Applications of Measure Theory to Statistics

A certain modified version of Kolmogorov’s strong law of large numbers is used for an extension of the result of C. Baxa and J. Schoi\(\beta \)engeier (2002) to a maximal set of uniformly distributed sequences in (0, 1) that strictly contains the set of all sequences having the form \((\{\alpha n\})_{n \in \mathbf{N}}\) for some irrational number \(\alpha \) and having the full \(\ell _1^{\infty }\)-measure, where \(\ell _1^{\infty }\) denotes the infinite power of the linear Lebesgue measure \(\ell _1\) in (0, 1).

By using the main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in \(R^{\infty }\) described in [P2], an infinite-dimensional Monte Carlo integration is elaborated and the validity of some new strong law type theorems are obtained in this chapter.

In [P5], it was shown that \(\mu \) almost every element of \(\mathbf {R}^{\infty }\) is uniformly distributed in \([-\frac{{1}}{{2}}, \frac{{1}}{{2}}]\), where \(\mu \) denotes the Moore–Yamasaki–Kharazishvili measure in \(\mathbf {R}^{\infty }\) for which \(\mu ([-\frac{1}{2},\frac{1}{2}]^{\infty }) = 1\). In the present chapter the same set is studied from the point of view of shyness and it is demonstrated that it is shy in \(\mathbf {R}^{\infty }\). In the Solovay model, the structure of the set of all sequences uniformly distributed modulo 1 in \([-\frac{{1}}{{2}}, \frac{{1}}{{2}}]\) is studied from the point of view of shyness and it is shown that it is the prevalent set in \(\mathbf {R}^{\infty }\).

This chapter contains a brief description of Yamasaki’s remarkable investigation (1980) of the relationship between Moore–Yamasaki–Kharazishvili type measures and infinite powers of Borel diffused probability measures on \(\mathbf{R}\). More precisely, there is given Yamasaki’s proof that no infinite power of the Borel probability measure with a strictly positive density function on R has an equivalent Moore–Yamasaki–Kharazishvili type measure. A certain modification of Yamasaki’s example is used for the construction of such a Moore–Yamasaki–Kharazishvili type measure that is equivalent to the product of a certain infinite family of Borel probability measures with a strictly positive density function on R. By virtue the properties of real-valued sequences equidistributed on the real axis, it is demonstrated that an arbitrary family of infinite powers of Borel diffused probability measures with strictly positive density functions on R is strongly separated and, accordingly, has an infinite-sample well-founded estimator of the unknown distribution function. This extends the main result established in the paper [ZPS].

By using the notion of a Haar ambivalent set introduced by Balka, Buczolich, and Elekes (2012), essentially new classes of statistical structures having objective and strong objective estimates of unknown parameters are introduced in a Polish nonlocally compact group admitting an invariant metric and relations between them are studied in this chapter. An example of a weakly separated statistical structure is constructed for which a question asking whether there exists a consistent estimate of an unknown parameter is not solvable within the theory \( (ZF)~ \& ~(DC)\). A question asking whether there exists an objective consistent estimate of an unknown parameter for any statistical structure in a nonlocally compact Polish group with an invariant metric when a subjective one exists, is answered positively when there exists at least one such parameter, the preimage of which under this subjective estimate is a prevalent. These results extend recent results of Pantsulaia and Kintsurashvili (2014). Some examples of objective and strong objective consistent estimates in a compact Polish group \(\{0; 1\}^N\) are considered in this chapter. At end of this chapter we present a certain claim for theoretical statisticians in which each consistent estimation with domain in a nonlocally compact Polish group equipped with an invariant metric must pass the certification exam on the objectivity before its practical application and we also give some recommendations.

The notion of Haar null sets was introduced by J. P. R. Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer, and Yorke. During the last 20 years this notion has been useful in studying exceptional sets in diverse areas. These include analysis, dynamical systems, group theory, and descriptive set theory. In the present chapter their notion of “prevalence” is used in studying structures of domains of some infinite sample statistics and in explaining why null hypothesis is rejected for almost every infinite sample by the hypothesis testing of a maximal reliability.