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OP-1643OpenHigh-dimensional statistics

A Nonparametric Framework for Statistical Modelling · 1986-stat-5

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

MSC 62 · Statistics

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Problem statement

Let (X,Y)(X,Y) be a random pair such that XX takes values in a known compact set C⊂RdC\subset\mathbb R^d, the distribution of XX has a density bounded above and bounded away from zero on CC, and Y∈{0,1}Y\in\{0,1\}. Define

f(x)=Pr⁡(Y=1∣X=x),g(x)=log⁡f(x)1−f(x), f(x)=\Pr(Y=1\mid X=x),\qquad g(x)=\log\frac{f(x)}{1-f(x)},

and assume that ff is continuous with 0<f(x)<10<f(x)<1 on CC. For each nn, let Sn\mathcal S_n be a pnp_n-dimensional vector space of functions on CC, with basis Bn1,…,BnpnB_{n1},\ldots,B_{np_n}, and write Bn(x)=(Bn1(x),…,Bnpn(x))⊤B_n(x)=(B_{n1}(x),\ldots,B_{np_n}(x))^\top. For θ∈Rpn\theta\in\mathbb R^{p_n}, set

gn,θ(x)=Bn(x)⊤θ,fn,θ(x)=egn,θ(x)1+egn,θ(x). g_{n,\theta}(x)=B_n(x)^\top\theta, \qquad f_{n,\theta}(x)=\frac{e^{g_{n,\theta}(x)}}{1+e^{g_{n,\theta}(x)}}.

Let θn\theta_n be the unique maximizer of

Λn(θ)=E ⁣[YBn(X)⊤θ−log⁡(1+eBn(X)⊤θ)], \Lambda_n(\theta)=E\!\left[YB_n(X)^\top\theta-\log\bigl(1+e^{B_n(X)^\top\theta}\bigr)\right],

and put gn=gn,θng_n=g_{n,\theta_n} and fn=fn,θnf_n=f_{n,\theta_n}. Given independent observations (X1,Y1),…,(Xn,Yn)(X_1,Y_1),\ldots,(X_n,Y_n) distributed as (X,Y)(X,Y), let θ^n\widehat\theta_n maximize

ℓn(θ)=∑i=1n[YiBn(Xi)⊤θ−log⁡(1+eBn(Xi)⊤θ)], \ell_n(\theta)=\sum_{i=1}^n\left[Y_iB_n(X_i)^\top\theta-\log\bigl(1+e^{B_n(X_i)^\top\theta}\bigr)\right],

when such a maximizer exists, and set g^n=gn,θ^n\widehat g_n=g_{n,\widehat\theta_n} and f^n=fn,θ^n\widehat f_n=f_{n,\widehat\theta_n}. Also define

δn=inf⁡s∈Sn∥s−g∥∞,In=nE ⁣[fn(X)(1−fn(X))Bn(X)Bn(X)⊤], \delta_n=\inf_{s\in\mathcal S_n}\|s-g\|_\infty, \qquad \mathcal I_n=nE\!\left[f_n(X)(1-f_n(X))B_n(X)B_n(X)^\top\right], I^n=∑i=1nf^n(Xi)(1−f^n(Xi))Bn(Xi)Bn(Xi)⊤, \widehat{\mathcal I}_n=\sum_{i=1}^n\widehat f_n(X_i)(1-\widehat f_n(X_i))B_n(X_i)B_n(X_i)^\top,

and, whenever the matrices are invertible,

SE⁡n(x)=fn(x)(1−fn(x))(Bn(x)⊤In−1Bn(x))1/2, \operatorname{SE}_n(x)=f_n(x)(1-f_n(x))\bigl(B_n(x)^\top\mathcal I_n^{-1}B_n(x)\bigr)^{1/2}, SE⁡^n(x)=f^n(x)(1−f^n(x))(Bn(x)⊤I^n−1Bn(x))1/2. \widehat{\operatorname{SE}}_n(x)=\widehat f_n(x)(1-\widehat f_n(x))\bigl(B_n(x)^\top\widehat{\mathcal I}_n^{-1}B_n(x)\bigr)^{1/2}.

