Abstract
We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator’s asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function.
Highlights
In the case of estimating a bimonotone pmf, i.e. a pmf which is monotone with respect to the usual matrix pre-order on Z2+, we state the limit distribution of the order restricted maximum likelihood estimator, thereby generalising previously obtained results by [16], who treated the one-dimensional case, i.e. the mle of a monotone pmf on Z+
In the case of estimating a bimonotone regression function, i.e. a function defined on Z2+ that is monotone with respect to the matrix pre-order on Z2+, we state the limit distribution of the isotonic regression estimator, again generalising previously known results for the isotonic regression on Z+, cf. [16]
The problem of estimation of a bimonotone regression function via least squares was studied in detail in [7], where the authors described an algorithm for minimization of a smooth function under bimonotone order constraints
Summary
In the paper [16] the authors studied the finite sample behaviour of the isotonized estimator of one-dimensional empirical pmf. The asymptotic behaviour of the regression estimates over a continuous setup under monotonic restriction was first studied in [9, 24], where it was shown that the difference of the regression function and its estimate multiplied by n1/3, at a point with a positive slope, has a nondegenerate limiting distribution. Theorem 3 describes the asymptotic behaviour of the isotonized estimator for the infinite dimensional case.
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