OpenFrontiers
Join
← Back to the index
OP-1642OpenHigh-dimensional statistics

A Nonparametric Framework for Statistical Modelling · 1986-stat-4

Nonparametric Inference · ICM 1986: A Nonparametric Framework for Statistical Modelling

MSC 62 · Statistics

Inferred from the community field; finer classification has not been reviewed.

Source assessment · not reviewedAs of 2026-08-18Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let C⊂RdC\subset\mathbb{R}^d be compact, and let ff be a continuous, strictly positive probability density on CC. Set g=log⁡fg=\log f. For each nn, let Sn\mathcal S_n be a pnp_n-dimensional linear space of functions on CC with basis Bn1,…,BnpnB_{n1},\ldots,B_{np_n}, where ∑j=1pnBnj=1\sum_{j=1}^{p_n}B_{nj}=1 on CC and no nonzero linear combination of the basis functions vanishes almost everywhere. Assume that pn→∞p_n\to\infty, pn=o(n1/2−ε)p_n=o(n^{1/2-\varepsilon}) for some ε>0\varepsilon>0, every continuous function on CC can be approximated uniformly by elements of Sn\mathcal S_n, and there is a constant M>0M>0 such that, for every nn:

  • the L2(C)L_2(C)-orthogonal projection Πn\Pi_n onto Sn\mathcal S_n satisfies ∥Πnh∥∞≤M∥h∥∞\|\Pi_nh\|_\infty\le M\|h\|_\infty for every h∈L∞(C)h\in L_\infty(C);
  • ∣Bnj∣≤M|B_{nj}|\le M on CC for 1≤j≤pn1\le j\le p_n;
  • each BnjB_{nj} is supported in a set Cnj⊂CC_{nj}\subset C of diameter at most Mpn−1/dMp_n^{-1/d}, and each CnjC_{nj} intersects at most MM of the sets CnkC_{nk};
  • for every θ=(θ1,…,θpn)∈Rpn\theta=(\theta_1,\ldots,\theta_{p_n})\in\mathbb R^{p_n} and every jj,
M−1∣θj∣2≤∥∑k=1pnθkBnk∥∞,Cnj2≤Mpn∫Cnj(∑k=1pnθkBnk(x))2 dx. M^{-1}|\theta_j|^2\le \left\|\sum_{k=1}^{p_n}\theta_kB_{nk}\right\|_{\infty,C_{nj}}^2 \le Mp_n\int_{C_{nj}}\left(\sum_{k=1}^{p_n}\theta_kB_{nk}(x)\right)^2\,dx.

For θ∈Rpn\theta\in\mathbb R^{p_n} define

cn(θ)=log⁡∫Cexp⁡ ⁣(∑j=1pnθjBnj(x))dx,fn(x;θ)=exp⁡ ⁣(∑j=1pnθjBnj(x)−cn(θ)).c_n(\theta)=\log\int_C\exp\!\left(\sum_{j=1}^{p_n}\theta_jB_{nj}(x)\right)dx, \qquad f_n(x;\theta)=\exp\!\left(\sum_{j=1}^{p_n}\theta_jB_{nj}(x)-c_n(\theta)\right).

Let Θn0={θ∈Rpn:∑jθj=0}\Theta_{n0}=\{\theta\in\mathbb R^{p_n}:\sum_j\theta_j=0\}, let θn\theta_n be the unique maximizer over Θn0\Theta_{n0} of

Λn(θ)=∑j=1pnθj∫CBnj(x)f(x) dx−cn(θ),\Lambda_n(\theta)=\sum_{j=1}^{p_n}\theta_j\int_CB_{nj}(x)f(x)\,dx-c_n(\theta),

and put fn=fn( ⋅ ;θn)f_n=f_n(\,\cdot\,;\theta_n). Given independent observations Y1,…,YnY_1,\ldots,Y_n with density ff, let θ^n\widehat\theta_n maximize

ℓn(θ)=∑i=1n∑j=1pnθjBnj(Yi)−ncn(θ)\ell_n(\theta)=\sum_{i=1}^n\sum_{j=1}^{p_n}\theta_jB_{nj}(Y_i)-nc_n(\theta)

over Θn0\Theta_{n0}, and put f^n=fn( ⋅ ;θ^n)\widehat f_n=f_n(\,\cdot\,;\widehat\theta_n). Define probability measures

P(A)=∫Af(x) dx,Pn(A)=∫Afn(x) dx,P^n(A)=∫Af^n(x) dx,P(A)=\int_Af(x)\,dx, \qquad P_n(A)=\int_Af_n(x)\,dx, \qquad \widehat P_n(A)=\int_A\widehat f_n(x)\,dx,

and set

δn=inf⁡s∈Sn∥s−g∥∞.\delta_n=\inf_{s\in\mathcal S_n}\|s-g\|_\infty.

For a class A\mathcal A of measurable subsets of CC, write

∥Q1−Q2∥A=sup⁡A∈A∣Q1(A)−Q2(A)∣.\|Q_1-Q_2\|_{\mathcal A}=\sup_{A\in\mathcal A}|Q_1(A)-Q_2(A)|.

Determine explicit additional regularity conditions on A\mathcal A, (Sn)(\mathcal S_n), and ff under which one can prove that

∥Pn−P∥A=O ⁣(pn−1/dδn)\|P_n-P\|_{\mathcal A}=O\!\left(p_n^{-1/d}\delta_n\right)

and, for every fixed A∈AA\in\mathcal A with 0<P(A)<10<P(A)<1,

L ⁣(P^n(A)−Pn(A)P(A)(1−P(A))/n)⟶N(0,1).\mathcal L\!\left( \frac{\widehat P_n(A)-P_n(A)}{\sqrt{P(A)(1-P(A))/n}} \right)\longrightarrow N(0,1).

