# Limit distribution theory for multiple isotonic regression

We study limit distributions for the tuning-free max-min block estimator originally proposed in [FLN17] in the problem of multiple isotonic regression, under both fixed lattice design and random design settings. We show that, if the regression function f_0 admits vanishing derivatives up to order α_k along the k-th dimension (k=1,...,d) at a fixed point x_0 ∈ (0,1)^d, and the errors have variance σ^2, then the max-min block estimator f̂_n satisfies (n_∗/σ^2)^1/2+∑_k ∈D_∗α_k^-1(f̂_n(x_0)-f_0(x_0))C(f_0,x_0). Here D_∗, n_∗, depending on {α_k} and the design points, are the set of all `effective dimensions' and the size of `effective samples' that drive the asymptotic limiting distribution, respectively. If furthermore either {α_k} are relative primes to each other or all mixed derivatives of f_0 of certain critical order vanish at x_0, then the limiting distribution can be represented as C(f_0,x_0) =_d K(f_0,x_0) ·D_α, where K(f_0,x_0) is a constant depending on the local structure of the regression function f_0 at x_0, and D_α is a non-standard limiting distribution generalizing the well-known Chernoff distribution in univariate problems. The above limit theorem is also shown to be optimal both in terms of the local rate of convergence and the dependence on the unknown regression function whenever explicit (i.e. K(f_0,x_0)), for the full range of {α_k} in a local asymptotic minimax sense.

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