On Consistency in Monotonic Regression

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.