In diagnostic test studies, one crucial task is to evaluate the diagnostic accuracy of a test. Currently, most studies focus on the Receiver Operating Characteristics Curve and the Area Under the Curve. On the other hand, the Youden index, widely applied in practice, is another comprehensive measurement for the performance of a diagnostic test. For a continuous-scale test classifying diseased and non-diseased groups, finding the Youden index of the test is equivalent to maximize the sum of sensitivity and specificity for all the possible values of the cut-point. This dissertation concentrates on statistical inferences for the Youden index. First, an auxiliary tool for the Youden index, called the diagnostic curve, is defined and used to evaluate the diagnostic test. Second, in the paired-design study to assess the diagnostic accuracy of two biomarkers, the difference in paired Youden indices frequently acts as an evaluation standard. We propose an exact confidence interval for the difference in paired Youden indices based on generalized pivotal quantities. A maximum likelihood estimate-based interval and a bootstrap-based interval are also included in the study. Third, for certain diseases, an intermediate level exists between diseased and non-diseased status. With such concern, we define the Youden index for three ordinal groups, propose the empirical estimate of the Youden index, study the asymptotic properties of the empirical Youden index estimate, and construct parametric and nonparametric confidence intervals for the Youden index. Finally, since covariates often affect the accuracy of a diagnostic test, therefore, we propose estimates for the Youden index with a covariate adjustment under heteroscedastic regression models for the test results. Asymptotic properties of the covariate-adjusted Youden index estimators are investigated under normal error and non-normal error assumptions.

In the field of diagnostic test studies, the accuracy of a diagnostic test is essential in evaluating the performance of the test. The receiver operating characteristic (ROC) curve and the area under the curve (AUC) are widely used in such evaluation procedures. Meanwhile, the Youden index is also introduced into practice to measure the accuracy of the diagnostic test from another aspect. The Youden index maximizes the sum of sensitivity and specificity, assuring decent true positive and negative rates. It draws one's attention due to its merit of finding the optimal cut-off points of biomarkers. Similar to Partial ROC, a new index, called "Partial Youden index" can be defined as an extension of Youden's Index. It is more meaningful than regular Youden index since the regular one is just a special case of the Partial Youden Index. In this thesis, we focus on construction of generalized confidence intervals for the Partial Youden Index and its corresponding optimal cut-off points. Extensive simulation studies are conducted to evaluate the finite sample performances of the new intervals.

In medical diagnostic studies, the area under the Receiver Operating Characteristic (ROC) curve (AUC) and Youden index are two summary measures widely used in the evaluation of the diagnostic accuracy of a medical test with continuous test results. The first half of this dissertation will highlight ROC analysis including extension of Youden index to the partial Youden index as well as novel confidence interval estimation for AUC and Youden index in the presence of covariates in induced linear regression models. Extensive simulation results show that the proposed methods perform well with small to moderate sized samples. In addition, some real examples will be presented to illustrate the methods. The latter half focuses on the application of empirical likelihood method in economics and finance. Two models draw our attention. The first one is the predictive regression model with independent and identically distributed errors. Some uniform tests have been proposed in the literature without distinguishing whether the predicting variable is stationary or nearly integrated. Here, we extend the empirical likelihood methods in Zhu, Cai and Peng (2014) with independent errors to the case of an AR error process. The proposed new tests do not need to know whether the predicting variable is stationary or nearly integrated, and whether it has a finite variance or an infinite variance. Another model we considered is a GARCH(1,1) sequence or an AR(1) model with ARCH(1) errors. It is known that the observations have a heavy tail and the tail index is determined by an estimating equation. Therefore, one can estimate the tail index by solving the estimating equation with unknown parameters replaced by Quasi Maximum Likelihood Estimation (QMLE), and profile empirical likelihood method can be employed to effectively construct a confidence interval for the tail index. However, this requires that the errors of such a model have at least finite fourth moment to ensure asymptotic normality with n1/2 rate of convergence and Wilk's Theorem. We show that the finite fourth moment can be relaxed by employing some Least Absolute Deviations Estimate (LADE) instead of QMLE for the unknown parameters by noting that the estimating equation for determining the tail index is invariant to a scale transformation of the underlying model. Furthermore, the proposed tail index estimators have a normal limit with n1/2 rate of convergence under minimal moment condition, which may have an infinite fourth moment, and Wilk's theorem holds for the proposed profile empirical likelihood methods. Hence a confidence interval for the tail index can be obtained without estimating any additional quantities such as asymptotic variance.

