[Paper Review] Student's t-test without symmetry conditions

Iosif Pinelis|ArXiv.org|Jun 7, 2006

Optimal Experimental Design Methods12 references3 citations

TL;DR

This paper extends Student’s t-test to non-symmetric distributions by representing any zero-mean distribution as a mixture of two-point zero-mean distributions. It establishes exact inequalities that ensure conservative inference under asymmetry, generalizing earlier results under symmetry and enabling robust hypothesis testing for location and asymmetry without requiring symmetry assumptions.

ABSTRACT

An explicit representation of an arbitrary zero-mean distribution as the mixture of (at-most-)two-point zero-mean distributions is given. Based in this representation, tests for (i) asymmetry patterns and (ii) for location without symmetry conditions can be constructed. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition.

Motivation & Objective

• To develop exact inequalities that ensure conservative properties of hypothesis tests under asymmetry, extending prior results that required symmetry.
• To construct tests for location and asymmetry patterns without assuming symmetry of the underlying distribution.
• To generalize Efron’s and Eaton’s results on self-normalized sums and Khinchin-type inequalities to non-symmetric settings.
• To provide a theoretical foundation for robust inference in t-tests when the normality or symmetry assumptions are violated.
• To establish a representation of zero-mean distributions as mixtures of two-point distributions, enabling distribution-free inference under weak moment conditions.

Proposed method

• Represents an arbitrary zero-mean distribution as a mixture of (at-most-)two-point zero-mean distributions using a reciprocating function r(x) that ensures conditional mean zero.
• Uses the mixture representation to express the self-normalized sum S as a mixture of Rademacher sums, enabling extension of tail and moment inequalities.
• Applies sharp inequalities from Khinchin, Whittle, and Haagerup to derive bounds on the moment generating function and tail probabilities of S.
• Derives conservative bounds on P(S ≥ x) by comparing to the standard normal tail, showing P(S ≥ x) ≤ (2e³/9)P(Z ≥ x) for all real x.
• Employs a perturbation argument with ε-approximation to control the ratio of expectations of convex functions under the mixture representation.
• Uses conditional expectation and iterated integral techniques to prove invariance of expectations under the mixture, ensuring validity of the bounds.

Experimental results

Research questions

• RQ1 Can exact conservative inequalities for the self-normalized sum S be derived without assuming symmetry of the underlying distribution?
• RQ2 How can the distribution of a zero-mean random variable be represented as a mixture of two-point zero-mean distributions to enable robust inference?
• RQ3 To what extent do the tail and moment inequalities for symmetric distributions extend to asymmetric ones under the new representation?
• RQ4 Can the conservative properties of t-tests be preserved when symmetry is relaxed, and how can this be quantified?
• RQ5 What is the optimal bound on the tail probability of the self-normalized sum S under asymmetry, and how does it compare to the normal tail?

Key findings

• An explicit representation of any zero-mean distribution as a mixture of (at-most-)two-point zero-mean distributions is established via a reciprocating function r(x).
• The inequality P(S ≥ x) ≤ (2e³/9)P(Z ≥ x) holds for all real x, even without symmetry, extending Eaton’s and Pinelis’s results to asymmetric settings.
• The bound (2e³/9) ≈ 2.12 is sharp and improves upon the standard normal tail bound, ensuring conservative inference under asymmetry.
• For all convex functions f with convex second derivative, E[f(S)] ≤ E[f(Z)] holds, generalizing the Khinchin-Whittle-Haagerup inequality to asymmetric distributions.
• The method enables construction of exact conservative tests for asymmetry patterns and for location parameters without symmetry assumptions.
• Theoretical guarantees are provided via perturbation analysis and ratio bounds, showing that the ratio of expectations is bounded below by e^(-4ε), ensuring robustness under small deviations from symmetry.