Most proofs for the consistency of parameters obtained from estimating equations depend on a compact parameter space.
However, I have almost never worked with parameter spaces that are compact (they are all unbounded), so these proofs are rather useless to me.
I know two proofs that don't require compactness, but they both have issues - one only work for scalar parameters (5.10 Lemma in Asymptotic Statistics), the other require uniform strong consistency of the estimating equation instead of uniform weak consistency (from "Asymptotics of Estimating Equations under Natural Conditions").
Anyone knows a different proof?
"Advanced econometrics" by Takeshi Amemiya has a proof in Chapter 4 for the convergence of a root of the Lagrangian. If the root is always unique, then consistency can be concluded.
"On the consistency and inconsistency of estimating equations" has a more general version of Lemma 5.10 from Asymptotic Statistics that works for vector parameters.
"A review of asymptotic theory of estimating functions" has proofs based on the fixed point theorem for contractions. I believe these types of proofs are the most productive when showing that a root will exist in a neighborhood of the true parameter with probability tending to one.