Biased Sampling and Confidence

DMS 9626347伍德罗夫该研究涉及有偏抽样模型和指数时间序列。在有偏抽样模型中，将受试者纳入研究的概率可能取决于感兴趣的变量--例如，受试者的大小。兴趣集中在其中包含概率以单调方式依赖于感兴趣的变量的情况，但不假设包含概率已知或直到少数参数已知。这项研究包括开发对偏差存在的测试，当存在偏差时开发估计程序，并研究测试和估计器的性质。利用似然和惩罚似然来建立检验和估计量，并通过渐近分析和模拟相结合的方法研究了它们的性质。指数时间序列是一个有限维分布形成指数族的随机过程。对于这类过程，在相当一般的条件下，归一化极大似然估计的抽样分布是渐近正态的。研究包括建立这些抽样分布的高阶近似，并使用改进的近似来形成校正的置信度集。在数学上，结果采用非常弱的展开形式，其中使用贝叶斯方法来获得对抽样分布的近似。在涉及搜索隐藏对象的调查中经常出现有偏差的样本，因为发现对象的概率可能取决于其属性。例如，天文学家更有可能发现一个明亮的大星系，而不是一个昏暗的小星系；地质学家更有可能发现一个大油井，而不是一个小油井。以前关于这类问题的工作主要集中在包含概率以一种已知的方式依赖于感兴趣的变量的情况。这项新的研究涉及开发数据分析，在后一种关系未知的情况下适用，并包括从观察数据估计关系的方法。这种关系很重要，因为找不到的对象的数量在很大程度上取决于它。这项研究考虑了指数时间序列，其中包括经典的时间序列，如股票价格和天气，但也有许多其他如适应性设计实验(自行设计的实验)和序贯临床试验。

DMS 9626347 Woodroofe The research concerns biased sampling models and exponential time series. In biased sampling models, the probability of including a subject in the study may depend on variables of interest--for example, the size of the subject. Interest centers on cases in which the inclusion probabilities depend on the variables of interest in a monotone way, but without assuming that the inclusion probabilities are known or known up to a few parameters. The research involves developing tests for the presence of bias, developing estimation procedures when bias is present and studying the properties of both tests and estimators. Likelihood and penalized likelihood are used to develop the tests and estimators, and the properties are studied through a combination of asymptotic analysis and simulation. An exponential time series is a stochastic process whose finite dimensional distributions form exponential families. For such process, the sampling distributions of normalized maximum likelihood estimators are asymptotically normal, under quite general conditions. The research involves developing higher order approximations to the these sampling distributions and using the refined approximations to form corrected confidence sets. Mathematically, the results take the form of very weak expansions in which a Bayesian approach is used to obtain approximations to sampling distributions. Biased samples arise frequently in investigations that involve searching for hidden objects, since the probability of finding an object may depend on its properties. For example, astronomers are more likely to find a large bright galaxy than a small dim one, and geologists are more likely to find a large oil well than a small one. Previous work on such problems has concentrated on the case in which the inclusion probabilities depend on the variables of interest in a known way. The new research involves developing data analyses that are appr opriate when the latter relationship is not known and includes methods for estimating the relationship from observed data. The relationship is important, because the number of objects not found depends on it in crucial way. The research considers exponential time series which include classical time series, like stock prices and weather, but also many others like adaptively designed experiments (experiments that design themselves) and sequential clinical trials.