Mathematical Sciences: Implementation of Accurate Methods for Practical Inference

尽管在开发精确的近似技术（如鞍点和改进的基于似然的程序）上花费了大量的研究努力，但在评估使用这些方法在实践中可以实现的推断类型方面投入的努力较少。在进行这种评估时，可以看到可用的推论在范围上受到统计问题和计算复杂性的严重限制。这两个限制的来源是交织在一起的，因为有效的频率论程序的固有要求可能产生计算困难。确保频率推断的正确性可能需要大量的计算，需要对源自高阶渐近近似的复杂表达式进行许多繁琐的求值。在本研究中，这些困难是通过频率论和贝叶斯推理的综合来克服的，因为后者的方法在前景和实现上更简单。特别是，计算问题是通过适应基于采样的技术，如马尔可夫链蒙特卡罗，以获得高阶近似。其结果将是在各种各样的实际问题中改进推理；例子包括逻辑回归、删节数据模型和方差成分估计。在更复杂的统计模型中，统计学家通常依靠近似的推理方法，主要是因为精确的方法既难以推导，又难以计算。然而，这些近似方法的有效性取决于样本量很大，这意味着这些方法在小样本问题中可能不准确。现在，廉价的计算能力已经广泛使用，统计学家正试图使用更现实、更复杂的模型。例如，用于分析全球环境变化或DNA评估的模型非常复杂。这项研究的重点是开发既能提供准确性又能提供计算可追溯性的统计方法。这项工作必须将高性能计算与现代统计方法相结合。＊＊＊

DMS 9625440 Casella Although much research effort has been expended on developing accurate approximation techniques such as saddlepoint and improved likelihood-based procedures, less effort has been devoted to assessing the types of inferences that can be achieved in practice by using these methods. When this assessment is made, it is seen that the available inferences are severely limited in scope by both statistical issues and computational complexity. These two sources of limitation are intertwined, as computational difficulties can arise from the inherent demands of valid frequentist procedures. Ensuring the correctness of frequentist inferences can be computationally intensive, requiring many cumbersome evaluations of the complex expressions that derive from higher-order asymptotic approximations. In this research, these difficulties are overcome by a synthesis of frequentist and Bayesian inference, as the latter approach is simpler in outlook and implementation. In particular, the computational problem is addressed by adapting sampling-based techniques, such as Markov Chain Monte Carlo, to attain the higher-order approximations. The result will be improved inferences in a wide variety of practical problems; examples include logistic regression, censored data models, and variance component estimation. %%% In more complicated statistical models, statisticians have typically relied on approximate methods of inference, primarily because exact methods can be both difficult to derive and complex to compute. However, the validity of these approximate methods rests on the sample size being large, which means that such methods may not be accurate in problems with small samples. Now that inexpensive computational power has become widely available, statisticians are attempting to use the more realistic and complex models. For example, models used in analyzing global environmental change, or DNA assessment, are quite complex. The focus of t his research is to develop statistical methods that offer both accuracy and computational tractability. This work necessarily blends high-performance computing with modern statistical methodology. ***