Consistency in Concave Regression

For each t in some subinterval T of the real line let Fₜ be a distribution function with mean $m(t)$. Suppose $m(t)$ is concave. Let t₁, t₂, ⋯ be a sequence of points in T and let Y₁, Y₂, ⋯ be an independent sequence of random variables such that the distribution function of Yₖ is Ftₖ. We consider estimators mₙ(t) = mₙ(t; Y₁, ⋯, Yₙ) which are concave in t and which minimize ∑ⁿᵢ₌₁ mₙ(tᵢ; Y₁, ⋯, Yₙ) - Yᵢ² over the class of concave functions. We investigate their consistency and the convergence of ₙ'(t)\ to $m'(t)$.