Statistical methods for finite mixture, hidden Markov and density ratio models.

木结构的强度很大程度上取决于木材的质量。确保绝大多数特定木制品超过预先规定的质量标准是至关重要的。为此，实验室每年随机抽取样本，找出其优势，并根据实验数据估计质量指标。这个过程既昂贵又费力；因此，需要有效的统计方法。我们的密度比模型（DRM）项目就是为此目的而设计的。DRM通过密度比连接了几个种群分布。与经验似然（EL）一起，DRM从多个独立的样本中汇集信息以提高效率。将开展更多的研究，以增强林业和其他工业应用。这种组合也适用于调查抽样中的小区域估计。经过调查后，可以在最高水平上进行适当精度的推断，但不能对个别地区进行推断。概率抽样计划的随机性质可能对许多感兴趣的区域产生很少或没有直接信息，导致需要进行小区域估计。统计分析必须以小区域的结构假设为基础，而这种假设的可行性至关重要。DRM提出了一个非限制性的“结构性假设”。它提供了一种新的方法，其优点是能够对平均值和分位数（如收入中位数）进行高质量的估计，而不是局限于平均值（如平均收入）。

The strength of a wood structure strongly depends on the quality of the lumber. It is vital to ensure that the vast majority of specific wood products exceed a prespecified quality standard. For this purpose, every year labs find the strengths of a random sample, and quality indices are estimated based on the lab data. This process is costly and laborious; efficient statistical methods are therefore in demand. Our density ratio model (DRM) project is designed for this purpose. DRM connects several population distributions through a density ratio. Together with the empirical likelihood (EL), DRM pools information from several independent samples to improve efficiency. More research will be carried out to enhance the forestry and other industrial applications. The combination is also useful for small-area estimation in survey sampling. After a survey, inferences with appropriate precisions are possible at the top level but not for individual regions. The random nature of the probability sampling plan may yield little or no direct information for many regions of interest, leading to a need for small-area estimation. Statistical analyses have to be based on structural assumptions for small areas, and the viability of the assumption is crucial. The DRM posts a nonrestrictive "structural assumption." It provides a fresh approach and has the advantage of enabling quality estimates for both means and quantiles (such as the median income) rather than being limited to means (such as the average income). Accurately predicting the ups and downs of a stock index is "mission impossible." A stochastic description of the movement is probably the best we can do. We aim to find the most appropriate mathematical models for financial times series and then to craft efficient analysis methods. A regime-switch model postulates that the day-to-day fluctuations of a time series are reflections of hidden states governed by a Markov chain. The structure of this chain sheds light on the volatility in the time series. The standard inference platform has been the full likelihood; we have argued that composite likelihoods offer an effective alternative. I have developed a specific composite likelihood that provides point estimators with a negligible efficiency loss. It has a simpler mathematical structure that facilitates thorough theoretical investigation. I aim to develop consistent variance estimation and to explore the potential of the composite likelihood ratio test for various aspects of the model and for the construction of confidence intervals. Patients with the same disease differ in many ways, and there is thus a need for personalized medicine. Population heterogeneity can often be discovered by testing the order of a finite mixture model. We have developed a number of tests for the order of mixture models. They have easy-to-use large-sample properties and fill a large void in statistical inference. I intend to vastly expand the horizon of the EM-test and to develop easy-to-use software packages. Last but not least, adding a pseudo-observation elegantly solves a technical issue in the application of the empirical likelihood. It also improves the precision of the resulting statistical inference. Since I introduced this idea, it has been applied by many researchers, particularly econometricians. There are many additional research problems to be explored.