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A Nonparametric Framework for Statistical Modelling · 1986-stat-7

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

MSC 62 · Statistics

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Problem statement

Let (Xi,Yi)i=1n(X_i,Y_i)_{i=1}^n be independent copies of (X,Y)(X,Y), where XX takes values in C=[0,1]dC=[0,1]^d, has a density bounded away from zero and infinity 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, 0<f<10<f<1 on CC, and gg is additive: g(x)=∑k=1dgk(xk)g(x)=\sum_{k=1}^d g_k(x_k).

For each nn and 1≤k≤d1\le k\le d, let Snk\mathcal S_{nk} be a pnkp_{nk}-dimensional vector space of functions on [0,1][0,1] containing the constants, and set

Sn={s(x)=∑k=1dsk(xk):sk∈Snk},pn=1+∑k=1d(pnk−1). \mathcal S_n=\left\{s(x)=\sum_{k=1}^d s_k(x_k):s_k\in\mathcal S_{nk}\right\}, \qquad p_n=1+\sum_{k=1}^d(p_{nk}-1).

Assume that pn→∞p_n\to\infty, that pn=o(n1/2−ε)p_n=o(n^{1/2-\varepsilon}) for some ε>0\varepsilon>0, and that every continuous additive function on CC can be approximated uniformly by elements of Sn\mathcal S_n. For each n,kn,k, let bnkjb_{nkj}, 1≤j≤pnk1\le j\le p_{nk}, be a basis of Snk\mathcal S_{nk}. Assume that there is a constant MM independent of n,k,jn,k,j and intervals Inkj⊂[0,1]I_{nkj}\subset[0,1] such that ∣bnkj∣≤M|b_{nkj}|\le M, bnkj=0b_{nkj}=0 outside InkjI_{nkj}, ∣Inkj∣≤Mpnk−1|I_{nkj}|\le M p_{nk}^{-1}, each InkjI_{nkj} intersects at most MM other such intervals, and

M−1∣aj∣2≤∥∑ℓ=1pnkaℓbnkℓ∥∞,Inkj2≤Mpnk∫Inkj(∑ℓ=1pnkaℓbnkℓ(u))2du M^{-1}|a_j|^2\le \left\|\sum_{\ell=1}^{p_{nk}}a_\ell b_{nk\ell}\right\|_{\infty,I_{nkj}}^2 \le M p_{nk}\int_{I_{nkj}}\left(\sum_{\ell=1}^{p_{nk}}a_\ell b_{nk\ell}(u)\right)^2du

for every coefficient vector aa. Assume also that the L2[0,1]L_2[0,1] orthogonal projections onto the spaces Snk\mathcal S_{nk} are uniformly bounded as operators on L∞[0,1]L_\infty[0,1].

Choose a linearly independent basis Bn1,…,BnpnB_{n1},\ldots,B_{np_n} of Sn\mathcal S_n and write Bn(x)=(Bn1(x),…,Bnpn(x))TB_n(x)=(B_{n1}(x),\ldots,B_{np_n}(x))^{\mathsf T}. For θ∈Rpn\theta\in\mathbb R^{p_n}, put

gθ(x)=Bn(x)Tθ,fθ(x)=egθ(x)1+egθ(x). g_\theta(x)=B_n(x)^{\mathsf T}\theta,\qquad f_\theta(x)=\frac{e^{g_\theta(x)}}{1+e^{g_\theta(x)}}.

Let θn\theta_n be the unique maximizer of

Λn(θ)=E[Ygθ(X)−log⁡(1+egθ(X))], \Lambda_n(\theta)=\mathbb E\left[Yg_\theta(X)-\log\bigl(1+e^{g_\theta(X)}\bigr)\right],

and let θ^n\widehat\theta_n maximize

ℓn(θ)=∑i=1n[Yigθ(Xi)−log⁡(1+egθ(Xi))]. \ell_n(\theta)=\sum_{i=1}^n\left[Y_i g_\theta(X_i)-\log\bigl(1+e^{g_\theta(X_i)}\bigr)\right].

Set gn=gθng_n=g_{\theta_n}, fn=fθnf_n=f_{\theta_n}, g^n=gθ^n\widehat g_n=g_{\widehat\theta_n}, and f^n=fθ^n\widehat f_n=f_{\widehat\theta_n}. Define

In(θ)=n E[fθ(X)(1−fθ(X))Bn(X)Bn(X)T] \mathcal I_n(\theta)=n\,\mathbb E\left[f_\theta(X)\bigl(1-f_\theta(X)\bigr)B_n(X)B_n(X)^{\mathsf T}\right]

and

SE⁡n(x)=fn(x)(1−fn(x))(Bn(x)TIn(θn)−1Bn(x))1/2. \operatorname{SE}_n(x)=f_n(x)\bigl(1-f_n(x)\bigr) \left(B_n(x)^{\mathsf T}\mathcal I_n(\theta_n)^{-1}B_n(x)\right)^{1/2}.

Let

I^n(θ^n)=∑i=1nf^n(Xi)(1−f^n(Xi))Bn(Xi)Bn(Xi)T, \widehat{\mathcal I}_n(\widehat\theta_n)=\sum_{i=1}^n\widehat f_n(X_i)\bigl(1-\widehat f_n(X_i)\bigr)B_n(X_i)B_n(X_i)^{\mathsf T},

and define SE⁡^n(x)\widehat{\operatorname{SE}}_n(x) by replacing fnf_n and In(θn)\mathcal I_n(\theta_n) in the preceding display by f^n\widehat f_n and I^n(θ^n)\widehat{\mathcal I}_n(\widehat\theta_n).

