The main purpose of this thesis is to compare the performance of the score test and the likelihood ratio test by computing type I errors and type II errors when the tests are applied to the geometric distribution and inflated binomial distribution. We first derive test statistics of the score test and the likelihood ratio test for both distributions. We then use the software package R to perform a simulation to study the behavior of the two tests. We derive the R codes to calculate the two types of error for each distribution. We create lots of samples to approximate the likelihood of type I error and type II error by changing the values of parameters. In the first chapter, we discuss the motivation behind the work presented in this thesis. Also, we introduce the definitions used throughout the paper. In the second chapter, we derive test statistics for the likelihood ratio test and the score test for the geometric distribution. For the score test, we consider the score test using both the observed information matrix and the expected information matrix, and obtain the score test statistic zO and zI . Chapter 3 discusses the likelihood ratio test and the score test for the inflated binomial distribution. The main parameter of interest is w, so p is a nuisance parameter in this case. We derive the likelihood ratio test statistics and the score test statistics to test w. In both tests, the nuisance parameter p is estimated using maximum likelihood estimator pˆ. We also consider the score test using both the observed and the expected information matrices. Chapter 4 focuses on the score test in the inflated binomial distribution. We generate data to follow the zero inflated binomial distribution by using the package R. We plot the graph of the ratio of the two score test statistics for the sample data, zI /zO , in terms of different values of n0, the number of zero values in the sample. In chapter 5, we discuss and compare the use of the score test using two types of information matrices. We perform a simulation study to estimate the two types of errors when applying the test to the geometric distribution and the inflated binomial distribution. We plot the percentage of the two errors by fixing different parameters, such as the probability p and the number of trials m. Finally, we conclude by briefly summarizing the results in chapter 6.

The DNA microarray biotechnology simultaneously monitors the expression of thousands of genes and aims to identify genes that are differently expressed under different conditions. From the statistical point of view, it can be restated as identify genes strongly associated with the response or covariant of interest. The Gene Set Enrichment Analysis (GSEA) method is one method which focuses the analysis at the functional related gene sets level instead of single genes. It helps biologists to interpret the DNA microarray data by their previous biological knowledge of the genes in a gene set. GSEA has been shown to efficiently identify gene sets containing known disease-related genes in the real experiments. Here we want to evaluate the statistical power of this method by simulation studies. The results show that the the power of GSEA is good enough to identify the gene sets highly associated with the response or covariant of interest.

This study was conducted to estimate power of tests in Analysis of Variance (ANOVA) in beef cattle experiments, determine sample size and classify the coefficient of variation. Data was collected from thesis and dissertations of the Program of Post-Graduation in Animal Science of Federal University of Santa Maria (PPGZ-UFSM) among the years of 1991 to 2012, in beef cattle production area using ANOVA. Power was estimated by assumption of non central F distribution to the alternative hypothesis in ANOVA. The number of replications for treatments was estimated by power of 80% and interval of [0,4 ; 2,0] to effect size. Classification of coefficient of variation was based on proposed by Garcia (1989) using mean and standard deviation, and Costa et al. (2002) using median and pseudo-sigma. Power of tests shows two frequency peaks of experiments with low and high power. The recommended average number of replication per treatments was among 7 and 10 replications. Classification of coefficient of variation was proposed by low, medium and high, differently from literature that considers the intervals low, medium, high and very high. / Este estudo foi realizado com o objetivo de determinar o poder do teste F em Análise de Variância (ANOVA) para experimentos com bovinos de corte, determinar o tamanho ideal de amostra e classificar o coeficiente de variação. Os dados foram utilizados de dissertações e teses publicadas do Programa de Pós-Graduação em Zootecnia da Universidade Federal de Santa Maria (PPGZ-UFSM) nos anos de 1991 à 2012, na área de bovinocultura de corte empregando ANOVA. O poder do teste foi determinado assumindo distribuição F de Fisher não central sob hipótese alternativa para a estatística de teste F na ANOVA. O número de repetições por tratamentos foi estimado com base no poder do teste de 80% e o intervalo [0,4 ; 2,0] para o tamanho do efeito. A classificação do coeficiente de variação foi baseada nas metodologias propostas por Garcia (1989) usando média e desvio padrão, e Costa et al. (2002) utilizando mediana e pseudo-sigma. As estimações de poder do teste apresentaram picos de frequências de alto e baixo poder. O número médio recomendado de repetições por tratamentos oscilou entre 7 e 10 repetições. Foi proposta a classificação do coeficiente de variação em baixo, médio e alto, diferindo da classificação da literatura que considera os intervalos baixo, médio, alto e muito alto.

