Quasi-Likelihood of Models: Modified Profile Likelihood for Model Selection

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。 大多数统计推断都是基于数据的统计模型，因此模型选择在统计推断中起着基础性的作用。虽然有大量的文献发展了模型选择的理论和方法，如AIC、BIC、自助准则、交叉验证准则等，但模型选择仍然是一个"未解决“的问题，因为没有一个神奇的方法可以得到最好的模型。该建议的目标是开发候选模型的准似然函数，作为非常准确和自然的模型选择标准。注意，这里的准似然函数是模型本身的函数，而不是模型中的参数。受修改的轮廓似然（MPLs）的启发，研究者将每个候选模型中的这些参数视为讨厌的参数，并将模型本身视为感兴趣的“参数”的值，以开发候选模型的准似然函数。然后，所选择的模型是使模型的准似然最大化的模型。一些模拟结果表明，建议MPL工程非常好的位置-尺度模型中的错误概率律的选择。对于复合变换模型，也得到了模型的MPL。然后，研究人员将开发模型的准似然函数，以选择回归中的变量和错误概率定律，并研究其理论特性。研究人员还将开发指数族模型的拟似然，研究其理论性质，证明其在模拟和应用中预期的良好性能，并探索将其应用于常规模型。除此之外，研究者可以进一步发展拟似然来选择变点问题中的变点数目，以及选择AR或阿尔马时间序列模型的阶数。研究人员还将比较拟议的准似然函数与AIC，BIC等，看看它的优点。模型选择是科学探究的基本任务之一，研究者可以研究统计学和其他学科中的其他模型选择问题，并进行一些重要的实际应用。提出的模型的拟似然提供了一种新颖的，非常自然的，普遍的和非常好的方法来选择模型。这一新的模型选择标准将是解决统计学和其他学科中“未解决”的模型选择问题的一个重大进展。由于模型选择问题存在于几乎每个学科中，因此所提出的模型的拟似然可以广泛地应用于各种学科，如统计学，信号处理，计量经济学，医学，生物学，计算机科学，通信，工程，物理学甚至定量化学。调查员将与其他学科合作，以解决他们的一些真实的问题。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Most of statistical inferences are based on statistical models for data, so model selection plays a fundamental role in statistical inferences. There is huge amount of literature to develop model selection theory and methodologies such as AIC, BIC, bootstrap criteria, cross-validation criteria and so on. However, model selection is still an ``unsolved'' problem in the sense that there are no magic procedures to get the best model. The goal of this proposal is to develop quasi-likelihood functions of candidate models as a very accurate and natural model selection criterion. Note that the quasi-likelihood functions here are functions of models themselves instead of parameters in the models. Motivated by the modified profile likelihoods (MPLs), the investigator treats those parameters in each candidate model as nuisance parameters, and the models themselves as the values of the ``parameter'' of interest, to develop the quasi-likelihood functions of candidate models. The selected model is then the one maximizing the quasi-likelihood of models. Some simulations have shown that the proposed MPL works very well for the selection of error probability laws in location-scale models. The MPL of models has also been obtained for composite transformation models. The investigator will then develop the quasi-likelihood function of models to select variables and error probability laws in regressions and study its theoretical properties. The investigator will also develop the quasi-likelihood of models in exponential family, study its theoretical properties justifying its good performances expected in simulations and applications, and explore to apply it to regular models. Other than these, the investigator may go further to develop the quasi-likelihood to select the number of change points in the change point problems, and to select the order of AR or ARMA time series models. The investigator will also compare the proposed quasi-likelihood function with AIC, BIC, and so on to see its advantages. The investigator may study the other model selection problems in statistics and the other disciplines and carry out some practical and important applications.Model selection is one of the fundamental tasks of scientific inquiry. The proposed quasi-likelihood of models provides a novel, very natural, universal and extraordinarily good way to select models. This novel model selection criterion would be a significant progress in solving the ``unsolved'' model selection problems in statistics and the other disciplines. Since model selection problems exist arguably in almost every discipline, the proposed quasi-likelihood of models can be broadly used in various disciplines such as statistics, signal processing, econometrics, medicine, biology, computer sciences, communication, engineering, physics and even quantitative chemistry. The investigator will collaborate with the other disciplines to solve some of their real problems.