Prove that θ^n\widehat\theta_n exists with probability tending to one and that

sup⁡x∈C∣SE⁡^n(x)SE⁡n(x)−1∣→Pr⁡0,SE⁡n(x)≍(pn/n)1/2 \sup_{x\in C}\left|\frac{\widehat{\operatorname{SE}}_n(x)}{\operatorname{SE}_n(x)}-1\right|\xrightarrow{\Pr}0, \qquad \operatorname{SE}_n(x)\asymp(p_n/n)^{1/2}

uniformly for x∈Cx\in C, and

sup⁡x∈Csup⁡t∈R∣Pr⁡(f^n(x)−fn(x)SE⁡n(x)≤t)−Φ(t)∣⟶0, \sup_{x\in C}\sup_{t\in\mathbb R}\left|\Pr\left(\frac{\widehat f_n(x)-f_n(x)}{\operatorname{SE}_n(x)}\le t\right)-\Phi(t)\right|\longrightarrow0,

where Φ\Phi is the standard normal distribution function. Deduce that

f^n(x)±z1−α/2SE⁡^n(x) \widehat f_n(x)\pm z_{1-\alpha/2}\widehat{\operatorname{SE}}_n(x)

is an asymptotic (1−α)(1-\alpha)-level pointwise confidence interval for fn(x)f_n(x), uniformly in xx, and also for f(x)f(x) whenever

inf⁡s∈Sn∥s−g∥∞=o((pn/n)1/2). \inf_{s\in\mathcal S_n}\|s-g\|_\infty=o\bigl((p_n/n)^{1/2}\bigr).

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The exact problem remains unresolved in the published literature located as of July 31, 2026. Later work proves each of its main ingredients in related settings, but I found no citable result that simultaneously treats the stated unpenalized additive logistic sieve MLE, proves existence with probability tending to one, establishes the standard-error order and uniform consistency of the empirical-information plug-in estimator, and proves the displayed normal approximation uniformly over all x∈[0,1]dx\in[0,1]^d under the statement's localized-basis and growth assumptions. Consequently, the asserted uniform Wald coverage is not completely verified by the available results.

Key literature and established partial progress:

  • Charles J. Stone, “A Nonparametric Framework for Statistical Modelling,” in Proceedings of the International Congress of Mathematicians, Berkeley 1986, 1987, pp. 1052–1056. Direct source: https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1986.2/ICM1986.2.ocr.pdf. Section 4 explicitly proposes the additive-logistic analogue of its Theorem 2; it does not prove it.

  • Charles J. Stone, “The Dimensionality Reduction Principle for Generalized Additive Models,” The Annals of Statistics 14(2), 1986, 590–606. DOI: https://doi.org/10.1214/aos/1176349940. This proves the one-dimensional optimal convergence-rate principle for maximum-likelihood additive spline estimation in generalized additive models. It supplies approximation and stochastic-rate theory, but not the pointwise normal limit, plug-in standard-error consistency, or uniform Wald coverage required here.

  • Stephen Portnoy, “Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity,” The Annals of Statistics 16(1), 1988, 356–366. DOI: https://doi.org/10.1214/aos/1176350710. Portnoy proves increasing-dimensional likelihood expansions and asymptotic normality under moment and dimension-growth conditions for regular exponential families. These general parametric results are relevant to the finite-dimensional logistic sieves, but the paper does not verify the localized additive-spline design conditions, the approximation-centered target fnf_n, uniformity over xx, or consistency of the particular empirical-information standard error in the statement.

  • Xuming He and Qi-Man Shao, “On Parameters of Increasing Dimensions,” Journal of Multivariate Analysis 73(1), 2000, 120–135. DOI: https://doi.org/10.1006/jmva.1999.1873. This establishes componentwise asymptotic normality for broad increasing-dimensional MM-estimators, including logistic regression examples. It does not provide the function-indexed uniform normal approximation or the uniform plug-in variance conclusion for additive spline evaluation functionals required by the exact problem.

  • Charles J. Stone, Mark H. Hansen, Charles Kooperberg, and Young K. Truong, “Polynomial Splines and Their Tensor Products in Extended Linear Modeling: 1994 Wald Memorial Lecture,” The Annals of Statistics 25(4), 1997, 1371–1470. DOI: https://doi.org/10.1214/aos/1031594728. This develops polynomial-spline likelihood modeling for conditional probabilities and generalized additive structures and proves broad L2L_2 convergence-rate results. Its general theory does not establish the exact uniform pointwise studentized limit and Wald statement posed here.

  • Takuma Yoshida and Kanta Naito, “Asymptotics for Penalised Splines in Generalised Additive Models,” Journal of Nonparametric Statistics 26(2), 2014, 269–289. DOI: https://doi.org/10.1080/10485252.2014.899360; arXiv:1208.3920. Their Theorems 1–2 derive pointwise bias, variance of order Kn/nK_n/n, and asymptotic normality for ridge-corrected penalized B-spline estimators in generalized additive models, including Bernoulli/logistic models; a corollary gives approximate pointwise confidence intervals at fixed interior points. This is the closest direct progress, but it concerns a penalized and ridge-corrected estimator, includes explicit bias correction, is stated at fixed interior points rather than uniformly over [0,1]d[0,1]^d, and does not prove the exact empirical-information standard-error ratio in this problem.

What remains open is a complete theorem under the statement's precise assumptions for the unpenalized additive logistic maximum-likelihood sieve, especially the uniform-in-xx studentized normal approximation and uniform consistency of the stated observed-information standard error, including boundary points. The cited results strongly support pointwise analogues under additional spline, smoothness, penalization, or dimension-growth hypotheses, but they do not collectively exhaust the exact claim.

References
  1. [1]ICM Conjectures · 1986-stat-7 link
  2. [2]ICM 1986: A Nonparametric Framework for Statistical Modelling link

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