## The Annals of Statistics

### Consistency in Concave Regression

#### Abstract

For each $t$ in some subinterval $T$ of the real line let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is concave. 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 $F_{t_k}$. We consider estimators $m_n(t) = m_n(t; Y_1, \cdots, Y_n)$ which are concave in $t$ and which minimize $\sum^n_{i=1} \lbrack m_n(t_i; Y_1, \cdots, Y_n) - Y_i\rbrack^2$ over the class of concave functions. We investigate their consistency and the convergence of $\{m_n'(t)\}$ to $m'(t)$.

#### Article information

Source
Ann. Statist., Volume 4, Number 6 (1976), 1038-1050.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176343640

Digital Object Identifier
doi:10.1214/aos/1176343640

Mathematical Reviews number (MathSciNet)
MR426273

Zentralblatt MATH identifier
0341.62034

JSTOR