Separation problem for a family of Borel and Baire G-powers of shift measures on $ \mathbb{R} $

On Moore-Yamasaki-Kharazishvili Type Measures and the Infinite Powers of Borel Diffused Probability Measures on R

“…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 of the properties of equidistributed sequences 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 M. KINTSURASHVILI et al 84 separated and, accordingly, has an infinite-sample well-founded estimator of the unknown distribution function. This extends the main result established in [20]. …”

Objective and Strong Objective Consistent Estimates of Unknown Parameters for Statistical Structures in a Polish Group Admitting an Invariant Metric

“…In Section 2 we give some notions and facts from the theory of Haar null sets in complete metric linear spaces and equidistributed sequences on the real axis R. Concepts of objective and strong objective infinite sample consistent estimates for statistical structures are introduced also in this section. Section 3 presents a certain construction of the objective infinite sample consistent estimate of an unknown distribution function which generalises the recent results obtained in [27]. There is proved an existence of the infinite sample consistent estimate of an unknown distribution function F (F ∈ F ) for the family of Borel probability measures {p N F : F ∈ F }, where F denotes the family of all strictly increasing and continuous distribution functions on R and p N F denotes an infinite power of the Borel probability measure p F on R defined by F .…”

“…It was proved in [27] that T n : R n → R (n ∈ N ) defined by T n (x 1 , · · · , x n ) = −F −1 (n −1 #({x 1 , · · · , x n } ∩ (−∞; 0])) (1.5) for (x 1 , · · · , x n ) ∈ R n , is a consistent estimator of a useful signal θ in onedimensional linear stochastic model (1.6) where #(·) denotes a counting measure, ∆ k is a sequence of independent identically distributed random variables on R with strictly increasing continuous distribution function F and expectation of ∆ 1 does not exist. In this direction the following two examples of simulations of linear one-dimensional stochastic models have been considered.…”

“…This observations lead us to additional conditions imposed on the estimate T 1 which assumes that T −1 1 (θ) must be neither Haar null nor prevalent for each θ ∈ R. Following [1], a set which is neither Haar null nor prevalent is called a Haar ambivalent set. Such estimates firstly were adopted as objective infinite sample consistent estimates of a useful signal in the linear one-dimensional stochastic model(see, [27], Theorem 4.1, p. 482).…”

“…In [27] has been established that the estimators lim T n := inf n sup m≥n T m and lim T n := sup n inf m≥n T m are consistent infinite sample estimates of a useful signal θ in the model (1.6) (see, [27], Theorem 4.2, p. 483). When we begin to study properties of these infinite sample estimators from the point of view of the theory of Haar null sets in R N , we observed a surprising and an unexpected fact for us that these both estimates are objective (see, [20], Theorem 3.1).…”