## The Annals of Statistics

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

### Consistency in Concave Regression

D. L. Hanson and Gordon Pledger

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 90C20: Quadratic programming

**Keywords**

Concave convex nonparametric regression concave regression convex regression consistency regression

#### Citation

Hanson, D. L.; Pledger, Gordon. Consistency in Concave Regression. Ann. Statist. 4 (1976), no. 6, 1038--1050. doi:10.1214/aos/1176343640. https://projecteuclid.org/euclid.aos/1176343640