Abstract
For each $t$ in some subset $T$ of $N$-dimensional Euclidean space let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is non-decreasing in each of the coordinates of $t$. Let $t_1, t_2,\cdots$ be a sequence of points in $T$ and let $Y_1, Y_2,\cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. Estimators $\hat{m}_n(t; Y_1,\cdots, Y_n)$ of $m(t)$ which are monotone in each coordinate of $t$ and which minimize $\sum^n_{i=1} \lbrack\hat{m}_n(t_i; Y_1,\cdots, Y_n) - Y_i\rbrack^2$ are already known. Brunk has investigated their consistency when $N = 1$. In this paper additional consistency results are obtained when $N = 1$ and some results are obtained in the case $N = 2$. In addition, we prove several lemmas about the law of large numbers which we believe to be of independent interest.
Citation
D. L. Hanson. Gordon Pledger. F. T. Wright. "On Consistency in Monotonic Regression." Ann. Statist. 1 (3) 401 - 421, May, 1973. https://doi.org/10.1214/aos/1176342407
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