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Eduardo Nakano

Publications and source records attributed to Eduardo Nakano.

5 recordsLinked to original sources
arXiv2026

Tail representations for absolute, signed-power, inverse, and logarithmic moments

Let $X$ be a real-valued random variable and let $p>0$. When $X$ can take negative values, $X^p$ is not generally real-valued for non-integer $p$; formulas for moments of arbitrary positive order must therefore distinguish the absolute power $|X|^p$ from the signed power $\operatorname{sgn}(X)|X|^p$. Using the layer-cake identity, we give a self-contained treatment of tail representations for the positive and negative parts of $X$, including absolute and signed-power moments, centered forms, and exact formulas on ordered discrete supports and integer lattices. The basic identities require neither a density nor continuity and apply to continuous, discrete, singular, and mixed distributions. For strictly positive random variables, we also give representations and finiteness criteria for inverse moments, together with distribution-function and Frullani--Laplace formulas for logarithmic moments. Examples illustrate two-sided support, atoms, lattice tails, and behavior near zero. The presentation places these classical identities in a common notation and states the domain and integrability conditions required in each case.

stat.ME↗
arXiv2026

Closed-form estimators for an exponential family derived from likelihood equations

In this paper, we derive closed-form estimators for the parameters of some probability distributions belonging to the exponential family. A bootstrap bias-reduced version of these proposed closed-form estimators are also derived. A Monte Carlo simulation is performed for the assessment of the estimators. The results are seen to be quite favorable to the proposed bootstrap bias-reduce estimators.

stat.ME↗
arXiv2026

Unified formulas for conditional quantities and transportation functionals

This paper develops a unified probabilistic framework based on distributional derivatives and Dirac delta representations for the analysis of conditional and transportation-related quantities. General identities are established for arbitrary random variables, encompassing absolutely continuous, discrete, and mixed distributions. The proposed approach yields unified formulas for conditional expectations, conditional distributions, hazard functions, and improper distributions, revealing a common localization mechanism underlying these classical concepts. The framework is further combined with copula methods to investigate transportation and dispersion functionals through dependence structures. Exploiting the extremal properties of the Fréchet--Hoeffding bounds together with expectation inequalities induced by $Δ$-antitonic functions, sharp bounds are derived for absolute difference moments under fixed marginals. These results lead to concise derivations of quantile representations for the Wasserstein distance and a corresponding upper transportation functional, as well as survival-function representations and bounds for generalized absolute difference moments. As a particular case, new representations are obtained for the bivariate Gini mean difference and the associated bivariate Gini index. Applications are given to Wasserstein-type functionals arising in the normal approximation of standardized counting distributions, including Poisson, Binomial, and Negative Binomial models, for which explicit quantile representations are derived. Overall, the results establish explicit links among conditional structures, dependence modeling, dispersion measures, normal approximation, and optimal transport, providing a unified perspective on several fundamental constructions in probability and mathematical statistics.

math.ST↗
arXiv2025

Closed-form solutions for parameter estimation in exponential families based on maximum a posteriori equations

In this paper, we derive closed-form estimators for the parameters of certain exponential family distributions through the maximum a posteriori (MAP) equations. A Monte Carlo simulation is conducted to assess the performance of the proposed estimators. The results show that, as expected, their accuracy improves with increasing sample size, with both bias and mean squared error approaching zero. Moreover, the proposed estimators exhibit performance comparable to that of traditional MAP and maximum likelihood (ML) estimators. A notable advantage of the proposed method lies in its computational simplicity, as it eliminates the need for numerical optimization required by MAP and ML estimation.

stat.ME↗