On the pole and zero locations of rational Laplace transformations of non-negative functions. II

Zemanian, A. H.

doi:10.1090/s0002-9939-1961-0136938-8

Proc. Amer. Math. Soc.

1961

DOI: 10.1090/s0002-9939-1961-0136938-8

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On the pole and zero locations of rational Laplace transformations of non-negative functions. II

A. H. Zemanian¹

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Cited by 9 publications

References 8 publications

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An alternative characterization for matrix exponential distributions

Fackrell

2009

Adv. Appl. Probab.

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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

Fackrell

2009

Adv. Appl. Probab.

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Classes of probability density functions having Laplace transforms with negative zeros and poles

Sumita

Masuda

1987

Adv. Appl. Probab.

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We consider a class of functions on [0,∞), denoted by Ω, having Laplace transforms with only negative zeros and poles. Of special interest is the class Ω+ of probability density functions in Ω. Simple and useful conditions are given for necessity and sufficiency of f ∊ Ω to be in Ω+. The class Ω+ contains many classes of great importance such as mixtures of n independent exponential random variables (CMn), sums of n independent exponential random variables (PF∗ n ), sums of two independent random variables, one in CMr and the other in PF ∗ 1 (CMPFn with n = r + l) and sums of independent random variables in CMn (SCM). Characterization theorems for these classes are given in terms of zeros and poles of Laplace transforms. The prevalence of these classes in applied probability models of practical importance is demonstrated. In particular, sufficient conditions are given for complete monotonicity and unimodality of modified renewal densities.

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Phase-type distributions and representations: Some results and open problems for system theory

Commault

Mocanu²

2003

International Journal of Control

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On the pole and zero locations of rational Laplace transformations of non-negative functions. II

Cited by 9 publications

References 8 publications

An alternative characterization for matrix exponential distributions

An alternative characterization for matrix exponential distributions

Classes of probability density functions having Laplace transforms with negative zeros and poles

Phase-type distributions and representations: Some results and open problems for system theory

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