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Asymptotics and Well-Posedness of the Derived Distribution Density in a Study of Biovariability

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Abstract: In our recent work (Wang, Burgei, and Zhou, 2018) we studied the hearingloss injury among subjects in a crowd with a wide spectrum of heterogeneousindividual injury susceptibility due to biovariability. The injury risk of acrowd is defined as the average fraction of injured. We examined mathematicallythe injury risk of a crowd vs the number of acoustic impulses the crowdis exposed to, under the assumption that all impulses act independently incausing injury regardless of whether one is preceded by another. We concludedthat the observed dose-response relation can be explained solely onthe basis of biovariability in the form of heterogeneous susceptibility. We derivedan analytical solution for the distribution density of injury susceptibility,as a power series expansion in terms of scaled log individual non-injuryprobability. While theoretically the power series converges for all argumentvalues, in practical computations with IEEE double precision, at large argumentvalues, the numerical accuracy of the power series summation is completelywiped out by the accumulation of round-off errors. In this study, wederive a general asymptotic approximation at large argument values, for thedistribution density. The combination of the power series and the asymptoticsprovides a practical numerical tool for computing the distribution density.We then use this tool to verify numerically that the distribution obtainedin our previous theoretical study is indeed a proper density. In addition, wewill also develop a very efficient and accurate Pade approximation for thedistribution density.

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