相依竞争风险场合密度函数的核估计

Strong Uniform Consistency and Kernel Estimates of Density Functions under the Competing Risks Case

1非随机窗宽情形下的核估计设T_k表示某系统中第k个部件的寿命,整个系统的寿命为T=min{T_1,…,T_r},用R={1,2,…,r}表示风险集合。用下述ζ(T)表示系统的失效模式ζ(T)=,若T=T_j,j∈I,且T≠T_j,jI ,其它这里T=(T_1,T_2,…,T_r),I为R的一个子集。又设φ为R的非空子集的集合,φ_1={J∈φ,J∩I≠={J∈φ,J∩I=},则竞争风险问题即为估计2′-1个生存函数:S_1(l)

The kernel estimates of density functions is discussed under the competing risks case where T_1, T_2 …, T, may not be independent random variables. Asymptotic properties of two estimators, arising naturally as a result of considering two types of band widths, are investigated. In particular we show that the strong uniform consistency of the kernel estimates.