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗
Remaining questions in this record

Remaining questions recorded by the source

These are retained within the parent record and are not counted as additional index entries.

Question 1

Let I⊂RI\subset\mathbb R be a compact interval, let Y1,…,YnY_1,\ldots,Y_n be independent observations from a density ff that is continuous and strictly positive on II, and let Sn\mathcal S_n be a univariate log-spline space of dimension Jn→∞J_n\to\infty generated by a quasi-uniform B-spline basis. For θ∈RJn\theta\in\mathbb R^{J_n}, define

fn(x;θ)=exp⁡ ⁣(∑jθjBnj(x)−log⁡∫Iexp⁡ ⁣(∑jθjBnj(u))du). f_n(x;\theta)=\exp\!\left(\sum_j\theta_jB_{nj}(x)-\log\int_I\exp\!\left(\sum_j\theta_jB_{nj}(u)\right)du\right).

Let θn\theta_n maximize the expected log likelihood over the identifiable parameter subspace, let fn=fn(⋅;θn)f_n=f_n(\cdot;\theta_n), and let f^n=fn(⋅;θ^n)\widehat f_n=f_n(\cdot;\widehat\theta_n) be the maximum-likelihood log-spline estimate. Write FF, FnF_n, and F^n\widehat F_n for the distribution functions of ff, fnf_n, and f^n\widehat f_n, respectively, and let F^nemp(x)=n−1∑i=1n1{Yi≤x}\widehat F_n^{\mathrm{emp}}(x)=n^{-1}\sum_{i=1}^n\mathbf 1\{Y_i\le x\}. If

δn:=inf⁡s∈Sn∥s−log⁡f∥∞satisfiesδn/Jn=o(n−1/2), \delta_n:=\inf_{s\in\mathcal S_n}\|s-\log f\|_\infty \quad\text{satisfies}\quad \delta_n/J_n=o(n^{-1/2}),

prove or disprove that

∥F^n−F^nemp∥∞=Opr(n−1/2). \|\widehat F_n-\widehat F_n^{\mathrm{emp}}\|_\infty=O_{\mathrm{pr}}(n^{-1/2}).

Source reference: Charles J. Stone, “Large-Sample Inference for Log-Spline Models,” The Annals of Statistics, vol. 18, no. 2, 1990, conjecture stated in the informal summary preceding the quantile-function discussion, DOI: 10.1214/aos/1176347622.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

unresolved

The exact problem is not completely resolved in the literature. Stone’s 1986 proceedings paper introduces the loglinear density model, defines the approximation error δn\delta_n, and asks for a corresponding result beyond the special one-dimensional case; it does not specify sufficient general conditions on an arbitrary class A\mathcal A, the spaces Sn\mathcal S_n, and the density ff that establish both conclusions in the statement.

The main directly relevant partial result is:

Charles J. Stone, “Large-Sample Inference for Log-Spline Models,” The Annals of Statistics 18 (1990), no. 2, DOI: 10.1214/aos/1176347622. For one-dimensional compact intervals with a univariate log-spline model, Stone proves the distribution-function approximation bounds ∥F∗−F∥∞=O(δ/J)\|F^*-F\|_\infty=O(\delta/J) and ∥F^−F∗∥∞=Opr(n−1/2)\|\widehat F-F^*\|_\infty=O_{\mathrm{pr}}(n^{-1/2}), and proves pointwise asymptotic normality for F^(y)−F∗(y)\widehat F(y)-F^*(y) with its asymptotic standard deviation, uniformly over compact subsets of the interior. Here JJ is the spline dimension, FF is the true distribution function, F∗F^* is the expected-log-likelihood projection, and F^\widehat F is the fitted logspline distribution function. These results cover a special one-dimensional spline/distribution-function setting, not the original arbitrary-dimensional class-of-sets problem.

The same paper explicitly conjectures, in that one-dimensional setting, that ∥F^−F^emp∥∞=Opr(n−1/2)\|\widehat F-\widehat F^{\mathrm{emp}}\|_\infty=O_{\mathrm{pr}}(n^{-1/2}) if δ/J=o(n−1/2)\delta/J=o(n^{-1/2}), where F^emp\widehat F^{\mathrm{emp}} is the empirical distribution function. The conjecture is a narrower open component related to the original question; the literature search found no citable proof or counterexample resolving it through July 31, 2026.

The original source is Charles J. Stone, “A Nonparametric Framework for Statistical Modelling,” in Proceedings of the International Congress of Mathematicians, Berkeley, California, 1986, American Mathematical Society, 1987, pp. 188–202. It states the target bounds under unspecified “reasonable conditions” and notes only the special case proved in Stone’s 1986 Berkeley technical report, “Asymptotic Properties of Logspline Density Estimation,” Technical Report No. 69, Department of Statistics, University of California, Berkeley.

Thus the literature establishes important special-case approximation and pointwise-inference results, but it does not provide a complete answer covering the exact general statement’s arbitrary dd, arbitrary admissible A\mathcal A, and general localized spaces Sn\mathcal S_n.

References
  1. [1]ICM Conjectures · 1986-stat-4 link
  2. [2]ICM 1986: A Nonparametric Framework for Statistical Modelling link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.