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RATIO MONOTONICITY FOR TAIL PROBABILITIES IN THE RENEWAL RISK MODEL

Published online by Cambridge University Press:  31 March 2011

Georgios Psarrakos and
Michael Tsatsomeros
Georgios Psarrakos
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece E-mail: gpsarr@unipi.gr
Michael Tsatsomeros
Affiliation:
Department of Mathematics, Washington State University, Pullman, WA 99164-3113, E-mail: tsat@wsu.edu
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Abstract

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A renewal model in risk theory is considered, where $\overline{H}(u,y)$ is the tail of the distribution of the deficit at ruin with initial surplus u and $\overline{F}(y)$ is the tail of the ladder height distribution. Conditions are derived under which the ratio $\overline{H}(u,y)/\overline{F}(u+y)$ is nondecreasing in u for any y≥0. In particular, it is proven that if the ladder height distribution is stable and DFR or phase type, then the above ratio is nondecreasing in u. As a byproduct of this monotonicity, an upper bound and an asymptotic result for $\overline{H}(u,y)$ are derived. Examples are given to illustrate the monotonicity results.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 25 , Issue 2 , April 2011, pp. 171 - 185
Copyright
Copyright © Cambridge University Press 2011

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