Marc Hallin is Emeritus Professor at the Université libre de Bruxelles, where he obtained his PhD in Mathematics and has been teaching Statistics and Mathematical Statistics at the Departments of Mathematics and Economics for more than thirty years. He also held invited positions in Lille 1, Paris 6, the Université Catholique de Louvain, and, from to 2010 to 2015, at Princeton University. The author of about 250 research papers, he received several awards, among which the 2022 Senior Gottfried Noether Award of the American Statistical Association, the 2022 Pierre-Simon de Laplace Award of the Société française de Statistique, a Humboldt-Forschungspreis (Humboldt Research Award) of the Alexander von Humboldt Foundation, and the Medal of the Faculty of Mathematics and Physics of Charles University in Prague. He also was the 2017 recipient of the Hermann Otto Hirschfeld Lecture Series at the Humboldt-Universität zu Berlin, the 2018 Mahalanobis Memorial Lecturer at the Indian Statistical Institute, and the holder of a Cátedra de Excelencia at the Universidad Carlos III de Madrid. A Fellow of the IMS and the ASA, he is an elected member of the “Classe des Sciences” of the Royal Academy of Belgium.

This Medallion Lecture will be given at the 11th World Congress in Probability and Statistics in Bochum, Germany.

Ancillarity, Maximal Ancillarity, and Semiparametric Efficiency

This lecture is based on joint work with Bas Werker (Tilburg University, the Netherlands) and Bo Zhou (Virginia Tech, Blacksburg, USA).

The concept of ancillarity was first introduced by Ronald Fisher (1925), who coined the expression (see Stigler, 2001); but formal definitions and profound properties were only provided by Debabrata Basu (1955, 1958, 1959, 1964) who subsequently (Basu 1977) emphasized the role of ancillarity in the elimination of nuisance parameters—an old and ubiquitous problem in statistical inference. In a semiparametric model involving a family of distributions {P⁽ⁿ⁾_ϑ,f | ϑ ∈ Θ, f ∈ F}, say, indexed by a finite-dimensional parameter of interest ϑ and an infinite-dimensional nuisance f, we will call ancillary—more precisely, nuisance-ancillary—a function ζ⁽ⁿ⁾(ϑ) of ϑ and the observation whose distribution under P⁽ⁿ⁾_ϑ,f  does not depend on the nuisance f. Similarly, we call nuisance-ancillary a sigma-field B⁽ⁿ⁾(ϑ) of Borel sets whose probability under P⁽ⁿ⁾_ϑ,f only depends on ϑ. A natural way to conduct valid nuisance-free inference on ϑ, then, is to restrict to (nuisance-) ancillary statistics, hence to maximal (in the sense of set inclusion) ancillary sigma-
fields.

Throughout, we will concentrate on the example of models with unspecified (absolutely continuous) innovation densities—models under which an n-tuple X⁽ⁿ⁾ := (X₁, … , X_n) of d-dimensional observations X_t has distribution P⁽ⁿ⁾_ϑ,f if and only if an n-tuple
Z⁽ⁿ⁾(ϑ) := (Z₁ (ϑ), … , Z_n(ϑ)) of residuals is i.i.d. with density f (typical examples are regression or VARMA models). For d = 1 and
concentrating, for the sake of simplicity, on the problem of testing hypotheses of the form H₀⁽ⁿ⁾:= {P⁽ⁿ⁾_ϑ0,f | f ∈ F}, the sigma-field B⁽ⁿ⁾(ϑ₀) generated by the residual ranks R⁽ⁿ⁾(ϑ₀) :=(R₁⁽ⁿ⁾(ϑ₀), …, R_n⁽ⁿ⁾(ϑ₀)), then, is maximal ancillary at ϑ₀ (this follows from a theorem by Basu (1959); see Section E in Hallin et al. (2021b)) and B⁽ⁿ⁾(ϑ₀)-
measurable tests—that is, rank tests based on R⁽ⁿ⁾(ϑ₀)—are a natural way of eliminating the nuisance f.

A major problem, however, is that maximal ancillary and nuisance-ancillary sigma-fields, typically, are not unique. For instance, if d > 1 in the example above, the sigma-field B₁⁽ⁿ⁾(ϑ₀) generated by the ranks of the residuals’ first components is maximal ancillary, but so is the sigma-field B₂⁽ⁿ⁾(ϑ₀) generated by the ranks of the residuals’ second components (this follows from the same theorem in Basu (1959)). Although both are maximal, hence cannot be enlarged without losing ancillarity, they both are abandoning significant ancillary information. In a different context, this non-uniqueness has sparked (see, e.g., Chapter 10 in Lehmann and Romano (2022)) quite an amount of discussion about the concept of ancillarity which, for that reason, has sometimes been disregarded.

While this non-uniqueness problem generally arises in finite-sample problems, however, it often disappears asymptotically: in Locally Asymptotically Normal (LAN) families, limiting experiments (Gaussian shifts or equivalent) yield unique maximal nuisance-ancillary sigma-fields. These maximal nuisance-ancillary sigma-fields, moreover, are generated by the L₂ tangent projections traditionally performed in semiparametric inference (cf. the classical monograph by Bickel et al. (1993)), hence, are measuring weak limits of all semiparametrically optimal procedures. Tangent projections, however, depend on the nuisance. Performing (approximate) tangent projections, thus, requires the estimation of the nuisance.

