This paper proposes a unified semiparametric method for the additive risk model under general biased sampling. By using the estimating equation approach, we propose both estimators of the regression parameters and nonparametric function. An advantage is that our approach is still suitable for the lengthbiased data even without the information of the truncation variable. Meanwhile, large sample properties of the proposed estimators are established, including consistency and asymptotic normality. In addition, the finite sample behavior of the proposed methods and the analysis of three groups of real data are given.
In this paper, we consider a two-dimensional perturbed risk model with stochastic premiums and certain dependence between the two marginal surplus processes. We obtain the Lundberg-type upper bound for the infinite-time ruin probability by martingale approach, discuss how the dependence affects the obtained upper bound and give some numerical examples to illustrate our results. For the heavy-tailed claims case, we derive an explicit asymptotic estimation for the finite-time ruin probability.
