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The two-trials rule for drug approval requires "at least two adequate and well-controlled studies, each convincing on its own, to establish effectiveness". This is usually employed by requiring two significant pivotal trials and is the standard regulatory requirement to provide evidence for a new drug's efficacy. However, there is need to develop suitable alternatives to this rule for a number of reasons, including the possible availability of data from more than two trials. I consider the case of up to 3 studies and stress the importance to control the partial Type-I error rate, where only some studies have a true null effect, while maintaining the overall Type-I error rate of the two-trials rule, where all studies have a null effect. Some less-known $p$-value combination methods are useful to achieve this: Pearson's method, Edgington's method and the recently proposed harmonic mean $\chi^2$-test. I study their properties and discuss how they can be extended to a sequential assessment of success while still ensuring overall Type-I error control. I compare the different methods in terms of partial Type-I error rate, project power and the expected number of studies required. Edgington's method is eventually recommended as it is easy to implement and communicate, has only moderate partial Type-I error rate inflation but substantially increased project power.

We investigate the statistical evidence for the use of `rough' fractional processes with Hurst exponent $H< 0.5$ for the modeling of volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized $p$-th variation along a sequence of partitions. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional processes. We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than $0.5$. Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the roughness of the spot volatility process, realized volatility always exhibits `rough' behaviour with an apparent Hurst index $\hat{H}<0.5$. These results suggest that the origin of the roughness observed in realized volatility time-series lies in the microstructure noise rather than the volatility process itself.

Property-based testing (PBT), while an established technique in the software testing research community, is still relatively underused in real-world software. Pain points in writing property-based tests include implementing diverse random input generators and thinking of meaningful properties to test. Developers, however, are more amenable to writing documentation; plenty of library API documentation is available and can be used as natural language specifications for property-based tests. As large language models (LLMs) have recently shown promise in a variety of coding tasks, we explore the potential of using LLMs to synthesize property-based tests. We call our approach PBT-GPT, and propose three different strategies of prompting the LLM for PBT. We characterize various failure modes of PBT-GPT and detail an evaluation methodology for automatically synthesized property-based tests. PBT-GPT achieves promising results in our preliminary studies on sample Python library APIs in $\texttt{numpy}$, $\texttt{networkx}$, and $\texttt{datetime}$.

We study how to release summary statistics on a data stream subject to the constraint of differential privacy. In particular, we focus on releasing the family of symmetric norms, which are invariant under sign-flips and coordinate-wise permutations on an input data stream and include $L_p$ norms, $k$-support norms, top-$k$ norms, and the box norm as special cases. Although it may be possible to design and analyze a separate mechanism for each symmetric norm, we propose a general parametrizable framework that differentially privately releases a number of sufficient statistics from which the approximation of all symmetric norms can be simultaneously computed. Our framework partitions the coordinates of the underlying frequency vector into different levels based on their magnitude and releases approximate frequencies for the "heavy" coordinates in important levels and releases approximate level sizes for the "light" coordinates in important levels. Surprisingly, our mechanism allows for the release of an arbitrary number of symmetric norm approximations without any overhead or additional loss in privacy. Moreover, our mechanism permits $(1+\alpha)$-approximation to each of the symmetric norms and can be implemented using sublinear space in the streaming model for many regimes of the accuracy and privacy parameters.

It is known that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error measured in the square norm. In this paper we demonstrate how known results on universal sampling discretization of the uniform norm and recent results on universal sampling representation allow us to provide good universal methods of sampling recovery for anisotropic Sobolev and Nikol'skii classes of periodic functions of several variables. The sharpest results are obtained in the case of functions on two variables, where the Fibonacci point sets are used for recovery.

