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.Comment: 19 pages; to appear in Lobachevskii Journal of Mathematic

Measure theoretical approach to recurrent properties for quantum dynamics

Poincare's recurrence theorem, which states that every Hamiltonian dynamics enclosed in a finite volume returns to its initial position as close as one wishes, is a mathematical basis of statistical mechanics. It is Liouville's theorem that guarantees that the dynamics preserves the volume on the state space. A quantum version of Poincare's theorem was obtained in the middle of the 20th century without any volume structures of the state space (Hilbert space). One of our aims in this paper is to establish such properties of quantum dynamics from an analog of Liouville's theorem, namely, we will construct a natural probability measure on the Hilbert space from a Hamiltonian defined on the space. Then we will show that the measure is invariant under the corresponding Schrodinger flow. Moreover, we show that the dynamics naturally causes an infinite-dimensional Weyl transformation. It also enables us to discuss the ergodic properties of such dynamics.ArticleJOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 44(46):465209 (2011)journal articl

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.Comment: 43 pages; v3: Section 2 for backgrounds and Section 8 for the mixture Cauchy model added. Introduction shortened. To appear in Acta Math. Hun

Confidence disc for Cauchy distributions

We will construct a confidence region of parameters for $N$ samples from Cauchy distributed random variables. Although Cauchy distribution has two parameters, a location parameter $\mu \in \mathbb{R}$ and a scale parameter $\sigma > 0$, we will infer them at once by regarding them as a single complex parameter $\gamma := \mu +I \sigma$. Therefore 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.Comment: 13 pages, 6 figure

Stochastic Partial Differential Equations with Two Reflecting Walls

We study stochastic partial differential equations (SPDEs) driven by space-time white noise with two reflecting smooth walls $h_1$ and $h_2$. If the solution stays in the open interval $(h_1(x,t), h_2(x,t))$, the dynamics obeys a usual type of SPDEs, and at a point where the value of the solution is $h_1$ or $h_2$, we add forces in order to prevent it from exiting the interval $[h_1, h_2]$. We will first show the existence and uniqueness of the solutions, and secondly study the stationary distribution of the dynamics and corresponding Dirichlet forms

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

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