Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018 ◽  
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.


1986 ◽  
Vol 23 (4) ◽  
pp. 1013-1018 ◽  
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.


Author(s):  
Phil Diamond

AbstractCompetition between a finite number of searching insect parasites is modelled by differential equations and birth-death processes. In the one species case of intraspecific competition, the deterministic equilibrium is globally stable and, for large populations, approximates the mean of the stationary distribution of the process. For two species, both inter- and intraspecific competition occurs and the deterministic equilibrium is globally stable. When the birth-death process is reversible, it is shown that the mean of the stationary distribution is approximated by the equilibrium. Confluent hypergeometric functions of two variables are important to the theory.


2012 ◽  
Vol 49 (4) ◽  
pp. 1036-1051 ◽  
Author(s):  
Damian Clancy

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.


1992 ◽  
Vol 29 (4) ◽  
pp. 781-791 ◽  
Author(s):  
Masaaki Kijima

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn}, where µ0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ < μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that limn→∞λn = λ and limn→∞µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.


2012 ◽  
Vol 49 (04) ◽  
pp. 1036-1051
Author(s):  
Damian Clancy

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.


1992 ◽  
Vol 29 (04) ◽  
pp. 781-791 ◽  
Author(s):  
Masaaki Kijima

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn }, where µ 0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ &lt; μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that lim n→∞ λn = λ and lim n→∞ µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.


1985 ◽  
Vol 17 (3) ◽  
pp. 514-530 ◽  
Author(s):  
Erik A. Van Doorn

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.


Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 147
Author(s):  
Félix Belzunce ◽  
Carolina Martínez-Riquelme ◽  
Magdalena Pereda

In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.


1985 ◽  
Vol 17 (03) ◽  
pp. 514-530 ◽  
Author(s):  
Erik A. Van Doorn

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.


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