALT for side-by-side tin-lead and lead-free solder black box designs are life tested. Commercial models from strain data do not exist at the low levels associated with life testing and need to be developed because testing performed and presented here indicate that both tin-lead and lead-free solders are similar. In addition, earlier failures due to vibration like connector failure modes will occur before solder interconnect failures. / Dissertation/Thesis / Ph.D. Industrial Engineering 2012

abstract: Signal processing techniques have been used extensively in many engineering problems and in recent years its application has extended to non-traditional research fields such as biological systems. Many of these applications require extraction of a signal or parameter of interest from degraded measurements. One such application is mass spectrometry immunoassay (MSIA) which has been one of the primary methods of biomarker discovery techniques. MSIA analyzes protein molecules as potential biomarkers using time of flight mass spectrometry (TOF-MS). Peak detection in TOF-MS is important for biomarker analysis and many other MS related application. Though many peak detection algorithms exist, most of them are based on heuristics models. One of the ways of detecting signal peaks is by deploying stochastic models of the signal and noise observations. Likelihood ratio test (LRT) detector, based on the Neyman-Pearson (NP) lemma, is an uniformly most powerful test to decision making in the form of a hypothesis test. The primary goal of this dissertation is to develop signal and noise models for the electrospray ionization (ESI) TOF-MS data. A new method is proposed for developing the signal model by employing first principles calculations based on device physics and molecular properties. The noise model is developed by analyzing MS data from careful experiments in the ESI mass spectrometer. A non-flat baseline in MS data is common. The reasons behind the formation of this baseline has not been fully comprehended. A new signal model explaining the presence of baseline is proposed, though detailed experiments are needed to further substantiate the model assumptions. Signal detection schemes based on these signal and noise models are proposed. A maximum likelihood (ML) method is introduced for estimating the signal peak amplitudes. The performance of the detection methods and ML estimation are evaluated with Monte Carlo simulation which shows promising results. An application of these methods is proposed for fractional abundance calculation for biomarker analysis, which is mathematically robust and fundamentally different than the current algorithms. Biomarker panels for type 2 diabetes and cardiovascular disease are analyzed using existing MS analysis algorithms. Finally, a support vector machine based multi-classification algorithm is developed for evaluating the biomarkers' effectiveness in discriminating type 2 diabetes and cardiovascular diseases and is shown to perform better than a linear discriminant analysis based classifier. / Dissertation/Thesis / Ph.D. Electrical Engineering 2012

The main objective of this work will be to examine the hypothesis that all the treatment means are the same and equal to some unknown quantity, when we know that the variance is the same for each sample, and to determine if the conventional method for making this test (the F-test) applicable when the sample sizes are assumed to be random variables. This hypothesis can be tested by using a likelihood-ration test. To do this, a density function or distribution has to be found for this ratio, thus permitting us to make probability statements about the occurrence of this ration under the null hypothesis.

Testing the equality between two high-dimensional covariance matrices is challenging. As the most efficient way to measure evidential discrepancies in observed data, the likelihood ratio test is expected to be powerful when the null hypothesis is violated. However, when the data dimensionality becomes large and potentially exceeds the sample size by a substantial margin, likelihood ratio based approaches face practical and theoretical challenges. To solve this problem, this study proposes a method by which we first randomly project the original high-dimensional data into lower-dimensional space, and then apply the corrected likelihood ratio tests developed with random matrix theory. We show that testing with a single random projection is consistent under the null hypothesis. Through evaluating the power function, which is challenging in this context, we provide evidence that the test with a single random projection based on a random projection matrix with reasonable column sizes is more powerful when the two covariance matrices are unequal but component-wise discrepancy could be small -- a weak and dense signal setting. To more efficiently utilize this data information, we propose combined tests from multiple random projections from the class of meta-analyses. We establish the foundation of the combined tests from our theoretical analysis that the p-values from multiple random projections are asymptotically independent in the high-dimensional covariance matrices testing problem. Then, we show that combined tests from multiple random projections are consistent under the null hypothesis. In addition, our theory presents the merit of certain meta-analysis approaches over testing with a single random projection. Numerical evaluation of the power function of the combined tests from multiple random projections is also provided based on numerical evaluation of power function of testing with a single random projection. Extensive simulations and two real genetic data analyses confirm the merits and potential applications of our test. / Statistics