Concave regression: value-constrained estimation and likelihood ratio-based inference

Abstract

We propose a likelihood ratio statistic for forming hypothesis tests and confidence intervals for a nonparametrically estimated univariate regression function, based on the shape restriction of concavity (alternatively, convexity). Dealing with the likelihood ratio statistic requires studying an estimator satisfying a null hypothesis, that is, studying a concave least-squares estimator satisfying a further equality constraint. We study this null hypothesis least-squares estimator (NLSE) here, and use it to study our likelihood ratio statistic. The NLSE is the solution to a convex program, and we find a set of inequality and equality constraints that characterize the solution. We also study a corresponding limiting version of the convex program based on observing a Brownian motion with drift. The solution to the limit problem is a stochastic process. We study the optimality conditions for the solution to the limit problem and find that they match those we derived for the solution to the finite sample problem. This allows us to show the limit stochastic process yields the limit distribution of the (finite sample) NLSE. We conjecture that the likelihood ratio statistic is asymptotically pivotal, meaning that it has a limit distribution with no nuisance parameters to be estimated, which makes it a very effective tool for this difficult inference problem. We provide a partial proof of this conjecture, and we also provide simulation evidence strongly supporting this conjecture.

A Appendix: Technical formulas and other results

Here is a statement of an integration by parts formulas for functions of bounded variation. See, e.g., page 102 of [19] for the definition of bounded variation.

Lemma 5

([19]) Assume that F and G are of bounded variation on a set [a, b] where $-\infty< a< b < \infty$. If at least one of F and G is continuous, then

$$\begin{aligned} \int _{(a,b]} FdG + \int _{(a,b]} GdF = F(b)G(b) - F(a)G(a). \end{aligned}$$

Theorem 6

([23, Theorem 2.1]) Let $\sigma , a > 0$. Let $X(t) = \sigma W(t) - 4a t^3$ where W(t) is standard two-sided Brownian motion starting from 0, and let Y be the integral of X satisfying $Y(0)=0$. Thus $Y_{a,\sigma }(t) = \sigma \int _0^t W(s) ds - a t^4$ for $t \in \mathbb {R}$. Then, with probability 1, there exists a uniquely defined random continuous function $H_{a,\sigma }$ satisfying the following:

• 1. The function $H_{a,\sigma }$ satisfies $H_{a,\sigma }(t) \le Y(t)$ for all $t\in \mathbb {R}$.

• 2. The function $H_{a,\sigma }$ has a concave second derivative, $\widehat{r}_{a,\sigma } := H_{a,\sigma }''$.

• 3. The function $H_{a,\sigma }$ satisfies $\int _{\mathbb {R}} ( H_{a,\sigma }(t)-Y_{a,\sigma }(t) ) dH_{a,\sigma }^{(3)}(t)=0$.

Theorem 7

([24, Theorem 6.3]) Suppose that the regression model (9) holds, that $\epsilon _{n,1},\ldots ,\epsilon _{n,n}$ are i.i.d. with $E^{\epsilon _{n,1}^2 t} < \infty$ for some $t > 0$, that $r_0 \in \mathcal {C}$, that $r_0''(x_0)<0$, and that $r_0''$ is continuous in a neighborhood of $x_0$. Let Assumptions 1 and 2 hold. Let $a := |r_0''(x_0)|/ 24$ and $\sigma ^2 := {{\mathrm{Var}}}( \epsilon _{n,i})$. Then

$$\begin{aligned} n^{2/5} \left( \widehat{r}_n\left( x_0 + tn^{-1/5}\right) - r_0(x_0) - r_0'(x_0) t n^{-1/5}\right) \rightarrow _d \widehat{r}_{a,\sigma }(t) \end{aligned}$$

in $L^p[-K,K]$ for all $K > 0$.