高维拟似然模型下的广义似然比检验

大规模数据集存在于生物学、化学、经济学、金融学、遗传学、神经科学和物理学等许多领域。它给数据科学家带来了更多的挑战和新的机遇。大规模数据带来的主要挑战是维度灾难和内存限制。维数灾难意味着维数可能随样本量呈指数增长，需要在非多项式维度下研究估计及假设检验问题。内存限制意味着一台机器的内存无法容纳所有数据集或者勉强能存下数据集但不能进行任何计算。因此，许多传统的估计和推断方法需要重新研究。为了应对大数据的挑战，研究者提出了许多创新算法来解决大规模数据集下的统计推断问题。然而，这些算法不是依据统计学原理而开发的，这导致计算结果的准确性和稳定性没有理论保证。本论文主要在高维拟似然模型下研究单个参数或一组参数特定线性结构的显著性检验，并且提出了海量数据下检验统计量的分布式算法。

首先，对于非多项式维数的指数族广义线性模型，本文提出了一种高维普
通最小二乘投影（HOLP）来降低备择假设空间的维度。基于HOLP选择的子模型，本文构造了似然比统计量。然而，HOLP选择的子模型可能是一个随机子模型，子模型的随机性会降低似然比检验的效果。为了处理HOLP选择的子模型的随机性，本文提出了重新拟合似然比及其相应的Wald和Score检验。模拟和实际数据分析结果表明似然比和重新拟合检验在有限样本下具有较高的检验功效。

其次，对于非多项式维数的拟似然模型，基于高维惩罚拟似然估计，本文提出了广义似然比检验。虽然HOLP在指数族广义线性模型中表现良好，但是HOLP 难以应用到拟似然模型中。广义似然比的思想是首先通过惩罚拟似然估计选择子模型，从而降低备择假设空间的维度，第二步是在降维后的空间中进行假设检验。如果惩罚拟似然估计达到了变量选择的强相合性，广义似然比检验具有接近Oracle广义似然比检验的功效。然而，在超高维模型中，惩罚估计可能会产生虚假变量，从而降低检验的性能。为了解决这个问题，本文提出了重新拟合似然比及其相应的广义Wald和Score检验。这些重新拟合后的检验不仅不受虚假变量效应的影响，而且对调整参数也很稳健。模拟和实际数据分析研究表明广义似然比及其重新拟合检验在有限样本下是可行的、有效的。

最后，本文给出了广义似然比及其重新拟合统计量的分布式计算方法。由于HOLP具有封闭形式，可以使用分块矩阵递归算法来求解大规模矩阵的逆。因此，本文主要关注广义似然比统计量及其重新拟合统计量的计算。正如上面所说，广义似然比及其重新拟合统计量的核心点是变量选择。基于分治法（Divide-and-Conquer）和投票加权法， 本文提出了惩罚拟似然分布式估计。具体来说，将整个数据分成多个子数据集，存储在不同机器上，并在每台机器上拟合数据，然后通过投票加权法来整合所有机器的结果。基于分布式惩罚拟似然估计变量选择的结果，本文采用牛顿-拉夫森（Newton-Raphson）方法计算全数据下的极大拟似然估计量。由于牛顿-拉夫森方法每次迭代都有封闭形式，该封闭形式只涉及Score函数和Fisher信息矩阵的计算。故可以通过整合每台机器上的Score函数和Fisher信息矩阵来获得全数据下的极大拟似然估计量。基于极大拟似然估计，采用类似的平均整合方法就可以得到广义似然比及其重新拟合统计量。模拟试验和实际数据分析结果表明分布式惩罚拟似然估计与完全样本的惩罚拟似然估计具有相同的渐近效率。

There exist massive high-dimensional data sets in various fields such as biology, chemistry, economics, finance, genetics, neuroscience, and physics. Meanwhile, data scientists are facing more challenges and new opportunities. The major challenges are the curse of dimensionality and memory constraints. The curse of dimensionality means that the dimensionality could grow exponentially with the sample size,  and hypothesis tests are conducted under non-polynomial dimensionality. The memory constraints mean that the memory of one machine cannot contain the whole data set or may barely hold the data set but have no extra room to do computation. Thus, improving the traditional estimation and inference methods is necessary. In order to overcome the challenges of big data, lots of innovative algorithms have been proposed to address statistical inference problems under high-dimensional data sets.
However, their developments are not followed by statistical principles, and their computational results' accuracy and stability are without theoretical supports. Thus, there always exists a gap between statistical theory and computational performance. Based on the high dimensional quasi-likelihood models, this dissertation focuses on the significant tests of a single parameter or the linear structure of a group of parameters, and proposes a distributed algorithm of test statistics in massive data.

First, for exponential generalized linear models with non-polynomial, this thesis propose a new method to reduce the dimension of alternative hypothesis space, referred to as the high dimensional ordinary least square projection(HOLP). Based on the sub-model selected by HOLP, the likelihood ratio statistic is constructed. However, the sub-model selected by HOLP may be  random, which deteriorates the performance of the test. To improve this situation, the refitted likelihood ratio test and the corresponding Score and Wald tests are proposed. The result of the simulation and real data demonstrates the refitted likelihood ratio test and the corresponding Score and Wald tests are powerful under finite samples.

Second, for quasi-likelihood models with non-polynomial dimensionality, the generalized likelihood ratio test is proposed based on the high dimensional penalized quasi-likelihood estimators. Although HOLP performs well in exponential generalized linear models, HOLP is hard to apply to quasi-likelihood models. The idea of the generalized likelihood ratio test is to reduce the dimension of the alternative hypothesis space by the high dimensional penalized quasi-likelihood estimation and then to conduct hypothesis testing in the dimension-reduced space. If the variable selection consistency is achieved, the generalized likelihood ratio test's performance is nearly oracle's. Whereas, in ultrahigh-dimensional models, the penalized estimation may cause misleading, which may include nuisance variables and let the performance of tests to become worse. To tackle this problem, we introduce refitted generalized likelihood ratio test and the corresponding generalized Score and Wald tests. These tests are free of spurious effects and robust against the tuning parameter. And the simulation studies and real data analysis show that the methodology is practicable and effective for finite samples.

Finally, the distributed computational method of the generalized likelihood ratio and the corresponding refitted statistics is proposed in this dissertation. Since HOLP has a closed form, the block-recursive algorithm is an efficient way to solve the inverse of a large-scale matrix. Then, we focus on the calculation of generalized likelihood ratio statistic and its refitted version. As mentioned above, the key issue of the generalized likelihood ratio statistics is the variable selection. Based on the divide-and-conquer and weighted voting method, we propose the distributed penalized quasi-likelihood estimation (DPQE). Specifically, we split the whole data into many subsets, which are stored in different machines. And then, we run each subset on one of the machines, and aggregate the results from all machines via weighted voting. By the sub-model selected by the DPQE, we adopt the Newton-Raphson method to compute the maximum quasi-likelihood estimators. Since there exists the closed form formula in each iteration of the Newton-Raphson method, and the closed form formula only depends on Score function and Fisher information matrix. The Score function and Fisher information matrix are obtained by averaging the corresponding Score function and Fisher information matrix of the subsample at each machine. Then MQLEs are obtained. Lastly, the generalized likelihood ratio statistic and the corresponding refitting statistics are obtained by the similar averaging method. Simulation and real data analysis  show that the distributed estimator shares the same asymptotic efficiency as the estimator based on the full sample.

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