Optimal designs for estimating individual coefficients in polynomial regression with no intercept

Holger Dette; Petr Shpilev; Viatcheslav B. Melas

arxiv: 1906.08343 · v1 · pith:6WO6PEP7new · submitted 2019-06-19 · 🧮 math.ST · stat.TH

Optimal designs for estimating individual coefficients in polynomial regression with no intercept

Holger Dette , Viatcheslav B. Melas , Petr Shpilev This is my paper

Pith reviewed 2026-05-25 19:42 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords optimal designc-optimal designpolynomial regressionno interceptChebyshev systemexperimental designindividual coefficients

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The pith

Explicit optimal designs are identified for estimating each individual coefficient in polynomial regression without an intercept on [-1,1].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Studden's 1968 characterization of c-optimal designs, which required the regression functions to form a Chebyshev system, to the case of polynomial regression without an intercept where this property does not hold. It determines the optimal designs explicitly for estimating the individual coefficients in such models. This matters for experimental planning because many regression applications omit the intercept term, and the result supplies concrete point allocations and weights that minimize the variance for each target coefficient. A sympathetic reader cares because the work shows the characterization of optimality can be carried through even when the usual Chebyshev assumption fails.

Core claim

In a seminal paper Studden characterized c-optimal designs in regression models where the regression functions form a Chebyshev system and used these results to determine the optimal design for estimating the individual coefficients in a polynomial regression model on the interval [-1,1] explicitly. In this note the optimal design is identified for estimating the individual coefficients in a polynomial regression model with no intercept where the regression functions do not form a Chebyshev system.

What carries the argument

The c-optimal design for each individual coefficient vector in the no-intercept polynomial model, obtained by extending Studden's characterization to the non-Chebyshev case.

If this is right

The identified designs give the minimal possible variance for estimating any single coefficient in the no-intercept model.
The designs are explicit, so they can be written down and used directly for any fixed degree.
The same support-point structure applies across all individual coefficients once the degree is fixed.
Efficiency gains are realized relative to uniform or arbitrary designs when the intercept is omitted.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same extension technique could be tested on other regression families that lack the Chebyshev property but still admit explicit optimal designs.
Numerical verification for small degrees would provide independent confirmation that the derived weights and nodes are indeed optimal.
The result suggests that optimal-design theory may remain tractable for certain structured departures from classical assumptions without requiring entirely new machinery.

Load-bearing premise

The regression functions on [-1,1] allow characterization of c-optimal designs even though they do not form a Chebyshev system.

What would settle it

For a concrete low-degree case such as quadratic regression without intercept, compute the c-criterion value for the claimed optimal design and for a competing design with different support points; if any competing design yields strictly smaller variance for the target coefficient, the claimed optimality is falsified.

read the original abstract

In a seminal paper \cite{studden1968} characterized $c$-optimal designs in regression models, where the regression functions form a Chebyshev system. He used these results to determine the optimal design for estimating the individual coefficients in a polynomial regression model on the interval $[-1,1]$ explicitly. In this note we identify the optimal design for estimating the individual coefficients in a polynomial regression model with no intercept (here the regression functions do not form a Chebyshev system).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note fills the no-intercept gap left by Studden (1968) with explicit c-optimal designs for the model on {x, x², …, xᵏ}.

read the letter

The paper works out the c-optimal designs for estimating each individual coefficient in the no-intercept polynomial regression on [-1,1]. Studden’s Chebyshev-system argument does not apply directly because these functions lack the required alternation property, so the authors supply the designs by other means, apparently through direct verification of the equivalence theorem conditions for this specific family. That is the concrete addition: the explicit support points and weights for each target coefficient c. The work is short, cites the relevant prior result cleanly, and stays within the bounds of what it claims. The constructions are presented as the main output, which is appropriate for a note of this type. The main limitation is that the argument is tied to this model and does not supply a broader replacement for the Chebyshev property; any extension to other non-Chebyshev systems would need separate justification. For higher degrees the patterns would also benefit from a clear statement of how far the explicit forms have been checked. The paper is aimed at readers already working in optimal design for regression models. It is a modest but legitimate extension rather than a reworking of the field, and the central claim is narrow enough that a referee can verify the derivations without excessive effort. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper extends Studden (1968) by identifying explicit c-optimal designs for estimating individual coefficients β_j (j=1,...,k) in the no-intercept polynomial model y = β_1 x + ... + β_k x^k + ε on [-1,1], where the regression functions {x, x², ..., x^k} do not form a Chebyshev system. It claims these designs can be characterized despite the failure of the standard Chebyshev assumption used in the original work.

