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

Research output: Contribution to journalArticle

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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.

Original languageEnglish (US)
Pages (from-to)5-39
Number of pages35
JournalMathematical Programming
Volume174
Issue number1-2
DOIs
StatePublished - Mar 1 2019

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Constrained Estimation
Likelihood Ratio Statistic
Least Squares Estimator
Likelihood Ratio
Null hypothesis
Regression
Convex Program
Statistics
Limit Distribution
Equality Constraints
Stochastic Processes
Random processes
Brownian Motion with Drift
Nuisance Parameter
Concavity
Hypothesis Test
Regression Function
Optimality Conditions
Inequality Constraints
Brownian movement

Cite this

Concave regression : value-constrained estimation and likelihood ratio-based inference. / Doss, Charles R.

In: Mathematical Programming, Vol. 174, No. 1-2, 01.03.2019, p. 5-39.

Research output: Contribution to journalArticle

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