The $H^m$-conforming virtual elements of any degree $k$ on any shape of polytope in $\mathbb R^n$ with $m, n\geq1$ and $k\geq m$ are recursively constructed by gluing conforming virtual elements on faces in a universal way. For the lowest degree case $k=m$, the set of degrees of freedom only involves function values and derivatives up to order $m-1$ at the vertices of the polytope. The inverse inequality and several norm equivalences for the $H^m$-conforming virtual elements are rigorously proved. The $H^m$-conforming virtual elements are then applied to discretize a polyharmonic equation with a lower order term. With the help of the interpolation error estimate and norm equivalences, the optimal error estimates are derived for the $H^m$-conforming virtual element method.

The motion of glaciers can be simulated with the p-Stokes equations. We present an algorithm that solves these equations faster than the Picard iteration. We do that by proving q-superlinear global convergence of the infinite-dimensional Newton's method with Armijo step sizes to the solution of these equations. We only have to add an arbitrarily small diffusion term for this convergence result. We also consider approximations of exact step sizes. Exact step sizes are possible because we reformulate the problem as minimizing a convex functional. Next, we prove that the additional diffusion term only causes minor differences in the solution compared to the original p-Stokes equations. Finally, we test our algorithms on a reformulation of the experiment ISMIP-HOM B. The approximation of exact step sizes for the Picard iteration and Newton's method is superior in the experiment compared to the Picard iteration. Also, Newton's method with Armijo step sizes converges faster than the Picard iteration. However, the reached accuracy of Newton's method with Armijo step sizes depends more on the resolution of the domain.

The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning and the most classical comparison methods assume that the distributions occur in spaces of the same dimension. Recently, a new geometric solution has been proposed to address this problem when the measures live in Euclidean spaces of differing dimensions. Here, we study the same problem of comparing probability distributions of different dimensions in the tropical geometric setting, which is becoming increasingly relevant in computations and applications involving complex, geometric data structures. Specifically, we construct a Wasserstein distance between measures on different tropical projective tori - the focal metric spaces in both theory and applications of tropical geometry - via tropical mappings between probability measures. We prove equivalence of the directionality of the maps, whether starting from the lower dimensional space and mapping to the higher dimensional space or vice versa. As an important practical implication, our work provides a framework for comparing probability distributions on the spaces of phylogenetic trees with different leaf sets.

Entity-level fine-grained sentiment analysis in the financial domain is a crucial subtask of sentiment analysis and currently faces numerous challenges. The primary challenge stems from the lack of high-quality and large-scale annotated corpora specifically designed for financial text sentiment analysis, which in turn limits the availability of data necessary for developing effective text processing techniques. Recent advancements in large language models (LLMs) have yielded remarkable performance in natural language processing tasks, primarily centered around language pattern matching. In this paper, we propose a novel and extensive Chinese fine-grained financial sentiment analysis dataset, FinChina SA, for enterprise early warning. We thoroughly evaluate and experiment with well-known existing open-source LLMs using our dataset. We firmly believe that our dataset will serve as a valuable resource to advance the exploration of real-world financial sentiment analysis tasks, which should be the focus of future research. Our dataset and all code to replicate the experimental results will be released.

Graph convolution networks (GCN) are increasingly popular in many applications, yet remain notoriously hard to train over large graph datasets. They need to compute node representations recursively from their neighbors. Current GCN training algorithms suffer from either high computational costs that grow exponentially with the number of layers, or high memory usage for loading the entire graph and node embeddings. In this paper, we propose a novel efficient layer-wise training framework for GCN (L-GCN), that disentangles feature aggregation and feature transformation during training, hence greatly reducing time and memory complexities. We present theoretical analysis for L-GCN under the graph isomorphism framework, that L-GCN leads to as powerful GCNs as the more costly conventional training algorithm does, under mild conditions. We further propose L^2-GCN, which learns a controller for each layer that can automatically adjust the training epochs per layer in L-GCN. Experiments show that L-GCN is faster than state-of-the-arts by at least an order of magnitude, with a consistent of memory usage not dependent on dataset size, while maintaining comparable prediction performance. With the learned controller, L^2-GCN can further cut the training time in half. Our codes are available at //github.com/Shen-Lab/L2-GCN.