Abstract
For any continuous zero-mean random variable $X$, a reciprocating function $r$ is constructed, based only on the distribution of $X$, such that the conditional distribution of $X$ given the (at-most-)two-point set $\{X,r(X)\}$ is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two-point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations - of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) - go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. 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.
Highlights
Efron [11] observed that the tail of the distribution of the self-normalized sumS := X1 + · · · + X n, X + · 2 n (1.1)is bounded in certain sense by the tail of the standard normal distribution — provided that the Xi’s satisfy a certain symmetry condition; it is enough that the Xi’s be independent and symmetrically distributed
Under the symmetry condition, the distribution of the self-normalized sum is a mixture of the distributions of the normalized Khinchin-Rademacher sums, and can be nicely bounded, say by using an exponential bound by Hoeffding [17], or more precise bounds by Eaton [8; 9] and Pinelis [24]
Exact inequalities implying conservative properties of such tests will be given in this paper
Summary
Efron [11] observed that the tail of the distribution of the self-normalized sum. is bounded in certain sense by the tail of the standard normal distribution — provided that the Xi’s satisfy a certain symmetry condition; it is enough that the Xi’s be independent and symmetrically (but not necessarily identically) distributed. Exact inequalities implying conservative properties of such tests will be given in this paper These developments extend the mentioned results established earlier by Efron, Eaton, and Pinelis under the symmetry condition. In testing for the mean of the unknown distribution, the asymmetry pattern as represented by the reciprocating function should be considered as a nuisance parameter. The reciprocating functions in the expressions for SW and SY are usually to be estimated based on the sample Such a step — replacing nuisance parameters by their empirical counterparts — is quite common in statistics. Modeling with reciprocating functions appears to provide more flexibility It allows the mentioned conservative properties of the tests to be preserved without a symmetry condition. A brief account of results of [27] (without proofs) was presented in [28]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
