  To accommodate numerous practical scenarios, in this paper we extend
statistical inference for smoothed quantile estimators from finite domains to
infinite domains. We accomplish the task with the help of a newly designed
truncation methodology for discrete loss distributions with infinite domains. A
simulation study illustrates the methodology in the case of several
distributions, such as Poisson, negative binomial, and their zero inflated
versions, which are commonly used in insurance industry to model claim
frequencies. Additionally, we propose a very flexible bootstrap-based approach
for the use in practice. Using automobile accident data and their
modifications, we compute what we have termed the conditional five number
summary (C5NS) for the tail risk and construct confidence intervals for each of
the five quantiles making up C5NS, and then calculate the tail probabilities.
The results show that the smoothed quantile approach classifies the tail
riskiness of portfolios not only more accurately but also produces lower
coefficients of variation in the estimation of tail probabilities than those
obtained using the linear interpolation approach.
