Asymptotic Distributions of Slope of Greatest Convex Minorant Estimators.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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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 in, X sub ni, where EX sub ni mu in. In the second example, in 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
- Statistics and Probability