A natural question is, then, “Can we characterize sequences of maximal ancillary sigma-fields converging, as the sample size increases, and in a sense to be defined, to the unique maximal ancillary sigma-field of the limiting experiments?” These sequences, then, can be considered as “asymptotically strongly maximal ancillary.”

The classical way to construct semiparametrically optimal inference consists in performing the tangent projection defined in the limiting experiment on the corresponding finite-sample central sequence. Both, however, depend on the unspecified value of the nuisance f which, therefore, has to be estimated—an estimation problem which, in an infinite-dimensional space, may be an uneasy one. In the above particular case of models with unspecified innovation densities, this implies that the density f of the residuals has to be estimated—via kernel methods, which even for moderate values of d is difficult, and sample splitting, which may be costly in terms of finite-sample efficiency.

When a strongly maximal ancillary sequence of sigma-fields B_Anc⁽ⁿ⁾ exists, L₂ tangent space projections can be replaced by conditional expectations E [· | B_Anc⁽ⁿ⁾ ]. Contrary to the former projection, such conditional expectations and their distributions, by construction, do not depend on the nuisance f, allowing, e.g., for nuisance-free testing of H₀⁽ⁿ⁾ without the need for a painful estimation of f.

In the particular case of models with unspecified innovation densities, we show that the sigma-field generated by the recently proposed measure-transportation-based center-outward ranks and signs (Chernozhukov et al. (2017); Hallin et al. (2021)) is asymptotically strongly maximal ancillary. In dimension d = 1, that sigma-field reduces to the sigma-field of traditional univariate residual ranks.

This has far-reaching consequences:

(i) semiparametric efficiency bounds can be reached by B_Anc⁽ⁿ⁾ -measurable (hence fully distribution-free) center-outward rank- and sign-based procedures;

(ii) these procedures do not require the estimation of the nuisance (of the innovation density) f;

(iii) if, however, this nuisance f is estimated, the distribution of E [Δ⁽ⁿ⁾_ϑ,fˆ(n) | B_Anc⁽ⁿ⁾ ], where Δ⁽ⁿ⁾_ϑ,f denotes the central sequence computed at (ϑ, f) and f⁽ⁿ⁾ only depends on the order statistic of the residuals (a very natural assumption), remains conditionally (on that order statistic) nuisance-free, yielding uniformly semiparametrically efficient nuisance-free testing procedures; in the special case of models with unspecified innovation densities, nuisance-freeness means (finite-sample) distribution-freeness.

These properties demonstrate the considerable finite-sample advantages of E [· | B_Anc⁽ⁿ⁾ ] over classical tangent space projections.

References

[1] Basu, D. (1955). On statistics independent of a complete sufficient statistic, Sankhyā: the Indian Journal of Statistics Series A 15, 377–380.

[2] Basu, D. (1958). On statistics independent of sufficient statistics, Sankhyā: the Indian Journal of Statistics Series A 20, 223–226.

[3] Basu, D. (1959). The family of ancillary statistics, Sankhyā: the Indian Journal of Statistics Series A 21, 247–256.

[4] Basu, D. (1964). Recovery of ancillary information, Sankhyā: the Indian Journal of Statistics Series A 26, 3–16.

[5] Basu, D. (1977). On the elimination of nuisance parameters, Journal of the American Statistical Association 72, 355–66.

[6] Bickel, P.J., C.A.J. Klaassen, Y. Ritov, and J.A. Wellner (1993). Efficient and Adaptive Statistical Inference for Semiparametric Models, Johns Hopkins University Press, Baltimore. Revised edition (1998), Springer-Verlag, New York.

[7] Chernozhukov, V., A. Galichon, M. Hallin, and M. Henry (2017). Monge–Kantorovich depth, quantiles, ranks, and signs, Annals of Statistics 45, 223–256.

[8] Fisher, R. A. (1925). Theory of statistical estimation, Mathematical Proceedings of the Cambridge Philosophical Society 22, 700–725.

[9] Hájek, J. and Z. Šidák (1967). Theory of Rank Tests, Academic Press, New York.

[10] Hallin, M., E. del Barrio, J. Cuesta-Albertos, and C. Matrán, C. (2021a). Center-outward distribution and quantile functions, ranks, and signs in dimension d: a measure transportation approach, Annals of Statistics 49, 1139–1165.

[11] Hallin, M., E. del Barrio, J. Cuesta-Albertos, and C. Matrán (2021b). Online Supplement to “Distribution and Quantile Functions, Ranks, and Signs in Dimension d: a measure transportation approach,” Annals of Statistics 49, 1139–1165, 32 pp.

[12] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag, New York.

[13] Lehmann, E.L. and J.P. Romano (2022). Testing Statistical Hypotheses (4th edition), Springer.

[14] Stigler, S. M. (2001). Ancillary history, in M. de Gunst, C. Klaassen, and A. van der Vaart, State of the Art in Probability and Statistics, Festschrift for Willem van Zwet, Institute of Mathematical Statistics Lecture Notes–Monograph Series, Beachwood, OH, pages 555–567.