The variance is the measure of spread from the center. Therefore, how to accurately estimate variance has always been an important topic in recent years. In this paper, we consider a linear regression model which is the most popular model in practice. We use jackknife empirical likelihood method to obtain the interval estimate of variance in the regression model. The proposed jackknife empirical likelihood ratio converges to the standard chi-squared distribution. The simulation study is carried out to compare the jackknife empirical likelihood method and standard method in terms of coverage probability and interval length for the confidence interval of variance from linear regression models. The proposed jackknife empirical likelihood method has better performance. We also illustrate the proposed methods using two real data sets.

In this thesis, we are interested in empirical likelihood (EL) methods for two-sample problems, with focus on the difference of the two population means. A weighted empirical likelihood method (WEL) for two-sample problems is developed. We also consider a scenario where sample data on auxiliary variables are fully observed for both samples but values of the response variable are subject to missingness. We develop an adjusted empirical likelihood method for inference of the difference of the two population means for this scenario where missing values are handled by a regression imputation method. Bootstrap calibration for WEL is also developed. Simulation studies are conducted to evaluate the performance of naive EL, WEL and WEL with bootstrap calibration (BWEL) with comparison to the usual two-sample t-test in terms of power of the tests and coverage accuracies. Simulation for the adjusted EL for the linear regression model with missing data is also conducted.

In this thesis, we are interested in empirical likelihood (EL) methods for two-sample problems, with focus on the difference of the two population means. A weighted empirical likelihood method (WEL) for two-sample problems is developed. We also consider a scenario where sample data on auxiliary variables are fully observed for both samples but values of the response variable are subject to missingness. We develop an adjusted empirical likelihood method for inference of the difference of the two population means for this scenario where missing values are handled by a regression imputation method. Bootstrap calibration for WEL is also developed. Simulation studies are conducted to evaluate the performance of naive EL, WEL and WEL with bootstrap calibration (BWEL) with comparison to the usual two-sample t-test in terms of power of the tests and coverage accuracies. Simulation for the adjusted EL for the linear regression model with missing data is also conducted.

Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 119-126).

In spite of modern technological advancements that can convert wastewater into potable water, the acceptability of recycled water is generally low. This study examined strategies for increasing the public acceptability of recycled water. Based on the elaboration likelihood model, I hypothesized that issue relevance, argument quality, and delivery type would interact to produce differing levels of support for potable recycled water. Undergraduate students took part in a 2 (issue relevance: low, high) x 2 (argument quality: weak, strong) x 2 (delivery: textual, pictorial) online study relating to their opinion and support for the potential implementation of a potable recycled water system on campus. Issue relevance was manipulated by varying the completion date of implementing the system (low: five years; high: one year). Argument quality was manipulated by varying the complexity of the message presented (weak: point-form; strong: paragraph form). Delivery was manipulated by presenting water recycling processes in a textual or pictorial format. The hypotheses were not supported, although the means were in the predicted direction. Limitations and future directions are discussed. / Graduate

In this thesis we study the problem of estimation of parametric covariance and variogram functions for spatial and spatio- temporal random processes. It has the following principal parts. Variogram Estimation: We consider the "weighted" least squares criterion of fitting a parametric variogram function to second order stationary geo-statistical processes. Two new weight functions are investigated as alternative to the commonly used weight function proposed by Cressie (1985). We discuss asymptotic convergence properties of the sample variogram estimator and estimators of unknown parameters of parametric variogram functions, under a "mixed increasing domain" sampling design as proposed by Lahiriet al. While empirical results of Mean Square Errors, for parameter estimation, obtained using both the proposed functions are found to be comparatively better, we also theoretically establish that under general conditions one of the proposed weight functions give estimates with better asymptotic effciency. Spatio-Temporal Covariance Estimation: Over the past decade, there have been some important advances in methods for constructing valid spatiotemporal covariance functions; but not much attention has been given - so far - on methods of parameter estimation. In this thesis we propose a new frequency domain approach to estimating parameters of spatio-temporal covariance functions. We derive asymptotic strong consistency properties of the estimators using the concept of stochastic equicontinuity. The theory is illustrated with a simulation. Non-Linearity of Geostatistical Data: Linear prediction theory for spatial data is well established and substantial literature is available on the subject. Relatively less is known about non-linearity. In our final and ongoing, research problem we propose a non-linear predictor for geostatistical data. We demonstrate that the predictor is a function of higher order moments. This leads us to construct spatial bispectra for parametric third order moments.

