Least Squares Estimation of a Monotone Quasiconvex Regression Function
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We develop a new approach for the estimation of a multivariate function based on the economic axioms of monotonicity and quasiconvexity. We prove the existence of the nonparametric least squares estimator (LSE) for a monotone and quasiconvex function and provide two characterizations for it. One of these characterizations is useful from the theoretical point of view, while the other helps in the computation of the estimator. We show that the LSE is almost surely unique and is the solution to a mixed-integer quadratic optimization problem. We prove consistency and find finite sample risk bounds for the LSE under both fixed lattice and random design settings for the covariates. We illustrate the superior performance of the LSE against existing estimators via simulation. Finally, we use the LSE to estimate the production function for the Japanese plywood industry and the cost function for hospitals across the US.
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