arXiv subjects

Tianxiao Pang

Publications and source records attributed to Tianxiao Pang.

5 recordsLinked to original sources

arXiv2026

Complex-Valued Probability Measures and Their Applications in Information Theory

This paper introduces a comprehensive framework for complex-valued probability measures and explores their novel applications in information theory and statistical analysis. We define a complex probability measure as a phase-modulated extension of a classical probability measure. Building upon this foundation, we propose three fundamental information-theoretic quantities: complex entropy, which quantifies distribution uniformity through phase coherence; complex divergence, an asymmetric measure of dissimilarity between distributions; and the complex metric, a symmetric distance function satisfying the triangle inequality. We establish these concepts rigorously for both continuous and discrete probability distributions, proving key properties such as boundedness, continuity under total variation convergence, and clear extremal behaviors. A detailed comparative analysis with classical measures (Shannon entropy and Kullback-Leibler divergence) highlights the unique geometric and interpretive advantages of the proposed framework, particularly its sensitivity to distributional shape via a tunable phase parameter. We elucidate a profound formal analogy between the complex entropy integral and Feynman's path integral formulation of quantum mechanics, suggesting a deeper conceptual bridge. Finally, we demonstrate the practical utility of the complex metric through a detailed application in nonparametric two-sample hypothesis testing, outlining the testing procedure, advantages, limitations, and providing a conceptual simulation. This work opens new avenues for analyzing probability distributions through the lens of complex analysis and interference phenomena, with potential impacts across information theory, statistical inference, and machine learning.

cs.IT↗

arXiv2026

Bayesian Smoothed Quantile Regression

The standard asymmetric Laplace framework for Bayesian quantile regression (BQR) suffers from a fundamental decision-theoretic misalignment, yielding biased finite-sample estimates, and precludes gradient-based computation due to non-smoothness. We propose Bayesian smoothed quantile regression (BSQR), a principled framework built on a kernel-smoothed, fully differentiable likelihood. Methodologically, the symmetrizing property of our objective reduces inferential bias and aligns the posterior mean with the true conditional quantile. Theoretically, we establish posterior consistency and a Bernstein--von Mises theorem under misspecification, delivering asymptotic normality and valid frequentist coverage via a generalized Wilks phenomenon, while guaranteeing global posterior existence unlike empirical likelihood approaches. Computationally, BSQR enables Hamiltonian Monte Carlo for BQR, alleviating high-dimensional mixing bottlenecks. In simulations, BSQR reduces out-of-sample prediction error by up to 50% and improves sampling efficiency by up to 80% relative to asymmetric Laplace benchmarks, with uniform and triangular kernels performing particularly well. In a financial application to asymmetric systemic risk, BSQR uncovers distinct regime shifts around the COVID-19 period and yields sharper yet well-calibrated predictive quantiles, underscoring its practical relevance.

stat.ME↗

arXiv2025

Efficient importance sampling for copula models

In this paper, we propose an efficient importance sampling algorithm for rare event simulation under copula models. In the algorithm, the derived optimal probability measure is based on the criterion of minimizing the variance of the importance sampling estimator within a parametric exponential tilting family. Since the copula model is defined by its marginals and a copula function, and its moment-generating function is difficult to derive, we apply the transform likelihood ratio method to first identify an alternative exponential tilting family, after which we obtain simple and explicit expressions of equations. Then, the optimal alternative probability measure can be calculated under this transformed exponential tilting family. The proposed importance sampling framework is quite general and can be implemented for many classes of copula models, including some traditional parametric copula families and a class of semiparametric copulas called regular vine copulas, from which sampling is feasible. The theoretical results of the logarithmic efficiency and bounded relative error are proved for some commonly-used copula models under the case of simple rare events. Monte Carlo experiments are conducted, in which we study the relative efficiency of the crude Monte Carlo estimator with respect to the proposed importance-sampling-based estimators, such that substantial variance reductions are obtained in comparison to the standard Monte Carlo estimators.

stat.CO↗

arXiv2023

Kullback-Leibler Divergence and Akaike Information Criterion in General Hidden Markov Models

To characterize the Kullback-Leibler divergence and Fisher information in general parametrized hidden Markov models, in this paper, we first show that the log likelihood and its derivatives can be represented as an additive functional of a Markovian iterated function system, and then provide explicit characterizations of these two quantities through this representation. Moreover, we show that Kullback-Leibler divergence can be locally approximated by a quadratic function determined by the Fisher information. Results relating to the Cramér-Rao lower bound and the Hájek-Le Cam local asymptotic minimax theorem are also given. As an application of our results, we provide a theoretical justification of using Akaike information criterion (AIC) model selection in general hidden Markov models. Last, we study three concrete models: a Gaussian vector autoregressive-moving average model of order $(p,q)$, recurrent neural networks, and temporal restricted Boltzmann machine, to illustrate our theory.

math.ST↗

arXiv2016

Asymptotic Inference for AR(1) Penal Data

A general asymptotic theory is given for the panel data AR(1) model with time series independent in different cross sections. The theory covers the cases of stationary process, nearly non-stationary process, unit root process, mildly integrated, mildly explosive and explosive processes. It is assumed that the cross-sectional dimension and time-series dimension are respectively $N$ and $T$. The results in this paper illustrate that whichever the process is, with an appropriate regularization, the least squares estimator of the autoregressive coefficient converges to a normal distribution with rate at least $O(N^{-1/3})$. Since the variance is the key to characterize the normal distribution, it is important to discuss the variance of the least squares estimator. We will show that when the autoregressive coefficient $ρ$ satisfies $|ρ|<1$, the variance declines at the rate $O((NT)^{-1/2})$, while the rate changes to $O(N^{-1/2}T^{-1})$ when $ρ=1$ and $O(N^{-1/2}ρ^{-T+2})$ when $|ρ|>1$. $ρ=1$ is the critical point where the convergence rate changes radically. The transition process is studied by assuming $ρ$ depending on $T$ and going to $1$. An interesting phenomenon discovered in this paper is that, in the explosive case, the least squares estimator of the autoregressive coefficient has a standard normal limiting distribution in panel data case while it may not has a limiting distribution in univariate time series case.

stat.AP↗