arXiv subjects

Yoshiki Otobe

Publications and source records attributed to Yoshiki Otobe.

6 recordsLinked to original sources

arXiv2024

Power means of random variables and characterizations of distributions via fractional calculus

We investigate fractional moments and expectations of power means of complex-valued random variables by using fractional calculus. We deal with both negative and positive orders of the fractional derivatives. The one-dimensional distributions are characterized in terms of the fractional moments without any moment assumptions. We explicitly compute the expectations of the power means for both the univariate Cauchy distribution and the Poincaré distribution on the upper-half plane. We show that for these distributions the expectations are invariant with respect to the sample size and the value of the power.

math.PR↗

arXiv2023

Properties of complex-valued power means of random variables and their applications

We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is an example of a homogeneous quasi-arithmetic mean. Under certain conditions, several limit theorems hold for the power mean, similar to the case of the arithmetic mean of i.i.d. integrable random variables. Our feature is that the generators of the power means are allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. We establish integrabilities of the power mean of i.i.d. non-integrable random variables and a limit theorem for the variances of the power mean. We also consider the behavior of the power mean as the parameter of the power varies. The complex-valued power means are unbiased, strongly-consistent, robust estimators for the joint of the location and scale parameters of the Cauchy distribution.

math.PR↗

arXiv2022

Confidence disc and square for Cauchy distributions

We will construct a confidence region of parameters for a sample of size $N$ from Cauchy distributed random variables. Although Cauchy distribution has two parameters, a location parameter $μ\in \mathbb{R}$ and a scale parameter $σ> 0$, we will infer them at once by regarding them as a single complex parameter $γ:= μ+ iσ$. The region should be a domain in the complex plane, and we will give a simple and concrete formula to give the region as a disc and a square.

math.ST↗

arXiv2022

Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution

We establish some limit theorems for quasi-arithmetic means of random variables. This class of means contains the arithmetic, geometric and harmonic means. Our feature is that the generators of quasi-arithmetic means are allowed to be complex-valued, which makes considerations for quasi-arithmetic means of random variables which could take negative values possible. Our motivation for the limit theorems is finding simple estimators of the parameters of the Cauchy distribution. By applying the limit theorems, we obtain some closed-form unbiased strongly-consistent estimators for the joint of the location and scale parameters of the Cauchy distribution, which are easy to compute and analyze.

math.ST↗

arXiv2021

Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution

Some quasi-arithmetic means of random variables easily give unbiased strongly consistent closed-form estimators of the joint of the location and scale parameters of the Cauchy distribution. The one-step estimators of those quasi-arithmetic means of the Cauchy distribution are considered. We establish the Bahadur efficiency of the maximum likelihood estimator and the one-step estimators. We also show that the rate of the convergence of the mean-squared errors achieves the Cramer-Rao bound. Our results are also applicable to the circular Cauchy distribution.

math.ST↗

arXiv2021

Characterizations of the maximum likelihood estimator of the Cauchy distribution

This paper gives a new approach for the maximum likelihood estimation of the joint of the location and scale of the Cauchy distribution. We regard the joint as a single complex parameter and derive a new form of the likelihood equation of a complex variable. Based on the equation, we provide a new iterative scheme approximating the maximum likelihood estimate. We also handle the equation in an algebraic manner and derive a polynomial containing the maximum likelihood estimate as a root. This algebraic approach provides another scheme approximating the maximum likelihood estimate by root-finding algorithms for polynomials, and furthermore, gives non-existence of closed-form formulae for the case that the sample size is five. We finally provide some numerical examples to show our method is effective.

math.ST↗