Mixed Poisson distributions tail equivalent to their mixing distributions

Abstract

A mixed Poisson distribution can have an upper tail asymptotically equal to the upper tail of its mixing distribution. Two broad classes of mixing distributions that generate mixed Poisson distributions with this property are identified: unbounded, non-negative distributions with mild regularity conditions that satisfy either (a) the von Mises condition for the Fréchet extreme value domain of attraction; or (b) the von Mises condition for the Gumbel extreme value domain of attraction and have hazard rates f( t)/(1 − F( t)) of order o( t − b ) for some δ ⩾ 1 2 as t → ∞. The Pareto distribution is prototypical of class (a) and the lognormal of class (b).