In recent years, the RNA-sequencing (RNA-seq) method, which measures the transcriptome by counting short sequencing reads obtained by high-throughput sequencing, is replacing the microarray technology as the major platform in gene expression studies. The large amount of discrete data in RNA-seq experiments calls for effective analysis methods. In this dissertation, a new method to detect differentially expressed genes based on quasi-likelihood theory is developed in experiments with a completely randomized design with two experimental conditions. The proposed method estimates the variance function empirically and consequently it has similar sensitivities and FDRs across distributions with different variance functions. In a simulation study, the method is shown to have similar sensitivities and FDRs across the data with three different types of variance functions compared with some other popular methods. This method is applied to a real dataset with two experimental conditions along with some competing methods. The new method is then extended to more complex designs such as an experiment with multiple experimental conditions, an experiment with block design and an experiment with factorial design. The same advantages for the new method have been found in simulation studies. This method and some competing methods are applied to three real datasets with complex designs. The new method is also applied to analyze reads per kilobase per million mapped reads (RPKM) data. In the simulation, the method is compared with the Linear Models for Microarray Data (LIMMA) originally developed for microarray analysis (Smyth, 2004) and the question of normalization is also examined. It is shown that the new method and the LIMMA method have similar performance. Further normalization is required for the proper analysis of the RPKM data and the best such normalization is the scaling method. Analyzing raw count data properly has better performance than analyzing the RPKM data. Different normalization and statistical methods are applied to a real dataset with varied gene length across samples. / Thesis / Doctor of Philosophy (PhD)

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.

The use of doubly robust estimators is a key for estimating the population mean response in the presence of incomplete data. Cao et al. (2009) proposed an alternative doubly robust estimator which exhibits strong performance compared to existing estimation methods. In this thesis, we apply the jackknife empirical likelihood, the jackknife empirical likelihood with nuisance parameters, the profile empirical likelihood, and an empirical likelihood method based on the influence function to make an inference for the population mean. We use these methods to construct confidence intervals for the population mean, and compare the coverage probabilities and interval lengths using both the ``usual'' doubly robust estimator and the alternative estimator proposed by Cao et al. (2009). An extensive simulation study is carried out to compare the different methods. Finally, the proposed methods are applied to two real data sets.

Pietra ratio (Pietra index), also known as Robin Hood index, Schutz coefficient (Ricci-Schutz index) or half the relative mean deviation, is a good measure of statistical heterogeneity in the context of positive-valued data sets. In this thesis, two novel methods namely "adjusted jackknife empirical likelihood" and "extended jackknife empirical likelihood" are developed from the jackknife empirical likelihood method to obtain interval estimation of the Pietra ratio of a population. The performance of the two novel methods are compared with the jackknife empirical likelihood method, the normal approximation method and two bootstrap methods (the percentile bootstrap method and the bias corrected and accelerated bootstrap method). Simulation results indicate that under both symmetric and skewed distributions, especially when the sample is small, the extended jackknife empirical likelihood method gives the best performance among the six methods in terms of the coverage probabilities and interval lengths of the confidence interval of Pietra ratio; when the sample size is over 20, the adjusted jackknife empirical likelihood method performs better than the other methods, except the extended jackknife empirical likelihood method. Furthermore, several real data sets are used to illustrate the proposed methods.