86 research outputs found
A new generalization of the Takagi function
We consider a one-parameter family of functions on
and partial derivatives with respect to the
parameter . Each function of the class is defined by a certain pair of two
square matrices of order two. The class includes the Lebesgue singular
functions and other singular functions. Our approach to the Takagi function is
similar to Hata and Yamaguti. The class of partial derivatives
includes the original Takagi function and some
generalizations. We consider real-analytic properties of as a function of , specifically, differentiability, the Hausdorff
dimension of the graph, the asymptotic around dyadic rationals, variation, a
question of local monotonicity and a modulus of continuity. Our results are
extensions of some results for the original Takagi function and some
generalizations.Comment: 22 pages, 2 figures. The structure of paper has been changed
significantl
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
Confidence disc for Cauchy distributions
We will construct a confidence region of parameters for samples from
Cauchy distributed random variables. Although Cauchy distribution has two
parameters, a location parameter and a scale parameter
, we will infer them at once by regarding them as a single complex
parameter . 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
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