We examine the economic behavior of the regret-averse firm under price uncertainty. We show that the global and marginal effects of price uncertainty on production are both positive (negative) when regret aversion prevails if the random output price is positively (negatively) skewed. In this case, high (low) output prices are much more likely to be seen than low (high) output prices. To minimize regret, the firm is induced to raise (lower) its output optimal level. The skewness of the price distribution as such plays a pivotal role in determining the regret-averse firm\'s production decision.

Uncertainty analysis is an important part of system design. The formula for error propagation through a system model that is most-often cited in literature is based on a first-order Taylor series. This formula makes several important assumptions and has several important limitations that are often ignored. This thesis explores these assumptions and addresses two of the major limitations. First, the results obtained from propagating error through nonlinear systems can be wrong by one or more orders of magnitude, due to the linearization inherent in a first-order Taylor series. This thesis presents a method for overcoming that inaccuracy that is capable of achieving fourth-order accuracy without significant additional computational cost. Second, system designers using a Taylor series to propagate error typically only propagate a mean and variance and ignore all higher-order statistics. Consequently, a Gaussian output distribution must be assumed, which often does not reflect reality. This thesis presents a proof that nonlinear systems do not produce Gaussian output distributions, even when inputs are Gaussian. A second-order Taylor series is then used to propagate both skewness and kurtosis through a system model. This allows the system designer to obtain a fully-described non-Gaussian output distribution. The benefits of having a fully-described output distribution are demonstrated using the examples of both a flat rolling metalworking process and the propeller component of a solar-powered unmanned aerial vehicle.

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Percent within limits (PWL) is a commonly used quality control/quality assurance measure of highway pavement materials and construction, and it is a popular index for adjusting pay factors. However, PWL is based on the assumption of normal distribution of quality characteristics (e.g., concrete compressive strength and asphalt air voids). Skewness and kurtosis, which are common forms of statistical nonnormal distributions, can potentially bias the acceptance pay factor calculations. To examine this potential pay bias, simulations were performed to investigate the magnitude and the direction (overestimation or underestimation) of pay factor calculations. The study revealed that for both one-sided and two-sided specification limits, bias in pay factors not only did vary in magnitude but also reversed in direction over various ranges of PWL. These analyses showed that for a one-sided upper specification limit, on average, a positive skewness and kurtosis can underestimate the pay factor of an acceptable quality level population by 0.90%, and overestimates a rejectable quality level population by 3.8%. This leads to falsely penalizing acceptable products and rewarding bad products. The same was true for two-sided limits, which again varied based upon the percent of defective materials at the tails of the distribution. This is a very important issue because these biases in pay factors can easily upset the relative profit margins of the contractor. Furthermore, this may not be easily detectable without a detailed and sophisticated analysis as outlined in this paper. For multiple quality characteristics based pay factors, analyses showed that the combined magnitude of these biases was not linearly cumulative. Findings of the study indicate that bias in pay was higher for lots with fewer sublots and higher skewness and kurtosis.

Meta-analysis is a popular statistical method that synthesizes evidence from multiple studies. Conventionally, both the hierarchical and bivariate models for meta-analysis of diagnostic test accuracy (DTA) studies assume that the random-effects follow the bivariate normal distribution. However, this assumption is restrictive, and inferences could be misleading when it is violated. On the other hand, subjective methods such as inspection of forest plots are used to identify outlying studies in a meta-analysis of DTA studies. Moreover, inferences made using the well-established bivariate random-effects models, when outlying or influential studies are present, may lead to misleading conclusions. Thus, the aim of this thesis is to address these issues by introducing alternative and robust statistical methods. First, we extend the current bivariate linear mixed model (LMM) by assuming a flexible bivariate skew-normal distribution for the random-effects. The marginal distribution of the proposed model is analytically derived so that parameter estimation can be performed using standard likelihood methods. Overall, the proposed model performs better in terms of confidence interval width of the overall sensitivity and specificity, and with regards to bias and root mean squared error of the between-study (co)variances than the traditional bivariate LMM. Second, we propose objective methods based on solid statistical reasoning for identifying outlying and/or influential studies in a meta-analysis of DTA studies. The performances of the proposed methods are evaluated using a simulation study. The proposed methods outperform and avoid the subjectivity of the currently used ad hoc approaches. Finally, we develop a new robust bivariate random-effects model which accommodates outlying and influential observations and leads to a robust statistical inference by down-weighting the effect of outlying and influential studies. The proposed model produces robust point estimates of sensitivity and specificity compared to the standard models, and also generates a similar point and interval estimates of sensitivity and specificity as the standard models in the absence of outlying or influential studies. / Thesis / Doctor of Philosophy (PhD) / Diagnostic tests vary from the noninvasive rapid strep test used to identify whether a patient has a bacterial sore throat to the much complex and invasive biopsy test used to examine the presence, cause, and extent of a severe condition, say cancer. Meta-analysis is a widely used statistical method that synthesizes evidence from several studies. In this thesis, we develop novel statistical methods extending the traditional methods for meta-analysis of diagnostic test accuracy studies. Our proposed methods address the issue of modelling asymmetrical data, identifying outlier studies, and optimally accommodating these outlying studies in a meta-analysis of diagnostic test accuracy studies. Using both real-life and simulated datasets, we show that our proposed methods perform better than conventional methods in a wide range of scenarios. %Therefore, we believe that our proposed methods are essential for methodologists, clinicians and health policy professionals in the process of making a correct judgment to using the appropriate diagnostic test to diagnose patients.

