Proper econometric analysis should be informed by data structure. Many forms of financial data are recorded in discrete-time and relate to products of a finite term. If the data comes from a financial trust, it will often be further subject to random left-truncation. While the literature for estimating a distribution function from left-truncated data is extensive, a thorough literature search reveals that the case of discrete data over a finite number of possible values has received little attention. A precise discrete framework and suitable sampling procedure for the Woodroofe-type estimator for discrete data over a finite number of possible values is therefore established. Subsequently, the resulting vector of hazard rate estimators is proved to be asymptotically normal with independent components. Asymptotic normality of the survival function estimator is then established. Sister results for the left-truncating random variable are also proved. Taken together, the resulting joint vector of hazard rate estimates for the lifetime and left-truncation random variables is proved to be the maximum likelihood estimate of the parameters of the conditional joint lifetime and left-truncation distribution given the lifetime has not been left-truncated. A hypothesis test for the shape of the distribution function based on our asymptotic results is derived. Such a test is useful to formally assess the plausibility of the stationarity assumption in length-biased sampling. The finite sample performance of the estimators is investigated in a simulation study. Applicability of the theoretical results in an econometric setting is demonstrated with a subset of data from the Mercedes-Benz 2017-A securitized bond.
How Fast Do Signatures Learn? Statistical Theory and Applications for Path Regression
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Center for Data Science, Peking University（北京大学数据科学中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Oxford Man Institute, University of Oxford（牛津大学牛津人学研究所）；Center for Statistical Science, Peking University（北京大学统计科学中心）
On multipolar magnetic anomaly detection: multipolar signal subspaces, an analytical orthonormal basis, multipolar truncature and detection performance
词嵌入是自然语言处理中的一个基本工具。目前，词嵌入方法是基于基准数据集上的经验性能进行评估的，并且缺乏对其理论特性的严格理解。本文从统计理论的角度研究词嵌入，这是必不可少的形式化推理和不确定性量化。我们提出了一个基于Copula的文本数据的统计模型，并表明，在此模型下，现在经典的Word2Vec方法可以被解释为一个统计估计方法估计理论逐点互信息（PMI）。接下来，通过建立在Levy和Goldberg（2014）的工作基础上，我们开发了一个基于缺失值的估计器，作为Word2Vec方法的统计上易于处理和解释的替代方案。该估计器的估计误差与Word2Vec相当，并且改进了Levy和Goldberg（2014）提出的基于截断的方法。该估计器还在IMDb Movie Reviews数据集上的基准情感分析任务中执行Word2Vec。
