On approximating the renewal function with its linear asymptote: How large is large enough?
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.
scientific article; zbMATH DE number 3938201
scientific article