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Self-Normalized Sums and Directional ConclusionsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2012 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics , 2012. , p. 38
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 75
##### Keyword [en]

Self-normalized sums, heavy-tailedness, Student's t-statistic, distributional symmetry, exponential tilting, directional conclusions, reversal rates, multiple statistical inference, the Benjamini-Hochberg procedure
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-162168ISBN: 978-91-506-2260-7 (print)OAI: oai:DiVA.org:uu-162168DiVA, id: diva2:459190
##### Public defence

2012-01-20, Polhemsalen, Ångström Laboratory, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Opponent

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##### Supervisors

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#####

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Available from: 2011-12-14 Created: 2011-11-25 Last updated: 2011-12-14Bibliographically approved
##### List of papers

This thesis consists of a summary and five papers, dealing with self-normalized sums of independent, identically distributed random variables, and three-decision procedures for directional conclusions.

In Paper I, we investigate a general set-up for Student's t-statistic. Finiteness of absolute moments is related to the corresponding degree of freedom, and relevant properties of the underlying distribution, assuming independent, identically distributed random variables.

In Paper II, we investigate a certain kind of self-normalized sums. We show that the corresponding quadratic moments are greater than or equal to one, with equality if and only if the underlying distribution is symmetrically distributed around the origin.

In Paper III, we study linear combinations of independent Rademacher random variables. A family of universal bounds on the corresponding tail probabilities is derived through the technique known as exponential tilting. Connections to self-normalized sums of symmetrically distributed random variables are given.

In Paper IV, we consider a general formulation of three-decision procedures for directional conclusions. We introduce three kinds of optimality characterizations, and formulate corresponding sufficiency conditions. These conditions are applied to exponential families of distributions.

In Paper V, we investigate the Benjamini-Hochberg procedure as a means of confirming a selection of statistical decisions on the basis of a corresponding set of generalized p-values. Assuming independence, we show that control is imposed on the expected average loss among confirmed decisions. Connections to directional conclusions are given.

1. On the heavy-tailedness of Student's t-statistic$(function(){PrimeFaces.cw("OverlayPanel","overlay405660",{id:"formSmash:j_idt480:0:j_idt484",widgetVar:"overlay405660",target:"formSmash:j_idt480:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. On the quadratic moment of self-normalized sums$(function(){PrimeFaces.cw("OverlayPanel","overlay375021",{id:"formSmash:j_idt480:1:j_idt484",widgetVar:"overlay375021",target:"formSmash:j_idt480:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. On the tails of linear combinations of Rademacher random variables through exponential tilting$(function(){PrimeFaces.cw("OverlayPanel","overlay438136",{id:"formSmash:j_idt480:2:j_idt484",widgetVar:"overlay438136",target:"formSmash:j_idt480:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Characterizing optimality among three-decision procedures for directional conclusions$(function(){PrimeFaces.cw("OverlayPanel","overlay457260",{id:"formSmash:j_idt480:3:j_idt484",widgetVar:"overlay457260",target:"formSmash:j_idt480:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Applying the Benjamini–Hochberg procedure to a set of generalized *p*-values$(function(){PrimeFaces.cw("OverlayPanel","overlay457264",{id:"formSmash:j_idt480:4:j_idt484",widgetVar:"overlay457264",target:"formSmash:j_idt480:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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