Text watermarks in large language models (LLMs) are an increasingly important tool for detecting synthetic text and distinguishing human-written content from LLM-generated text. While most existing studies focus on determining whether entire texts are watermarked, many real-world scenarios involve mixed-source texts, which blend human-written and watermarked content. In this paper, we address the problem of optimally estimating the watermark proportion in mixed-source texts. We cast this problem as estimating the proportion parameter in a mixture model based on \emph{pivotal statistics}. First, we show that this parameter is not even identifiable in certain watermarking schemes, let alone consistently estimable. In stark contrast, for watermarking methods that employ continuous pivotal statistics for detection, we demonstrate that the proportion parameter is identifiable under mild conditions. We propose efficient estimators for this class of methods, which include several popular unbiased watermarks as examples, and derive minimax lower bounds for any measurable estimator based on pivotal statistics, showing that our estimators achieve these lower bounds. Through evaluations on both synthetic data and mixed-source text generated by open-source models, we demonstrate that our proposed estimators consistently achieve high estimation accuracy.

In this paper, we attempt to summarize monthly reports as investment reports. Fund managers have a wide range of tasks, one of which is the preparation of investment reports. In addition to preparing monthly reports on fund management, fund managers prepare management reports that summarize these monthly reports every six months or once a year. The preparation of fund reports is a labor-intensive and time-consuming task. Therefore, in this paper, we tackle investment summarization from monthly reports using transformer-based models. There are two main types of summarization methods: extractive summarization and abstractive summarization, and this study constructs both methods and examines which is more useful in summarizing investment reports.

Lexicon-based sentiment analysis (SA) in finance leverages specialized, manually annotated lexicons created by human experts to extract sentiment from financial texts. Although lexicon-based methods are simple to implement and fast to operate on textual data, they require considerable manual annotation efforts to create, maintain, and update the lexicons. These methods are also considered inferior to the deep learning-based approaches, such as transformer models, which have become dominant in various NLP tasks due to their remarkable performance. However, transformers require extensive data and computational resources for both training and testing. Additionally, they involve significant prediction times, making them unsuitable for real-time production environments or systems with limited processing capabilities. In this paper, we introduce a novel methodology named eXplainable Lexicons (XLex) that combines the advantages of both lexicon-based methods and transformer models. We propose an approach that utilizes transformers and SHapley Additive exPlanations (SHAP) for explainability to learn financial lexicons. Our study presents four main contributions. Firstly, we demonstrate that transformer-aided explainable lexicons can enhance the vocabulary coverage of the benchmark Loughran-McDonald (LM) lexicon, reducing the human involvement in annotating, maintaining, and updating the lexicons. Secondly, we show that the resulting lexicon outperforms the standard LM lexicon in SA of financial datasets. Thirdly, we illustrate that the lexicon-based approach is significantly more efficient in terms of model speed and size compared to transformers. Lastly, the XLex approach is inherently more interpretable than transformer models as lexicon models rely on predefined rules, allowing for better insights into the results of SA and making the XLex approach a viable tool for financial decision-making.

Nonconvex and nonsmooth optimization problems are important and challenging for statistics and machine learning. In this paper, we propose Projected Proximal Gradient Descent (PPGD) which solves a class of nonconvex and nonsmooth optimization problems, where the nonconvexity and nonsmoothness come from a nonsmooth regularization term which is nonconvex but piecewise convex. In contrast with existing convergence analysis of accelerated PGD methods for nonconvex and nonsmooth problems based on the Kurdyka-\L{}ojasiewicz (K\L{}) property, we provide a new theoretical analysis showing local fast convergence of PPGD. It is proved that PPGD achieves a fast convergence rate of $\cO(1/k^2)$ when the iteration number $k \ge k_0$ for a finite $k_0$ on a class of nonconvex and nonsmooth problems under mild assumptions, which is locally Nesterov's optimal convergence rate of first-order methods on smooth and convex objective function with Lipschitz continuous gradient. Experimental results demonstrate the effectiveness of PPGD.

Kernel-based tests provide a simple yet effective framework that use the theory of reproducing kernel Hilbert spaces to design non-parametric testing procedures. In this paper we propose new theoretical tools that can be used to study the asymptotic behaviour of kernel-based tests in several data scenarios, and in many different testing problems. Unlike current approaches, our methods avoid using lengthy $U$ and $V$ statistics expansions and limit theorems, that commonly appear in the literature, and works directly with random functionals on Hilbert spaces. Therefore, our framework leads to a much simpler and clean analysis of kernel tests, only requiring mild regularity conditions. Furthermore, we show that, in general, our analysis cannot be improved by proving that the regularity conditions required by our methods are both sufficient and necessary. To illustrate the effectiveness of our approach we present a new kernel-test for the conditional independence testing problem, as well as new analyses for already known kernel-based tests.

In this paper we extend the work of Owen (2007) by deriving a second order expansion for the slope parameter in logistic regression, when the size of the majority class is unbounded and the minority class is finite. More precisely, we demonstrate that the second order term converges to a normal distribution and explicitly compute its variance, which surprisingly once again depends only on the mean of the minority class points and not their arrangement under mild regularity assumptions. In the case that the majority class is normally distributed, we illustrate that the variance of the the limiting slope depends exponentially on the z-score of the average of the minority class's points with respect to the majority class's distribution. We confirm our results by Monte Carlo simulations.

We study the behavior of linear discriminant functions for binary classification in the infinite-imbalance limit, where the sample size of one class grows without bound while the sample size of the other remains fixed. The coefficients of the classifier minimize an expected loss specified through a weight function. We show that for a broad class of weight functions, the intercept diverges but the rest of the coefficient vector has a finite limit under infinite imbalance, extending prior work on logistic regression. The limit depends on the left tail of the weight function, for which we distinguish three cases: bounded, asymptotically polynomial, and asymptotically exponential. The limiting coefficient vectors reflect robustness or conservatism properties in the sense that they optimize against certain worst-case alternatives. In the bounded and polynomial cases, the limit is equivalent to an implicit choice of upsampling distribution for the minority class. We apply these ideas in a credit risk setting, with particular emphasis on performance in the high-sensitivity and high-specificity regions.