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OP-1659OpenHigh-dimensional statistics

Ueber eine Eigenschaft des Gaussschen Fehlergesetzes · 1928-stat-21

Statistical Distribution Theory · ICM 1928: Ueber eine Eigenschaft des Gaussschen Fehlergesetzes

MSC 62 · Statistics

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Problem statement

Classify all probability density functions φ:R→[0,∞)\varphi:\mathbb{R}\to[0,\infty) such that, for every integer n≥1n\geq 1 and every l1,…,ln∈Rl_1,\ldots,l_n\in\mathbb{R}, the function

L(x)=∏i=1nφ(x−li) L(x)=\prod_{i=1}^{n}\varphi(x-l_i)

is integrable, has a finite first moment, and has a unique global maximizer l′l', with this maximizer equal to the centroid l′′l'' of LL, where

l′′=∫−∞∞xL(x) dx∫−∞∞L(x) dx. l''=\frac{\int_{-\infty}^{\infty}xL(x)\,dx}{\int_{-\infty}^{\infty}L(x)\,dx}.

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

The exact problem is unresolved. The known classification in the original source is conditional and does not cover all probability densities allowed by the statement.

G. Pólya, “Über eine Eigenschaft des Gaußschen Fehlergesetzes,” Atti del Congresso Internazionale dei Matematici, Bologna 3–10 settembre 1928, vol. VI, 1932, pp. 63–64, direct proceedings PDF. Pólya proves that, after additionally assuming that φ>0\varphi>0, that φ\varphi is twice continuously differentiable, and that it is monotone on each side of its unique maximum, the only possibilities are normalized densities of the forms

Aexp⁡{−Bcosh⁡(Cx+D)}andAexp⁡{−(Cx+D)2}. A\exp\{-B\cosh(Cx+D)\} \quad\text{and}\quad A\exp\{-(Cx+D)^2\}.

He also states that these families do have the required mode-centroid coincidence. However, he explicitly introduces the extra assumptions as assumptions that might partly be dispensable; his result therefore does not classify nonsmooth, vanishing, or non-unimodal-on-each-side densities covered by the present statement.

E. J. G. Pitman, “The Estimation of the Location and Scale Parameters of a Continuous Population of Any Given Form,” Biometrika 30 (3/4), 1939, pp. 391–421, doi:10.1093/biomet/30.3-4.391. Pitman’s general location-family theory identifies the centroid in the statement as the minimum-risk equivariant location estimator (now called the Pitman estimator), but it does not classify the densities for which that estimator agrees exactly with the unique maximum-likelihood estimator for every sample.

No citable later result was found that removes Pólya’s auxiliary regularity and shape hypotheses or otherwise exhausts every density in the stated class. Thus the Gaussian and exponential-cosh densities are established solutions and are conditionally exhaustive, while exhaustive classification under only the hypotheses in the statement remains open.

References
  1. [1]ICM Conjectures · 1928-stat-21 link
  2. [2]ICM 1928: Ueber eine Eigenschaft des Gaussschen Fehlergesetzes link

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