Statistical inference for induced L-statistics: a random perturbation approach

Suppose that X and Y are two numerical characteristics defined for each individual in a population. In a random sample of (X, Y) with sample size n, denote the rth ordered X variate by Xr:n and the associated Y variate, the induced rth order statistics, by Y[r:n], respectively. Induced order statistics arise naturally in the context of selection where individuals ought to be selected by their ranks in a related X value due to difficulty or high costs of obtaining Y at the time of selection. The induced L-statistics, which take the form of , are very useful in regression analysis, especially when the observations are subject to a type-II censoring scheme with respect to the dependent variable, or when the regression function at a given quantile of the predictor variable is of interest. The limiting variance of the induced L-statistics involve the underlying regression function and inferences based on nonparametric estimation are often unstable. In this paper, we consider the distributional approximation of the induced L-statistics by the random perturbation method. Large sample properties of the randomly perturbed induced L-statistics are established. Numerical studies are also conducted to illustrate the method and to assess its finite-sample performance.