This dissertation consists of three essays. The first essay focuses on idiosyncratic volatility as a primary arbitrage cost for short sellers. Previous studies document (i) negative abnormal returns for high relative short interest (RSI) stocks, and (ii) positive abnormal returns for low RSI stocks. We examine whether these market inefficiencies can be explained by arbitrage limitations, especially firms' idiosyncratic risk. Consistent with limits to arbitrage hypothesis, we document an abnormal return of -1.74% per month for high RSI stocks (>=95th percentile) with high idiosyncratic volatility. However, for similar level of high RSI, abnormal returns are economically and statistically insignificant for stocks with low idiosyncratic volatility. For stocks with low RSI, the returns are positively related to idiosyncratic volatility. These results imply that idiosyncratic risk is a potential reason for the inability of arbitrageurs to extract returns from high and low RSI portfolios. The second essay investigates market efficiency in the absence of limits to arbitrage on short selling. Theoretical predictions and empirical results are ambiguous about the effect of short sale constraints on security prices. Since these constraints cannot be eliminated in equity markets, we use trades from futures markets where there is no distinction between short and long positions. With no external constraints on short positions, we document a weekend effect in futures markets which is a result of asymmetric risk between long and short positions around weekends. The premium is higher in periods of high volatility when short sellers are unwilling to accept higher levels of risk. On the other hand, riskiness of long positions does not seem to have a similar impact on prices. The third essay studies investor behaviors that generate mispricing by examining relationship between stock price and future returns. Based on traditional finance theory, valuation should not depend on nominal stock prices. However, recent literature documents that preference of retail investors for low price stocks results in their overvaluation. Motivated by this preference, we re-examine the relationship between stock price and expected return for the entire U.S. stock market. We find that stock price and expected returns are positively related if price is not confounded with size. Results in this paper show that, controlled for size, high price stocks significantly outperform low price stocks by an abnormal 0.40% per month. This return premium is attributed to individual investors' preference for low price stocks. Consistent with costly arbitrage, the return differential between high and low price stocks is highest for the stocks which are difficulty to arbitrage. The results are robust to price cut-off of $5, and in different sub-periods. / Ph. D.

市場上通常存在著跌幅大過漲幅的現象，更強烈的說法是，市場會在一夕之間崩盤，但卻不會在一夕之間漲上天，這造成了報酬負偏態的現象，而Horn and Stein的理論模型認為市場存在著兩群堅持己見、對股價有不同看法的投資人，再加上這群投資人面對放空的限制，是造成報酬負偏態的主要因素，若投資人之間看法差異愈大，則負偏態現象愈明顯。Chen, Horn and Stein根據他們的理論模型，他們將成交量定義週轉率，提出利用股票的週轉率來預測負偏態的概念，而本研究利用他們所提出的實證模型，應用在台灣股市上，並與美國實證結果相對照，實證結果顯示： 1. 在台灣，6個月期間週轉率愈高於平均的個股或大盤，下6個月報酬負偏態的情況會愈顯著，但其影響力和美國實證結果相對照小很多。 2. 市值愈大的股票，其報酬正偏態的情況愈顯著，這與美國的實證結果是相反的。 3. 依隨機泡沫模型理論，過去報酬率愈大的資產，愈有可能產生報酬負偏態的情況，而台灣的實證顯示，過去的報酬率無法有效的預測報酬負偏態，但美國的實證結果是成功的 / In stock market history, the very large movement are always decrease rather than increase. In other words, stock market tends to melt down, not melt up. This kind of return asymmetry causes the negative skewness of the stock return (either market portfolio or single stock). There are mainly three schools to explain mechanism behind the negative skewness of the return. They are leverage effect, assymmetry volatility, and stochastic bubble model. Chen, Horn and Stein states that stocks come through high turnover will later on go through the negative skewness of return. We use the empirical model proposed by Horn and Stein to inpsect if turnover can predict negative skewness of return in Taiwan stock market. we have three conclusions： 1. Negative skewness is greater in stocks and market portfolio that have experienced an increase in turnover rate relative to trend over the prior six month. This effect is smaller than that in America. 2. Negative skewness is greater in stocks that are larger in terms of market capitalization. This empirical evidence is contrary to those in America. 3. In view of stochastic bubble model, stocks that have high positive returns in the past are more likely to experience greater negative skewness in return. Empirical evidence in Taiwan shows that stochastic bubble does not apply to Taiwan stocks market, that is, past return in stocks can not predict the negative skewness in return.

自Markowitz(1952)提出平均數-變異數準則以來，對於該準則適宜性的討論即不曾停止過。許多實證上資料顯示資產報酬率分配不為常態，而越來越多學者也對於高於二階以上之高階動差對投資決策之影響提出證實。本文利用臺灣八大類股指數報酬率分配資料，運用多目標規劃求解法進行實證，發現臺灣股票市場呈現顯著峰態性質，此外，本文樣本外試驗結果亦指出，平均數-變異數-偏態-峰態架構下之最適投資組合的報酬率高於傳統平均數-變異數架構下之最適投資組合以及大盤報酬。

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In the present work we study families of skew-probability distributions. We will gradually build concept of families of more and more general distributions. For us the most important ones are skew normal distribution, elliptical distri- bution and skew elliptical distribution. On the each of them we will present basic properties and visualize particular examples. At the end we will generate realizations of variates and propose how to estimate the original distribution.