← Copula mathematical handbook

noncomputable def ProbabilityTheory.Copula.RankRegion.XiBlest.Support.checkerboardEstimator {n : ℕ} (A : CellMass (IntervalPartition.uniform (n + 1) ⋯) (IntervalPartition.uniform (n + 1) ⋯)) : ℝ

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theorem ProbabilityTheory.Copula.RankRegion.XiBlest.Support.checkerboardEstimator_eq_average {n : ℕ} (A : CellMass (IntervalPartition.uniform (n + 1) ⋯) (IntervalPartition.uniform (n + 1) ⋯)) : checkerboardEstimator A = (A.checkerboard.chatterjeeXi + A.checkMin.chatterjeeXi) / 2

theorem ProbabilityTheory.Copula.RankRegion.XiBlest.Support.checkerboardEstimator_mem {n : ℕ} (A : CellMass (IntervalPartition.uniform (n + 1) ⋯) (IntervalPartition.uniform (n + 1) ⋯)) : checkerboardEstimator A ∈ Set.Icc 0 1

theorem ProbabilityTheory.Copula.RankRegion.XiBlest.Support.checkerboardEstimator_correction {n : ℕ} (A : CellMass (IntervalPartition.uniform (n + 1) ⋯) (IntervalPartition.uniform (n + 1) ⋯)) : checkerboardEstimator A - A.checkerboard.chatterjeeXi = ((Matrix.of A.mass).transpose * Matrix.of A.mass).trace / 2

theorem ProbabilityTheory.Copula.RankRegion.XiBlest.Support.checkerboardEstimator_correction_bounds {n : ℕ} (A : CellMass (IntervalPartition.uniform (n + 1) ⋯) (IntervalPartition.uniform (n + 1) ⋯)) : 0 ≤ checkerboardEstimator A - A.checkerboard.chatterjeeXi ∧ checkerboardEstimator A - A.checkerboard.chatterjeeXi ≤ 1 / (↑n + 1) / 2

theorem ProbabilityTheory.Copula.RankRegion.XiBlest.Support.checkerboardEstimator_correction_tendsto (N : ℕ → ℕ) (hN : Filter.Tendsto N Filter.atTop Filter.atTop) (A : (k : ℕ) → CellMass (IntervalPartition.uniform (N k + 1) ⋯) (IntervalPartition.uniform (N k + 1) ⋯)) : Filter.Tendsto (fun (k : ℕ) => checkerboardEstimator (A k) - (A k).checkerboard.chatterjeeXi) Filter.atTop (nhds 0)

The finite-sample correction vanishes for every refining sequence of square matrices.

theorem ProbabilityTheory.Copula.RankRegion.XiBlest.Support.checkerboardEstimator_tendsto_iff (N : ℕ → ℕ) (hN : Filter.Tendsto N Filter.atTop Filter.atTop) (A : (k : ℕ) → CellMass (IntervalPartition.uniform (N k + 1) ⋯) (IntervalPartition.uniform (N k + 1) ⋯)) (l : ℝ) : Filter.Tendsto (fun (k : ℕ) => checkerboardEstimator (A k)) Filter.atTop (nhds l) ↔ Filter.Tendsto (fun (k : ℕ) => (A k).checkerboard.chatterjeeXi) Filter.atTop (nhds l)

Conditional transfer lemma; the statistical consistency hypothesis remains explicit.