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<p> In this work, optimal censoring schemes are investigated for the non-parametric confidence intervals of population quantiles under progressive Type-II right censoring. The proposed inference can be universally applied to any probability distributions for continuous random variables. By using the interval mass as an optimality criterion, the optimization process is also independent of the actual observed values from a sample as long as the initial sample size n and the number of observations m are predetermined. This study is based on the fact that each (uncensored) order statistic observed from progressive Type-II censoring can be represented as a mixture of underlying ordinary order statistics with exactly known weights [11, 12]. Using several sample sizes combined with various degrees of censoring, the results of the optimization are tabulated here for a wide range of quantiles with selected levels of significance (i.e., α = 0.01, 0.05, 0.10). With the optimality criterion under consideration, the efficiencies of the worst progressive Type-II censoring scheme and ordinary Type-II censoring scheme are also examined in comparison with the best censoring scheme obtained for a given quantile with fixed n and m.</p> / Thesis / Master of Science (MSc)

本文主要是研究統計理論中的聯立信賴區間估計法，針對雙特徵數的指數母體，研究 位置特徵數的差異狀況。也就是要找出位置特徵數的所有複合比，所有對比的聯立信 賴區間。全文共分四章。第一章緒論，說明研究動機、目的及方法。第二章指數分布 的順序統計式，由於本文所使用的估計方法會使用到順序統計式，所以在本章中特別 研究順序統計式的某些線性組合，找出某些特殊的性質；並且將Tanis （１９６３） 提出的理論擴展至第二型設限數據。第三章位置特徵數的聯立信賴區間，藉由第二章 中導出的定理以Tukey 法及Holder不等式法找出聯立信賴區間，並且比較這些方法的 優劣。第四章結論及建議。全文共一冊，約兩萬字。

Die Kapitel 1 bis 6 im ersten Teil dieses Skriptes beruhen auf einer Vorlesung Ökonometrie I, die zuletzt im WS 2001/02 gehalten wurde, die Kapitel 7 bis 16 beruhen auf einer Vorlesung Ökonometrie II, die zuletzt im SS 2006 gehalten wurde. Das achte Kapitel enthält eine komprimierte Zusammenfassung der Ergebnisse aus dem Teil Ökonometrie I.

Approved for public release; distribution in unlimited. / Budget limitations have reduced the number of military components available for testing, and time constraints have reduced the amount of time available for actual testing resulting in many items still operating at the end of test cycles. These two factors produce small test populations (small sample size) with "heavily" censored data. The assumption of "normal approximation" for estimates based on these small sample sizes reduces the accuracy of confidence bounds of the probability plots and the associated quantities. This creates a problem in acquisition analysis because the confidence in the probability estimates influences the number of spare parts required to support a mission or deployment or determines the length of warranty ensuring proper operation of systems. This thesis develops a method that simulates small samples with censored data and examines the error of the Fisher-Matrix (FM) and the Likelihood Ratio Bounds (LRB) confidence methods of two test populations (size 10 and 20) with three, five, seven and nine observed failures for the Weibull distribution. This thesis includes a Monte Carlo simulation code written in S-Plus that can be modified by the user to meet their particular needs for any sampling and censoring scheme. To illustrate the approach, the thesis includes a catalog of corrected confidence bounds for the Weibull distribution, which can be used by acquisition analysts to adjust their confidence bounds and obtain a more accurate representation for warranty and reliability work. / Civilian, Department of the Air Force

