This paper introduces the Tukey trimmed and Winsorized means for the transformed data based on a scaled deviation. The trimmed and Winsorized means and scale based on a scaled deviation are as special cases. Meanwhile, the trimmed and Winsorized skewness and kurtosis based on a scaled deviation are given. Furthermore, some of their robust properties (influence function, breakdown points) and asymptotic properties (asymptotic representation and limiting distribution) are also obtained.
A recursive formula of the Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims is obtained. In the discount-free case, an explicit formula is given. Utilizing such an explicit expression, we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.
In this paper, we prove a central limit theorem for m-dependent random variables under sublinear expectations. This theorem can be regarded as a generalization of Peng's central limit theorem.