On approximating the renewal function with its linear asymptote: How large is large enough?
type: Item
instance of: scholarly article
title: On approximating the renewal function with its linear asymptote: How large is large enough?
zbMATH Open document ID: 0585.60086
DOI: 10.1016/0167-6377(86)90003-9
author: Izzet Sahin
published in: Operations Research Letters
publication date: 1986-01-01T00:00:00Z
review text: The author defines a measure of convergence \(K(\epsilon)\) for the renewal function to its linear asymptote \(R(t)=t/\mu +(c^ 2-1)/2+o(1),\) \(t\to \infty\), where \(c=\sigma /\mu\). Using \textit{D. J. McConalogue}'s generalised cubic spline algorithm [Commun. Stat., Simulation Comput. B 10, 265-280 (1981; Zbl 0465.65008)], K(\(\epsilon)\) is computed for gamma, Weibull, truncated normal, inverse Gaussian and lognormal distributions. Then the measure \(K(\epsilon)\) is approximated by a second degree polynomial \(a_ 0(\epsilon)+a_ 1(\epsilon)c+a_ 2(\epsilon)c^ 2.\) It is observed in this paper that except for the lognormal distribution \(R(t)\) can be approximated by its linear asymptotic with small to moderate values of \(t\) as measured in units of \(\mu\). Also the variation in the convergence measure \(K(\epsilon)\) can be explained by the coefficient of variation c. The results of this paper are true only for a class of distributions which are parametrizable in terms of scale and shape parameters. The author has however concluded that outside this class the approximation suggested here is inadequate.
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60E99
zbMATH DE Number: 3938201
zbMATH Keywords: accuracy of asymptotic approximations
zbMATH Keywords: renewal function
reviewed by: V. Thangaraj
MaRDI profile type: Publication
full work available at URL: 0167-6377(86)90003-9
OpenAlex ID: W2077722162
cites work: On the Tabulation of the Renewal Function
cites work: Bounds, inequalities, and monotonicity properties for some specialized renewal processes
cites work: Remainder term estimates of the renewal function
cites work: A Quasi-Intrinsic Scheme for Passing a Smooth Curve through a Discrete Set of Points
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description: scientific article; zbMATH DE number 3938201