[pt] A apreciação ou crescimento de reservas de óleo e gás é importante para uma empresa petrolífera, pois ela indica quanto uma reserva provada crescerá desde o início da produção até um determinado ano. Cada campo apresenta uma apreciação própria que depende de diversos fatores que fazem com que as reservas que antes eram classificadas como prováveis e possíveis sejam reclassificadas agora como provadas. Esta dissertação propõe determinar a apreciação de reservas para óleo e gás com base em séries históricas de campos com características semelhantes, calculando a apreciação média e a elas associando as incertezas da previsão por meio de intervalo de confiança para cada ano após o início da produção, e assim, determinando curvas delimitadoras do intervalo. Dos modelos estudados que satisfazem à condição de monotonicidade não-crescente (a taxa de apreciação anual diminui com o tempo), têm-se o logarítmico e o hiperbólico, sendo que o logarítmico foi o que teve melhor ajuste aos dados observados. Entretanto não se deve descartar o modelo hiperbólico, pois ele tem um apelo teórico e pode ter melhor ajuste do que o modelo logarítmico dependendo dos dados. Esse estudo é de caráter teórico, pois não foi possível obter dados reais de reservas. / [en] Oil and gas reserve appreciation, or growth, is a phenomenon important to petroleum exploratory and producing companies. If well understood, the companies will improve their reserves projection into the future. Each field presents a particular appreciation that depends on various factors that move resources from an initial classification into proved reserve-classification. This thesis proposes a methodology for determining oil and gas reserves appreciation based on historical time series data from similar fields, calculating year to year average appreciation, and associating to each average a confidence interval. Two models were selected for adjustment (logarithmic and hyperbolic) both presenting monotonic non- increasing year to year growth. The logarithmic model presented better adjustment to a limited data-set but hyperbolic should not be discarded both because it has desirable features and may produce better future projections. The difficulties in obtaining adequate real historical data, and other kinds of information on reserves time series impeded further statistical analysis.

The problem of constructing a confidence interval for the noncentrality parameter of a noncentral t-distribution based upon one observation from the distribution is an interesting problem with important applications. A general theoretical approach to the problem is provided by the specification and inversion of acceptance sets for each possible value of the noncentrality parameter. The standard method is based upon the arbitrary assignment of equal tail probabilities to the acceptance set, while the choices of the shortest possible acceptance sets and UMP unbiased acceptance sets provide even worse confidence intervals, which means that since the standard confidence intervals are uniformly shorter than those of UMPU method, the standard method are "biased". However, with the correct choice of acceptance sets it is possible to provide an improvement in terms of confidence interval length over the confidence intervals provided by the standard method for all values of observation. The problem of testing the equality of the noncentrality parameters of two noncentral t-distributions is considered, which naturally arises from the comparison of two signal-to-noise ratios for simple linear regression models. A test procedure is derived that is guaranteed to maintain type I error while having only minimal amounts of conservativeness, and comparisons are made with several other approaches to this problem based on variance stabilizing transformations. In summary, these simulations confirm that the new procedure has type I error probabilities that are guaranteed not to exceed the nominal level, and they demonstrate that the new procedure has size and power levels that compare well with the procedures based on variance stabilizing transformations.

The capacitance-resistance model (CRM) has been continuously improved and tested on both synthetic and real fields. For a large waterflood, with hundreds of injectors and producers present in a reservoir, tens of thousands of model parameters (gains, time constants, and productivity indices) in a field must be determined to completely define the CRM. In this case obtaining a unique solution in history-matching large reservoirs by nonlinear regression is difficult. Moreover, this approach is more likely to produce parameters that are statistically insignificant. The nonlinear nature of the CRM also makes it difficult to quantify the uncertainty in model parameters. The analytical solutions of the two linear reservoir models, the linearly transformed CRM whose control volume is the drainage volume around each producer (ltCRMP) and integrated capacitance-resistance model (ICRM), are developed in this work. Both models are derived from the governing differential equation of the producer-based representation of CRM (CRMP) that represents an in-situ material balance over the effective pore volume of a producer. The proposed methods use a constrained linear multivariate regression (LMR) to provide information about preferential permeability trends and fractures in a reservoir. The two models’ capabilities are validated with simulated data in several synthetic case studies. The ltCRMP and ICRM have the following advantages over the nonlinear waterflood model (CRMP): (1) convex objective functions, (2) elimination of the use of solver when constraints are ignored, and (3) faster computation time in optimization. In both methods, a unique solution can always be obtained regardless of the number of parameters as long as the number of data points is greater than the number of unknowns (parameters). The methods of establishing the confidence limits on CRMP gains and ICRM parameters are demonstrated in this work. This research also presents a method that uses the ICRM to estimate the gains between newly introduced injectors and existing producers for a homogeneous reservoir without having to do additional simulations or regression on newly simulated data. This procedure can guide geoscientists to decide where to drill new injectors to increase future oil recovery and provide rapid solutions without having to run reservoir simulations for each scenario. / text

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.