estimate distribution moments

Conventional moment computation relies on analytic derivatives of probability density functions (PDFs) or moment-generating functions (MGFs), rendering it inapplicable to fractional, complex-order, and central/non-central moments when PDFs are unavailable or MGF derivatives are intractable. Method: This paper proposes the Complex-extended Moment Generating Function (CMGF) integration method—a novel framework that requires only integrability of the MGF over a contour in the complex plane. By leveraging analytic continuation and numerical contour integration, CMGF directly computes arbitrary real-order, complex-order, absolute, central, and non-central moments without invoking PDFs or MGF derivatives. Contribution/Results: As the first general-purpose MGF-based moment computation framework relying on integration rather than differentiation, CMGF is validated across three canonical scenarios where closed-form MGFs exist but PDFs or their derivatives do not. It significantly simplifies high-order and non-integer moment evaluation, enhancing efficiency in statistical inference and stochastic modeling.

The empirical statistical behavior of sample skewness and kurtosis under small-sample regimes remains poorly understood. Method: Leveraging asymptotic analysis, heavy-tailed distribution modeling, and experiments on both synthetic and real-world datasets, we derive rigorous theoretical bounds and examine moment estimation properties. Contribution/Results: We establish, for the first time, a strict asymptotic lower bound on sample kurtosis as a function of sample size and skewness. Extending the Taylor power law to higher-order moments, we reveal that the skewness–kurtosis 4/3 scaling law arises intrinsically from the asymptotic structure of heavy-tailed distributions. This scaling is robust only for heavy-tailed data and sufficiently large samples (n ≳ 100); under small samples, kurtosis exhibits a pronounced negative bias with a nontrivial lower bound, rendering classical moment estimators severely biased. Our results provide a theoretically grounded criterion for the validity domain of moment-based estimation and correct the misconception of universal power-law applicability.

This work investigates necessary and sufficient conditions for asymptotic variance reduction in Monte Carlo integration via moment matching. We establish theoretically that standard linear moment matching guarantees asymptotic variance reduction for the estimator $hat{E}[f(X)]$ of any integrable function $f$ if and only if the base distribution $X$ is Gaussian—revealing the uniqueness of the Gaussian distribution in moment-matching-based variance control. To generalize beyond Gaussianity, we propose a nonlinear moment matching framework applicable to arbitrary continuous distributions. This framework constructs control variates by optimizing higher-order moment constraints and derives an explicit, online-updatable formula for the simulated variance. Empirical results demonstrate that the proposed method significantly outperforms conventional approaches in both accuracy and stability of variance estimation.

This paper addresses the low finite-sample efficiency of L-moment estimation arising from fixing the number of L-moments equal to the number of parameters. We propose an adaptive scheme wherein the number of L-moments—and their associated weights—increases with sample size. We establish, for the first time, an asymptotic theoretical framework for L-moment order selection that grows with sample size, and introduce the Generalized L-Moment Estimator (GLME). GLME retains asymptotic efficiency while substantially improving small-sample accuracy. Monte Carlo simulations demonstrate that GLME achieves significantly lower mean squared error than maximum likelihood estimation (MLE) in small samples, and converges asymptotically to MLE’s efficiency. An empirical application to Brazilian ride-hailing expenditure data further confirms its robustness and practical utility. The core innovation lies in relaxing the conventional fixed-dimension L-moment matching constraint, thereby unifying finite-sample superiority with asymptotic optimality.

In structural estimation, objective functions are often noisy, nonsmooth, and nonconvex, causing conventional optimization methods to readily converge to local minima and hindering rigorous characterization of statistical properties. To address this, we propose a hybrid algorithm integrating an enhanced Gauss–Newton method with adaptive grid search: grid search ensures robust global exploration in early stages, followed by a seamless transition to the improved Gauss–Newton method for rapid local convergence. For the first time under purely econometric assumptions—without restrictive smoothness or convexity conditions—we simultaneously establish finite-sample optimization error bounds and the asymptotic distribution of the estimator. Simulation studies and empirical applications demonstrate that our method substantially improves estimation accuracy and convergence stability, attaining the true solution with high probability while avoiding exhaustive search. It thus achieves an optimal trade-off between computational efficiency and statistical reliability.

This work addresses the lack of a unified and efficient inference framework for models where exact likelihoods are intractable. It proposes the first general modeling and inference framework based on saddlepoint approximation, which preserves access to the moment-generating function through high-level operations to automatically construct the cumulant-generating function, its saddlepoint, and associated gradients. By integrating automatic differentiation, the framework optimizes the saddlepoint likelihood efficiently. It supports flexible modeling with complex distributional compositions and introduces diagnostic metrics that assess approximation error without requiring the true likelihood. Empirical results demonstrate that the method enables accurate parameter estimation, standard error computation, and error diagnosis with high computational efficiency and remarkable flexibility across multiple case studies.

本文针对GMM中存在局部误设问题，提出一种经验贝叶斯方法来估计并修正偏差，从而提高参数估计的精度。

The lack of efficient and stable methods for computing higher-order moments of the logit-normal distribution has hindered its application in inferential statistics. This work proposes an innovative approach based on a logistic function approximation, integrating probability integral transforms with numerical analysis techniques to circumvent direct numerical integration. For the first time, this method enables highly accurate and computationally efficient estimation of moments of any positive integer order. Evaluated on the first eight moments, it substantially outperforms existing Mordell integral-based schemes, eliminating numerical instabilities while achieving significantly faster computation than standard numerical integration in R. As practical demonstrations, the proposed method accelerates expectation propagation in logistic regression and improves integral approximations in logistic mixture models.

This study addresses the efficiency limitations of optimizers such as Adam, which arise from traditional stochastic approximation relying solely on first-order moments. For the first time, second-moment estimation is incorporated into the stochastic approximation framework, enabling a unified derivation of both the Adam and Muon algorithms from an optimal preconditioning perspective alongside a two-stage convergence analysis framework. Leveraging Dvoretzky's theorem, this work rigorously establishes almost sure convergence to a neighborhood of the optimum, thereby overcoming the theoretical constraints inherent in single-moment approaches. Furthermore, it develops a general theory encompassing mainstream deep learning optimizers and provides explicit bounds on the convergence radii for Muon and spectral variants of Adam, substantially deepening the theoretical understanding of optimization mechanisms.

This study addresses the bias arising from estimation errors in nuisance parameters within parametric moment condition models. To mitigate this issue, the paper proposes a high-order debiasing method that constructs moment functions exhibiting Neyman orthogonality of a specified order with respect to the nuisance parameters, thereby substantially reducing the sensitivity of the estimator to such errors. The approach is both unified and computationally tractable, with a key innovation being that the number of additional nuisance parameters required for orthogonality does not grow with the order of orthogonality—indeed, it can be reduced to a single scalar. Theoretical analysis and empirical evidence demonstrate that this method effectively diminishes estimation bias and significantly enhances robustness and precision across a broad class of econometric models.