Significance. If the explicit constructions and optimality proofs hold, the note supplies concrete optimal designs for a common model variant (no intercept) that falls outside the Chebyshev framework, thereby extending the applicability of c-optimality results to a practically relevant setting. The work is credited for directly addressing the non-Chebyshev case rather than assuming the property carries over.

major comments (2)

[Introduction / main result] Introduction and main result section: The manuscript asserts that optimal designs are identified even though {x^j}_{j=1}^k does not form a Chebyshev system, yet provides no referenced general theorem or alternative equioscillation property to replace Studden's characterization. The derivation therefore appears to rest on case-by-case verification whose scope and rigor must be shown explicitly for the claimed optimality to be load-bearing.
[Main result] Main theorem or construction (whichever states the explicit designs): Without an explicit sensitivity-function equioscillation argument or supporting lemma that holds for the no-intercept model, it is unclear whether the reported support points and weights remain optimal for arbitrary k and target coefficient index j; a concrete verification for at least one k>2 would be required to substantiate the general claim.

minor comments (2)

[References] The citation to Studden (1968) should include the full reference details (journal, volume, pages) in the bibliography.
[Model setup] Notation for the regression vector f(x) = (x, x², ..., x^k)^T should be introduced once and used consistently when stating the information matrix and c-optimality criterion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

Referee: Introduction and main result section: The manuscript asserts that optimal designs are identified even though {x^j}_{j=1}^k does not form a Chebyshev system, yet provides no referenced general theorem or alternative equioscillation property to replace Studden's characterization. The derivation therefore appears to rest on case-by-case verification whose scope and rigor must be shown explicitly for the claimed optimality to be load-bearing.

Authors: Our derivation does not rely on Studden's Chebyshev-system characterization or equioscillation property. Instead, optimality of the explicitly constructed designs is established via the general equivalence theorem for c-optimality (Kiefer and Wolfowitz, 1960), which applies to arbitrary regression functions. We will revise the introduction to reference this theorem explicitly and explain its direct application to the no-intercept model. revision: yes
Referee: Main theorem or construction (whichever states the explicit designs): Without an explicit sensitivity-function equioscillation argument or supporting lemma that holds for the no-intercept model, it is unclear whether the reported support points and weights remain optimal for arbitrary k and target coefficient index j; a concrete verification for at least one k>2 would be required to substantiate the general claim.

Authors: The main result states the support points and weights in closed form for arbitrary k and j. Optimality follows from verifying that the sensitivity function of the proposed design attains the required bound at the support points, as required by the equivalence theorem. We agree an illustrative check strengthens the presentation and will add an explicit numerical verification for k=3 (with the corresponding sensitivity function) in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; direct extension to non-Chebyshev case without reduction to inputs or self-citations

full rationale

The paper explicitly distinguishes its setting from Studden (1968) by noting that {x, x², ..., x^k} on [-1,1] does not form a Chebyshev system, then claims to identify the c-optimal designs anyway. This indicates case-by-case or direct verification rather than invoking the general theorem. No self-citations appear in the load-bearing steps, no parameters are fitted and renamed as predictions, and no ansatz or uniqueness result is smuggled via prior author work. The derivation chain is therefore self-contained against the external benchmark of Studden's result and does not reduce by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; no free parameters, invented entities, or additional axioms are visible. The central modeling choice (polynomial regression without intercept on [-1,1]) is treated as a domain assumption.

axioms (1)

domain assumption Polynomial regression model without intercept on the interval [-1,1]
Stated directly in the abstract as the setting for the new designs.

pith-pipeline@v0.9.0 · 5603 in / 1125 out tokens · 23087 ms · 2026-05-25T19:42:15.969829+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Chang, F.-C. (1999). Exact D -optimal designs for polynomial regression without intercept. Statistics & Probability Letters. , 44(2):131--136

work page 1999
[2]

and Heiligers, B

Chang, F.-C. and Heiligers, B. (1996). E -optimal designs for polynomial regression without intercept. Journal of Statistical Planning and Inference. , 55(3):371--387

work page 1996
[3]

Dette, H. (1990). A generalization of D - and D_1 -optimal designs in polynomial regression. Annals of Statistics , 18:1784--1805

work page 1990
[4]

