An alternative characterization for matrix exponential distributions

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

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An alternative characterization for matrix exponential distributions

Advances in Applied Probability ◽

10.1239/aap/1261669583 ◽

2009 ◽

Vol 41 (4) ◽

pp. 1005-1022 ◽

Cited By ~ 3

Author(s):

Mark Fackrell

Keyword(s):

Continuous Function ◽

Exponential Distribution ◽

Matrix Exponential ◽

Necessary Condition ◽

Stieltjes Transform ◽

Real Numbers ◽

Exponential Distributions ◽

Alternative Characterization

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Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

Open Mathematics ◽

10.1515/math-2020-0021 ◽

2020 ◽

Vol 18 (1) ◽

pp. 353-377 ◽

Cited By ~ 1

Author(s):

Zhien Li ◽

Chao Wang

Keyword(s):

Time Scales ◽

Quaternion Algebra ◽

Matrix Exponential ◽

Dynamic Equations ◽

Necessary Condition ◽

Quaternion Matrix ◽

Exponential Functions ◽

Cauchy Matrix ◽

Adjoint Matrix ◽

Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

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On the Resolvent of the Lévy Process with Matrix-Exponential Distribution of Jumps

Ukrainian Mathematical Journal ◽

10.1007/s11253-017-1336-4 ◽

2017 ◽

Vol 68 (12) ◽

pp. 1884-1899

Author(s):

E. V. Karnaukh

Keyword(s):

Exponential Distribution ◽

Lévy Process ◽

Matrix Exponential ◽

Levy Process

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Univariate and bivariate extensions of the generalized exponential distributions

Mathematica Slovaca ◽

10.1515/ms-2021-0073 ◽

2021 ◽

Vol 71 (6) ◽

pp. 1581-1598

Author(s):

Vahid Nekoukhou ◽

Ashkan Khalifeh ◽

Hamid Bidram

Keyword(s):

Em Algorithm ◽

Exponential Distribution ◽

Bivariate Distribution ◽

Generalized Exponential Distribution ◽

Exponential Distributions ◽

Unknown Parameters ◽

Data Set ◽

New Class ◽

Special Cases ◽

Univariate Distribution

Abstract The main aim of this paper is to introduce a new class of continuous generalized exponential distributions, both for the univariate and bivariate cases. This new class of distributions contains some newly developed distributions as special cases, such as the univariate and also bivariate geometric generalized exponential distribution and the exponential-discrete generalized exponential distribution. Several properties of the proposed univariate and bivariate distributions, and their physical interpretations, are investigated. The univariate distribution has four parameters, whereas the bivariate distribution has five parameters. We propose to use an EM algorithm to estimate the unknown parameters. According to extensive simulation studies, we see that the effectiveness of the proposed algorithm, and the performance is quite satisfactory. A bivariate data set is analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice.

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Fitting with Matrix-Exponential Distributions

Stochastic Models ◽

10.1081/stm-200056227 ◽

2005 ◽

Vol 21 (2-3) ◽

pp. 377-400 ◽

Cited By ~ 24

Author(s):

Mark Fackrell

Keyword(s):

Matrix Exponential ◽

Exponential Distributions

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Unbounded Positive Definite Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-143-4 ◽

1969 ◽

Vol 21 ◽

pp. 1309-1318 ◽

Cited By ~ 3

Author(s):

James Stewart

Keyword(s):

Abelian Group ◽

Positive Measure ◽

Positive Definite Function ◽

Positive Definite ◽

Locally Compact Abelian Group ◽

Definite Function ◽

Complex Numbers ◽

Stieltjes Transform ◽

Real Numbers ◽

Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

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