I’m studying a recursively defined estimator and want to prove consistency. The estimator is not merely computed by a recursive algorithm—the value on a sample is defined (also) through values on sub-samples.
Let's denote the estimator by $T_n$ and the parameter of interest by $\gamma$. So far, I've tried:
- Induction on $n$ for $\mathbb{P}(|T_n - \gamma| > \varepsilon)$, but it doesn't close without additional structure
- Covering $\{ |T_n - \gamma| > \varepsilon \} \subseteq E_n(\varepsilon)$ and bounding $\mathbb{P}(E_n(\varepsilon))$, but a tight enough cover is hard to quantify
- Solving a recursion relation of tail bounds, which only yields decay for $\varepsilon$ bounded away from $0$
I've also tried to use asymptotic properties of some of the quantities involved in the definition of $T_n$, but it seems especially difficult to use those with recursion.
Are there any other common proof techniques? Pointers to standard references are very welcome.
Edit: I can't share specifics, but think of $n$ i.i.d observations, some "base" estimator $T'_{i:j}$ computed on observations $i$ through $j$ and an aggregation function $A$, then setting $T_{i:j} := A(\{ T'_{i:j} \} \cup \bigcup_{i \leq k < j} \{ T_{i:k}, T_{k+1:j}\})$ and $T_n := T_{1:n}$.