Youden index, a main summary index for the Receiver Operating Characteristic (ROC) curve, is a comprehensive measurement for the effectiveness of a diagnostic test. For a continuous-scale diagnostic test, the optimal cut-point for the positive of disease is the cut-point leading to the maximization of the sum of sensitivity and specificity. Finding the Youden index of the test is equivalent to maximize the sum of sensitivity and specificity for all the possible values of the cut-point. In this thesis, we propose a new non-parametric confidence interval for the Youden index. Extensive simulation studies are conducted to compare the relative performance of the new interval with the existing intervals for the index. Our simulation results indicate that the newly developed non-parametric method performs as well as the existing parametric method but it has better finite sample performance than the existing non-parametric methods. The new method is flexible and easy to implement in practice. A real example is also used to illustrate the application of the proposed interval.

Many continuous medical tests often rely on a threshold for diagnosis. There are two sequential testing strategies of interest: Believe the Positive (BP) and Believe the Negative (BN). BP classifies a patient positive if either the first test is greater than a threshold θ1 or negative on the first test and greater than θ2 on the second test. BN classifies a patient positive if the first test is greater than a threshold θ3 and greater than θ4 on the second test. Threshold pairs θ = (θ1, θ2) or (θ3, θ4), depending on strategy, are defined as optimal if they maximized GYI = Se + r(Sp – 1). Of interest is to determine if these optimal threshold, or optimal operating point (OOP), estimates are “good” when calculated from a sample. The methods proposed in this dissertation derive formulae to estimate θ assuming tests follow a binormal distribution, using the Newton-Raphson algorithm with ridging. A simulation study is performed assessing bias, root mean square error, percentage of over estimation of Se/Sp, and coverage of simultaneous confidence intervals and confidence regions for sets of population parameters and sample sizes. Additionally, OOPs are compared to the traditional empirical approach estimates. Bootstrapping is used to estimate the variance of each optimal threshold pair estimate. The study shows that parameters such as the area under the curve, ratio of standard deviations of disease classification groups within tests, correlation between tests within a disease classification, total sample size, and allocation of sample size to each disease classification group were all influential on OOP estimation. Additionally, the study shows that this method is an improvement over the empirical estimate. Equations for researchers to use in estimating total sample size and SCI width are also developed. Although the models did not produce high coefficients of determination, they are a good starting point for researchers when designing a study. A pancreatic cancer dataset is used to illustrate the OOP estimation methodology for sequential tests.

Bankruptcies have been a sensitive topic all around the world for over 50 years. From their research, the authors have found that only a few bankruptcy studies have been conducted in Sweden and even less on the topic of bankruptcy prediction models. This thesis investigates the performance of the Altman, Ohlson and Zmijewski bankruptcy prediction models. This research investigates all Swedish companies during the years 2017 and 2018.  This study has the intention to shed light on some of the most famous bankruptcy prediction models. It is interesting to explore the predictive abilities and usability of those three models in Sweden. The second purpose of this study is to create two models from the most significant variable out of the three models studied and to test its prediction power with the aim to create two models designed for Swedish companies.  We identified a research gap in terms of Sweden, where bankruptcy prediction models have been rather unexplored and especially with those three models. Furthermore, we have identified a second research gap regarding the time period of the research. Only a few studies have been conducted on the topic of bankruptcy prediction models post the financial crisis of 2007/08.  We have conducted a quantitative study in order to achieve the purpose of the study. The data used was secondary data gathered from the Serrano database. This research followed an abductive approach with a positive paradigm. This research has studied all active Swedish companies between the years 2017 and 2018. Finally, this contributed to the current field of knowledge on the topic through the analysis of the results of the models on Swedish companies, using the liquidity theory, solvency and insolvency theory, the pecking order theory, the profitability theory, the cash flow theory, and the contagion effect. The results aligned with the liquidity theory, the solvency and insolvency theory and the profitability theory. Moreover, from this research we have found that the Altman model has the lowest performance out of the three models, followed by the Ohlson model that shows some mixed results depending on the statistical analysis. Lastly, the Zmijewski model has the best performance out of the three models. Regarding the performance and the prediction power of the two new models were significantly higher than the three models studied.