A Kernel-function-based Approach to Sequential Estimation with $\beta$-protection of Quantiles

Given a sequence ｛ $X_{n}$｝ of independent and identically distributed random variables with F, a sequential procedure for the p-th quantile ξ$_{P}$= $F^{-1}$ (P), 0$\beta$-protection. Some asymptotic properties for the proposed procedure and of an involved stopping time are proved: asymptotic consistency, asymptotic efficiency and asymptotic normality. From one of the results an effect of smoothing based on kernel functions is discussed. The results are also extended to the contaminated case.e.e.