Determine appropriate regularity conditions on CC, the distribution of XX, the sequence (Sn)(\mathcal S_n), its bases, and the growth of pnp_n, and prove under those conditions that

∥fn−f∥∞=O(δn), \|f_n-f\|_\infty=O(\delta_n),

that f^n\widehat f_n exists with probability tending to one, and that

∥f^n−fn∥2=OPr⁡ ⁣(pnn),∥f^n−fn∥∞=OPr⁡ ⁣(pnlog⁡pnn). \|\widehat f_n-f_n\|_2=O_{\Pr}\!\left(\sqrt{\frac{p_n}{n}}\right), \qquad \|\widehat f_n-f_n\|_\infty=O_{\Pr}\!\left(\sqrt{\frac{p_n\log p_n}{n}}\right).

Further prove, uniformly for xx in the region on which the regularity conditions apply, that

SE⁡n(x)∼pnn,SE⁡^n(x)SE⁡n(x)=1+oPr⁡(1), \operatorname{SE}_n(x)\sim\sqrt{\frac{p_n}{n}}, \qquad \frac{\widehat{\operatorname{SE}}_n(x)}{\operatorname{SE}_n(x)}=1+o_{\Pr}(1),

and

f^n(x)−fn(x)SE⁡n(x)→LN(0,1). \frac{\widehat f_n(x)-f_n(x)}{\operatorname{SE}_n(x)}\xrightarrow{\mathcal L}N(0,1).

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Progress & source assessment

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Status: unresolved.

The full problem asks for a single set of “appropriate regularity conditions” that yields all four approximation/existence and stochastic-error conclusions, together with a uniformly valid plug-in standard error of order pn/n\sqrt{p_n/n} and a pointwise asymptotic normal limit for the sieve logistic-regression MLE. The literature gives substantial partial progress, but no verified source found through July 31, 2026 proves this complete package for the general compact-domain, increasing-dimensional basis setting stated here. In particular, results proving convergence rates or pointwise inference under narrower assumptions do not establish the requested uniform variance equivalence and normal approximation in the same generality.

Key references and progress:

  • Charles J. Stone, “A Nonparametric Framework for Statistical Modelling,” Proceedings of the International Congress of Mathematicians (Berkeley, 1986), 1986, pp. 188–192. This is the source of the problem: it formulates logistic regression through the sieve likelihood and explicitly says that analogs of its density-estimation Theorems 1 and 2 should hold, without specifying the needed logistic-regression regularity conditions or proving the analogs.

  • Charles J. Stone, Mark H. Hansen, Charles Kooperberg, and Young K. Truong, “Polynomial Splines and Their Tensor Products in Extended Linear Modeling,” Annals of Statistics 25 (1997), 1371–1470, DOI: 10.1214/aos/1031594728. This develops a broad asymptotic theory for polynomial-spline extended linear models, including generalized-regression likelihoods and logistic-type models. It supplies important sieve approximation, consistency, and convergence-rate results under detailed spline and design assumptions, but it does not verify the entire theorem package in the statement, especially the stated uniform standard-error equivalence and normal limit.

  • Jianhua Z. Huang, “Functional ANOVA Models for Generalized Regression,” Journal of Multivariate Analysis 67 (1998), 49–71, DOI: 10.1006/jmva.1998.1753. This proves consistency and a rate-of-convergence result for maximum-likelihood generalized regression over tensor-product approximation spaces; logistic regression is explicitly included as a special case. The results are partial relative to the statement: they do not establish all of the requested global sup-norm stochastic bounds and uniformly standardized asymptotic-normal conclusions for the unrestricted multivariate sieve problem.

  • Jianhua Z. Huang, “Asymptotics for Polynomial Spline Regression under Weak Conditions,” Statistics & Probability Letters 65 (2003), 207–216, direct URL: https://doi.org/10.1016/S0167-7152(03)00253-0. This gives rate results for spline regression under weaker conditions and supports the broader rate program, but it concerns a narrower regression formulation and does not resolve the exact logistic-MLE statement above.

  • Jianhua Z. Huang, “Local asymptotics for polynomial spline regression,” Annals of Statistics 31 (2003), 1600–1635, DOI: 10.1214/aos/1074290331. This establishes strong local asymptotic results for polynomial-spline regression, relevant to pointwise inference, but it is not a proof of the stated multivariate logistic-regression result with the requested uniform standard-error formula and all four bounds.

Thus, established progress covers important special cases and components—especially sieve approximation, MLE consistency, and convergence rates for generalized or spline regression—but a genuine gap remains for the exact combined assertion in the statement.

References
  1. [1]ICM Conjectures · 1986-stat-5 link
  2. [2]ICM 1986: A Nonparametric Framework for Statistical Modelling link

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