The master's thesis provides a comprehensive overview of resampling methods for testing the correctness topology of the phylogenetic trees which estimate the process of phylogeny on the bases of biological sequences similarity. We focused on the possibility of errors creation in this estimate and the possibility of their removal and detection. These methods were implemented in Matlab for Bootstrapping, jackknifing, OTU jackknifing and PTP test (Permutation tail probability). The work aims to test their applicability to various biological sequences and also to assess the impact of the choice of input analysis parameters on the results of these statistical tests.

<div>Smart warehouses have become more popular in these days, with Automated Guided Vehicles (AGVs) being used for order pickups. They also allow efficient cost management with optimized storage and retrieval. Moreover, optimization of resources in these warehouses is essential to ensure maximum efficiency. In this thesis, we consider a three dimensional smart warehouse system equipped with heterogeneous AGVs (i.e., having different speeds). We propose scheduling and placement policies that jointly consider all the different design parameters including the scheduling decision probabilities and storage assignment locations. In order to provide differentiated service levels, we propose a prioritized probabilistic scheduling and placement policy to minimize a weighted sum of mean latency and latency tail probability (LTP). Towards this goal, we first derive closed-form expressions for the mean latency and LTP. Then, we formulate an optimization problem to jointly optimize a weighted sum of both the mean latency and LTP. The optimization problem is solved efficiently over the scheduling and decision variables. For a given placement of the products, scheduling decisions of customers’ orders are solved optimally and derived in closed forms. Evaluation results demonstrate a significant improvement of our policy (up to 32%) as compared to the state of other algorithms, such as the Least Work Left policy and Join the Shortest Queue policy, and other competitive baselines.</div>

衡量投資組合的稀有事件時，即使稀有事件違約的機率極低，但是卻隱含著高額資產違約時所帶來的重大損失，所以我們必須要精準地評估稀有事件的信用風險。本研究係在估計信用損失分配的尾端機率，模擬的模型包含同質模型與異質模型；然而蒙地卡羅法雖然在風險管理的計算上相當實用，但是估計機率極小的尾端機率時模擬不夠穩定，因此為增進模擬的效率，我們利用Glasserman and Li (Management Science, 51(11),2005)提出的重點取樣法，以及根據Chiang et al. (Joural of Derivatives, 15(2),2007)重點取樣法為基礎做延伸的改良式重點取樣法，兩種方法來對不同的投資組合做模擬，更是將改良式重點取樣法推廣至異質模型做討論，本文亦透過變異數縮減效果來衡量兩種方法的模擬效率。數值結果顯示，比起傳統的蒙地卡羅法，此兩種方法皆能達到變異數縮減，其中在同質模型下的改良式重點取樣法有很好的表現，模擬時間相當省時，而異質模型下的重點取樣法也具有良好的估計效率及模擬的穩定性。 / When measuring portfolio credit risk of rare-event, even though its default probabilities are low, it causes significant losses resulting from a large number of default. Therefore, we have to measure portfolio credit risk of rare-event accurately. In particular, our goal is estimating the tail of loss distribution. Models we simulate are including homogeneous models and heterogeneous models. However, Monte Carlo simulation is useful and widely used computational tool in risk management, but it is unstable especially estimating small tail probabilities. Hence, in order to improve the efficiency of simulation, we use importance sampling proposed by Glasserman and Li (Management Science, 51(11),2005) and modified importance sampling based on importance sampling which proposed by Chiang et al. (2007 Joural of Derivatives, 15(2),). Simulate different portfolios by these two of simulations. On top of that, we extend and discuss the modified importance sampling simulation to heterogeneous model. In this article, we measure efficiency of two simulations by variance reduction. Numerical results show that proposed methods are better than Monte Carlo and achieve variance reduction. In homogeneous model, modified importance sampling has excellent efficiency of estimating and saves time. In heterogeneous model, importance sampling also has great efficiency of estimating and stability.

In this thesis, we investigate several inferential issues regarding the lifetime data from exponential distribution under different censoring schemes. For reasons of time constraint and cost reduction, censored sampling is commonly employed in practice, especially in reliability engineering. Among various censoring schemes, progressive Type-I censoring provides not only the practical advantage of known termination time but also greater flexibility to the experimenter in the design stage by allowing for the removal of test units at non-terminal time points. Hence, we first consider the inference for a progressively Type-I censored life-testing experiment with k uniformly spaced intervals. For small to moderate sample sizes, a practical modification is proposed to the censoring scheme in order to guarantee a feasible life-test under progressive Type-I censoring. Under this setup, we obtain the maximum likelihood estimator (MLE) of the unknown mean parameter and derive the exact sampling distribution of the MLE through the use of conditional moment generating function under the condition that the existence of the MLE is ensured. Using the exact distribution of the MLE as well as its asymptotic distribution and the parametric bootstrap method, we discuss the construction of confidence intervals for the mean parameter and their performance is then assessed through Monte Carlo simulations. Next, we consider a special class of accelerated life tests, known as step-stress tests in reliability testing. In a step-stress test, the stress levels increase discretely at pre-fixed time points and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Here, we consider a k-step-stress accelerated life testing experiment with an equal step duration τ. In particular, the case of progressively Type-I censored data with a single stress variable is investigated. For small to moderate sample sizes, we introduce another practical modification to the model for a feasible k-step-stress test under progressive censoring, and the optimal τ is searched using the modified model. Next, we seek the optimal τ under the condition that the step-stress test proceeds to the k-th stress level, and the efficiency of this conditional inference is compared to the preceding models. In all cases, censoring is allowed at each change stress point iτ, i = 1, 2, ... , k, and the problem of selecting the optimal Tis discussed using C-optimality, D-optimality, and A-optimality criteria. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. Thus, we also consider the simple stepstress models under Type-I and Type-II censoring situations when the lifetime distributions corresponding to the different risk factors are independently exponentially distributed. Under this setup, we derive the MLEs of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. The exact distributions of the MLEs of the parameters are then derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions and the parametric bootstrap method, we discuss the construction of confidence intervals for the parameters and then assess their performance through Monte Carlo simulations. / Thesis / Doctor of Philosophy (PhD)