Why Null Hypothesis Is Rejected for Almost Every Infinite Sample by the Hypothesis Testing of a Maximal Reliability

Pantsulaia, Gogi

doi:10.1007/978-3-319-45578-5_6

Cited by 4 publications

(6 citation statements)

References 7 publications

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“…In order to explain a big gap between the theory of mathematical statistics and results of hypothesis testing, concepts of subjective and objective infinite sample consistent estimates of a useful signal in the linear one-dimensional stochastic model were introduced in [18]. This approach essentially uses the concept of Haar null sets in Polish topological vector spaces introduced by J.P.R.…”

Section: Introductioncontrasting

confidence: 43%

“…In [18], the examples of objective and strong objective infinite sample consistent estimates( [18], T ⋆ (p. 63), T • (p. 67)) of a useful signal in the linear onedimensional stochastic model were constructed by using the axiom of choice and a certain partition of the non-locally compact abelian Polish group R N constructed in [17].…”

Section: Auxiliary Notions and Facts From Functional Analysis And Meamentioning

confidence: 43%

“…Notice that the set S like ∪ z∈Z S θ+z is Borel shy set (see [18], Lemma 4.14, p. 60). Taking into account this fact, the results of Lemmas 2.4 and 2.7, invariance of Haar ambivalent sets under translations and symmetric transformation and the simple statement that difference of non shy and shy sets is non shy, we deduce that T −1 (θ) is a Borel measurable Haar ambivalent sets for each θ ∈ Θ.…”

Section: Auxiliary Notions and Facts From Functional Analysis And Meamentioning

confidence: 46%

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

Kintsurashvili

Kiria²,

Pantsulaia

2016

Applications of Measure Theory to Statistics

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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 non-locally-compact group admitting an invariant metric and relations between them are studied in this paper. An example of such 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 non-locally compact Polish group with an invariant metric when subjective one exists" is answered positively when there exists at least one such a parameter the pre-image of which under this subjective estimate is a prevalent. These results extend recent results of authors. Some examples of objective and strong objective consistent estimates in a compact Polish group {0; 1} N are considered in this paper.Primary 62-02; secondary 62D05.

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Section: Introductioncontrasting

confidence: 43%

Section: Auxiliary Notions and Facts From Functional Analysis And Meamentioning

confidence: 43%

Section: Auxiliary Notions and Facts From Functional Analysis And Meamentioning

confidence: 46%

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

Kintsurashvili

Kiria²,

Pantsulaia

2016

Applications of Measure Theory to Statistics

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“…In psychology, practically all null hypotheses are claimed to be false for sufficiently large samples so (...) it is nonsensical to perform an experiment with the sole aim of rejecting the null hypothesis''. According to Pantsulaia and Kintsurashvili (2014) the same concept is expressed by more than 200 authors. Clearly, to go beyond such a limitation of two-sided tests, this suggests considering a null hypothesis as made up of an interval of substantially equivalent points, rather than only one point.…”

Section: On Intersection-union and Union-intersection Approachesmentioning

confidence: 99%

A unified approach to permutation testing for equivalence

2020

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The notion of testing for equivalence of two treatments is widely used in clinical trials, pharmaceutical experiments, bioequivalence and quality control. It is traditionally operated within the intersection–union principle (IU). According to this principle the null hypothesis is stated as the set of effects the differences $$\delta$$ δ of which lie outside a suitable equivalence interval and the alternative as the set of $$\delta$$ δ that lie inside it. In the literature related solutions are essentially based on likelihood techniques, which in turn are rather difficult to deal with. A recently published paper goes beyond most of likelihood limitations by using the IU approach within the permutation theory. One more paper, based on Roy’s union–intersection principle (UI) within the permutation theory, goes beyond some limitations of traditional two-sided tests. Such UI approach, effectively a mirror image of IU, assumes a null hypothesis where $$\delta$$ δ lies inside the equivalence interval and an alternative where it lies outside. Since testing for equivalence can rationally be analyzed by both principles but, as the two differ in terms of the mirror-like roles assigned to the hypotheses under study, they are not strictly comparable. The present paper’s main goal is to look into these problems and provide a sort of comparative analysis of both by highlighting the related requirements, properties, limitations, difficulties, and pitfalls so as to get practitioners properly acquainted with their correct use in practical contexts.

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“…In [18], the examples of objective and strong objective infinite sample consistent estimates ([18],  T (p. 63), T (p. 67)) of a useful signal in the linear one-dimensional stochastic model were constructed by using the axiom of choice and a certain partition of the non-locally compact abelian…”

Section: Sample Consistent Estimate Of a Useful Signal In The Linear mentioning

confidence: 99%

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

2015

Jour. Stat. Adv. Theory and Appl.

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By using the notion of a Haar ambivalent set introduced by Balka et al. [1], 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 paper. An example of such 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 non-locally compact Polish group with an invariant metric when subjective one R ∈ θ 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).

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Why Null Hypothesis Is Rejected for Almost Every Infinite Sample by the Hypothesis Testing of a Maximal Reliability

Cited by 4 publications

References 7 publications

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

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

A unified approach to permutation testing for equivalence

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

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