Approximation Laws Of M-estimator On RSS

The idea of ranked set sampling(RSS) was first proposed by McIntyre in his effort to find a more efficient method to estimate the yield of pastures.The basic premise for RSS is an infinite population under study and the assumption that a set of sampling units drawn from the population can be ranked by certain means rather cheaply while the actual measurement of the variable of interest which is costly and/or time-consuming. The RSS procedure can be describles as that a sample of size k is drawn and ranked , but only the sample of interested order is measured.It is proved that when ranking is not random , the variance on RSS of is always smaller than under that under simple random sampling. There have been many research work about theoretical foundation and applications of RSS in recent years. But the research about M-estimator on RSS is still weak.The research about M-estimator on unbanlanced RSS is only about quantiles, and the perfect ranking is required. We provide M-estimator of unbanlanced RSS and only the consistency of ranking is needed. And we contruct an variable of indenpendent sums and prove the distance between the variable and M-estimator is converged to 0 in prob-ablity, hence the asymptotic normality of the M-estimator is established.Furthermore we propose the optimal design of it.And simulation results testify our conclusion.The asymptotic distribution of the M-estimator is usually related to the estimate of the nuisance parameters, which are not easy to be estimated accurately. We use the random weighting methods to solve the difficulty. It is different from others that the same weight is given to a group of samples. We propose random weighting estimator to approximate the distribution of M-estimator under the condition that the samples are given . We investigate the results by Monte Carlo simulations on balanced and unbalanced conditions. The proposed methods are shown to work well by the simulation studies.We provide two random weighting estimators of mean on SRS without given the distribution of weight and use the Edgeworth expansion to get convergence. And the convergency of estimator of mean on RSS is corollary of it. And those of M-estimator are also proposed.