Asymptotic Distributions of Slope-of-Greatest-Convex-Minorant Estimators

Abstract

Abstract : Isotonic estimation involves the estimator of a function which is known to be increasing with respect to a specified partial order. For the case of a linear order, a general theorem is given which simplifies and extends the techniques of Prakasa Rao (1966) and Brunk (1970). Sufficient conditions for a specified limit distribution to obtain are expressed in terms of a local condition and a global condition. The theorem is applied to several examples. The first example is estimation of a monotone function mu on (0,1) based on observations (i/n, X sub ni), where EX sub ni = mu (i/n). In the second example, i/n is replaced by random T sub ni. Robust estimators for this problem are described. Estimation of a monotone density function is also discussed. It is shown that the rate of convergence depends on the order of first non-zero derivative and that this result can obtain even if the function is not monotone over its entire domain. (Author)