When researchers began addressing the electrophysiology of conscious error awareness more than a decade ago, the role of the error-related negativity (ERN), alongside the subsequently occurring error positivity (Pe), was an obvious locus of attention given the fact that they are taken as indices of cortical error processing. In contrast to the clear-cut findings that link the amplitude of the Pe to error awareness, the association between the ERN amplitude and error awareness is vastly unclear, with a range of studies reporting significant differences in the ERN amplitude with respect to error awareness, while others observing no modulation of the ERN amplitude. One problem in the studies obtaining null findings is the fact that conclusions are drawn based on small sample sizes, increasing the probability of type II error, especially given the fact that the ERN elicited using various error awareness paradigms tends to be small. The aim of the present study was to therefore address the issue of type II error in order to draw more certain conclusions about the modulation of the ERN amplitude by conscious error awareness. Forty participants performed a manual response inhibition task optimised to examine error awareness. While the early and late Pe amplitudes showed the expected sensitivity to error awareness, the ERN results depicted a more complex picture. The ERN amplitude for unaware errors appeared more negative than that of aware errors, both numerically and on the grand average ERP. The unexpected findings were explained in terms of (a) latency issues in the present data, (b) characteristics of the manual response inhibition task used and the possibility that it elicits variation in neurocognitive processing, and (c), in relation to possible contamination by the contingent negative variation (CNV), an ERP component elicited during response preparation. Suggestions for future research on how to address the issues raised in the present paper are also discussed.

In introductory statistics courses, the concept of power is usually presented in the context of testing hypotheses about the population mean. We instead propose an exercise that uses a binomial probability table to introduce the idea of power in the context of testing a population proportion.

In introductory statistics courses, the concept of power is usually presented in the context of testing hypotheses about the population mean. We instead propose an exercise that uses a binomial probability table to introduce the idea of power in the context of testing a population proportion.

Competition between species is believed to lead to patterns of either competitive exclusion or limiting similarity within ecological communities; however, to date the amount of support for either as an outcome has been relatively weak. The two classes of null model commonly used to assess co-occurrence and limiting similarity have both been well studied for statistical performance; however, the methods used to evaluate their performance, particularly in terms of type II statistical errors, may have resulted in the underreporting of both patterns in the communities tested. The overall purpose of this study was to evaluate the efficacy of the negative co-occurrence and limiting similarity null models to detect patterns believed to result from competition between species and to develop an improved method for detecting said patterns. The null models were tested using synthetic but biologically realistic presence-absence matrices for both type I and type II error rate estimations. The effectiveness of the null models was evaluated with respect to community dimension (number of species × number of plots), and amount of pattern within the community. A novel method of subsetting species was developed to assess communities for patterns of co-occurrence and limiting similarity and four methods were assessed for their ability to isolate the species contributing signal to the pattern. Both classes of null model provided acceptable type I and type II error rates when matrices of more than 5 species and more than 5 plots were tested. When patterns of negative co-occurrence or limiting similarity were add to all species both null models were able to detect significant pattern (β > 0.95); however, when pattern was added to only a proportion of species the ability of the null models to detect pattern deteriorated rapidly with proportions of 80% or less. The use of species subsetting was able to detect significant pattern of both co-occurrence and limiting similarity when fewer than 80% of species were contributing signal but was dependent on the metric used for the limiting similarity null model. The ability of frequent pattern mining to isolate the species contributing signal shows promise; however, a more thorough evaluation is required in order to confirm or deny its utility.

Part I comprises three chapters (2-4) that analyse the optimal combination of a universal benefit (B≥0) and categorical benefit (C≥0) for an economy where individuals differ in both their ability to work and, if able to work, their productivity. C is ex-ante conditioned on applicants being unable to work, and ex-post conditioned on recipients not working. In Chapter 2 the benefit budget is fixed but the test awarding C makes Type I and Type II errors. Type I errors guarantee B > 0 at the optimum to ensure all unable individuals have positive consumption. The analysis with Type II errors depends on the enforcement of the ex-post condition. Under No Enforcement C > 0 at the optimum conditional on the awards test having some discriminatory power; whilst maximum welfare falls with both error propensities. Under Full Enforcement C > 0 at the optimum always; and whilst maximum welfare falls with the Type I error propensity it may increase with the Type II error propensity. Chapters 3 and 4 generalise the analysis to a linear-income tax framework. In Chapter 3 categorical status is perfectly observable. Optimal linear and piecewise-linear tax expressions are written more generally to capture cases where it is suboptimal to finance categorical transfers to eliminate inequality in the average social marginal value of income. Chapter 4 then derives the optimal linear income tax for the case with classification errors and Full Enforcement. Both equity and efficiency considerations capture the incentives an increase in the tax rate generates for able individuals to apply for C. Part II (Chapter 5) focuses on the decisions of individuals to work when receiving C, given a risk of being detected and fined proportional to C. Under CARA preferences the risk premium associated with the variance in benefit income is convex-increasing in C, thus giving C a role in enforcement.

