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.

翻译：为适应众多实际场景，本文将对平滑分位数估计量的统计推断从有限域扩展到无限域。我们借助一种新设计的截断方法，针对具有无限域的离散损失分布完成这一任务。模拟研究展示了该方法在若干分布下的应用，如泊松分布、负二项分布及其零膨胀版本——这些分布常用于保险业对索赔频率进行建模。此外，我们提出了一种高度灵活的基于自助法的实用方案。利用汽车事故数据及其修正版本，我们计算了所定义的尾部风险条件五数概括（C5NS），构建了构成C5NS的五个分位数各自的置信区间，并进而计算了尾部概率。结果表明，与线性插值方法相比，平滑分位数方法不仅更准确地分类了投资组合的尾部风险，而且在尾部概率估计中产生了更低的变异系数。