B., and Pepelyshev, A

Dette, H., Melas, V. B., and Pepelyshev, A. (2004). Optimal designs for estimating individual coefficients in polynomial regression—a functional approach. Journal of Statistical Planning and Inference , 118(1):201 -- 219

work page 2004
[5]

Elfving, G. (1952). Optimal allocation in linear regression theory. The Annals of Mathematical Statistics , 23:255--262

work page 1952
[6]

Fang, Z. (2002). D -optimal designs for polynomial regression models through origin. Statistics & Probability Letters , 57:343--351

work page 2002
[7]

L., Chang, F.-C., and K., W

Huang, M.-N. L., Chang, F.-C., and K., W. W. (1995). D -optimal designs for polynomial regression without an intercept. Statistica Sinica , 5(2):441--458

work page 1995
[8]

Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Annals of Statistics , 2:849--879

work page 1974
[9]

Li, K.-H., Lau, T.-S., and Zhang, C. (2005). A note on D -optimal designs for models with and without an intercept. Statistical Papers. , 46(3):451--458

work page 2005
[10]

and Rodr \' guez, C

Ortiz, I. and Rodr \' guez, C. (1998). D -optimal designs for weighted polynomial regression without any initial terms . In Atkinson, A. C., Pronzato, L., and Wynn, H. P., editors, MODA 5 - Advances in Model-Oriented Design Analysis and Experimental Design. , pages 67--74. Physica-Verlag, Heidelberg

work page 1998
[11]

Pukelsheim, F. (2006). Optimal Design of Experiments . SIAM, Philadelphia

work page 2006
[12]

Silvey, S. (1980). Optimal Design . Chapman and Hall, London

work page 1980
[13]

Studden, W. J. (1968). Optimal designs on T chebycheff points. Annals of Mathematical Statistics , 39(5):1435--1447

work page 1968
[14]

Szeg \"o , G. (1975). Orthogonal Polynomials. American Mathematical Society, Providence, R.I

work page 1975

[1] [1]

Chang, F.-C. (1999). Exact D -optimal designs for polynomial regression without intercept. Statistics & Probability Letters. , 44(2):131--136

work page 1999

[2] [2]

and Heiligers, B

Chang, F.-C. and Heiligers, B. (1996). E -optimal designs for polynomial regression without intercept. Journal of Statistical Planning and Inference. , 55(3):371--387

work page 1996

[3] [3]

Dette, H. (1990). A generalization of D - and D_1 -optimal designs in polynomial regression. Annals of Statistics , 18:1784--1805

work page 1990

[4] [4]

B., and Pepelyshev, A

Dette, H., Melas, V. B., and Pepelyshev, A. (2004). Optimal designs for estimating individual coefficients in polynomial regression—a functional approach. Journal of Statistical Planning and Inference , 118(1):201 -- 219

work page 2004

[5] [5]

Elfving, G. (1952). Optimal allocation in linear regression theory. The Annals of Mathematical Statistics , 23:255--262

work page 1952

[6] [6]

Fang, Z. (2002). D -optimal designs for polynomial regression models through origin. Statistics & Probability Letters , 57:343--351

work page 2002

[7] [7]

L., Chang, F.-C., and K., W

Huang, M.-N. L., Chang, F.-C., and K., W. W. (1995). D -optimal designs for polynomial regression without an intercept. Statistica Sinica , 5(2):441--458

work page 1995

[8] [8]

Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Annals of Statistics , 2:849--879

work page 1974

[9] [9]

Li, K.-H., Lau, T.-S., and Zhang, C. (2005). A note on D -optimal designs for models with and without an intercept. Statistical Papers. , 46(3):451--458

work page 2005

[10] [10]

and Rodr \' guez, C

Ortiz, I. and Rodr \' guez, C. (1998). D -optimal designs for weighted polynomial regression without any initial terms . In Atkinson, A. C., Pronzato, L., and Wynn, H. P., editors, MODA 5 - Advances in Model-Oriented Design Analysis and Experimental Design. , pages 67--74. Physica-Verlag, Heidelberg

work page 1998

[11] [11]

Pukelsheim, F. (2006). Optimal Design of Experiments . SIAM, Philadelphia

work page 2006

[12] [12]

Silvey, S. (1980). Optimal Design . Chapman and Hall, London

work page 1980

[13] [13]

Studden, W. J. (1968). Optimal designs on T chebycheff points. Annals of Mathematical Statistics , 39(5):1435--1447

work page 1968

[14] [14]

Szeg \"o , G. (1975). Orthogonal Polynomials. American Mathematical Society, Providence, R.I

work page 1975