Sverige är beroende av nystartade företag och entreprenörer för att finansiera landets välstånd. Riskkapitalbolag innehar ofta en betydande roll för de nya företagen när verksamheten ska utvecklas eller expandera. Bolagen assisterar de nya företagen med ett brett spektrum av nyckelaktiviteter som finansiering, operativt arbete, strategi eller kontaktnät. Tidigare studier visar att riskkapitalfinansierade bolag både växer snabbare och lyckas oftare. Samtidigt visar forskningen att riskkapitalbolag präglas av stort risktagande och övermod vid sina investeringar. Emellertid finns mindre djupgående forskning om vilka konkreta faktorer och omständigheter som resulterar i de olika utfallen. Syftet med examensarbetet är att skapa en bättre förståelse för specifika faktorer som påverkar svenska riskkapitalbolags beslutsfattande och resulterar i särskilda utfall. Undersökningen ämnar att betona likheter och skillnader i riskhanteringen av investeringsval mellan riskkapitalbolagen där utfallen antingen resulterar i nytta eller förlust för bolagen. För att uppfylla syftet har institutionella teorin, kognitiva bias och moderna portföljteorin aktualiserats. Undersökningen genomfördes genom intervjuer med beslutsfattare från åtta stycken konfidentiella riskkapitalbolag. Samtliga bolag kodades om för att säkerställa konfidentialitet i syfte att erhålla djupare inblickar i deras investeringsutfall. De erhållna resultaten visar på att framförallt kognitiva bias men även den institutionella teorin förklarade de olika investeringsutfallen väl. Riskkapitalbolagen präglades av stark övermod vid sina misslyckade investeringar och medelstark vid sina lyckade. Riskkapitalbolagen var också starkt påverkade från både extern tryck av samhället men även av interna riktlinjer vid sina investeringar. Flera branschmönster har också kartlagts vid både lyckade och misslyckade investeringar däribland syndikering av kapital, efterfrågan av serieentreprenörer och att det enligt de själva vanligtvis är entreprenören eller marknaden som är anledningen till att investeringen inte lyckas. Resultatet antyder också att en av de största svårigheterna för riskkapitalbolagen är att bedöma entreprenörernas möjligheter att utveckla företaget. Den mest tänkbara anledningen till att riskkapitalbolagen har starka tendenser till övermod är för att det krävs för att lyckas i branschen. Inget nystartat bolag är riskfritt och för att våga investera i branschen behövs stark tilltro till sin egen förmåga. / Sweden is dependent upon start-ups and entrepreneurs in order to successfully finance domestic prosperity. Venture capital (VC) companies often play a significant role for new companies when the business is to be developed or expanded. The VC companies support the new companies with a wide range of key features. Such as financing, operational work, strategy or contact networks. Previous studies show that VC-financed companies both grow faster and succeed more often than non VC-backed companies. At the same time, research shows that VC-companies are characterized by great risk-taking and audaciousness in their investments. However, there is less in-depth research on what explicit factors and circumstances result in the successful and unsuccessful investments. The purpose of the study is to create a better understanding of similarities and differences in different investment outcomes and highlight the patterns of their investments. Institutional theory, cognitive bias and modern portfolio theory are the theories used in this study. The survey was conducted through eight semi-structured qualitative interviews with different decision makers from VC-companies in Sweden. In order to gain a deeper understanding of their investment outcomes all company names were re-coded. The obtained results showed that cognitive biases was the primary theory of explanation. However, the institutional theory could also explain the different investment outcomes. The VC-companies were characterized by strong audaciousness with their failed investments and medium audaciousness with the successful. The requirement of success in the VC-industry explains why the companies have tendencies to be audacious in their decision making. A strong confidence is needed when investing in start-up companies since these companies imply a great risk. External societal pressure and internal guidelines strongly influence the investments of the VC. The study identified several industry patterns for both successful and unsuccessful investments.  The patterns include syndication of capital and demand for serial entrepreneurs. The study also showed a pattern of venture capitalists explaining their unsuccessful investment due to issues with the entrepreneur or the market. The results suggest that one of the greatest difficulties for VC-companies is to assess the ability of the entrepreneur to develop the company.

In this paper, we will investigate the important properties of three major parts of statistical inference: point estimation, interval estimation and hypothesis testing. For point estimation, we consider the two methods of finding estimators: moment estimators and maximum likelihood estimators, and three methods of evaluating estimators: mean squared error, best unbiased estimators and sufficiency and unbiasedness. For interval estimation, we consider the the general confidence interval, confidence interval in one sample, confidence interval in two samples, sample sizes and finite population correction factors. In hypothesis testing, we consider the theory of testing of hypotheses, testing in one sample, testing in two samples, and the three methods of finding tests: uniformly most powerful test, likelihood ratio test and goodness of fit test. Many examples are used to illustrate their applications.