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Self-Normalized Sums and Directional Conclusions
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2012. , 38 p.
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: diva2:459190
Public defence
2012-01-20, Polhemsalen, Ångström Laboratory, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2011-12-14 Created: 2011-11-25 Last updated: 2011-12-14Bibliographically approved
List of papers
1. On the heavy-tailedness of Student's t-statistic
Open this publication in new window or tab >>On the heavy-tailedness of Student's t-statistic
2011 (English)In: Bernoulli, ISSN 1350-7265, E-ISSN 1573-9759, Vol. 17, no 1, 276-289 p.Article in journal (Refereed) Published
Abstract [en]

Let {X-i}(i >= 1) be an i.i.d. sequence of random variables and define, for n >= 2, T-n={(n-1/2 (sigma) over capn-1Sn)(0), ((sigma) over capn > 0,)((sigma) over capn=0,) with S-n = Sigma(n)(i=1) X-i, (sigma) over cap (2)(n)=1/n-1 Sigma(n)(i=1)(X-i - n(-1)S(n))(2). We investigate the connection between the distribution of an observation X-i and finiteness of E vertical bar T-n vertical bar(r) for (n, r) is an element of N->= 2 x R+. Moreover, assuming T-n ->(d) T, we prove that for any r > 0, lim(n ->infinity) E vertical bar T-n vertical bar(r) = E vertical bar T vertical bar(r) < infinity, provided there is an integer n(0) such that E vertical bar T-n0 vertical bar(r) is finite.

Keyword
finiteness of moments, robustness, Student's t-statistic, t-distributions, t-test
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-149782 (URN)10.3150/10-BEJ262 (DOI)000288042800013 ()
Available from: 2011-03-23 Created: 2011-03-23 Last updated: 2017-12-11Bibliographically approved
2. On the quadratic moment of self-normalized sums
Open this publication in new window or tab >>On the quadratic moment of self-normalized sums
2010 (English)In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 80, no 17-18, 1289-1296 p.Article in journal (Refereed) Published
Abstract [en]

Let an integer n >= 2 and a vector of independent, identically distributed random variables X-1, ..., X-n be given with P(X = 0) = 0 and define the self-normalized sum Z(n) = (Sigma(n)(i=1) X-i)/(Sigma(n)(i=1) X-i(2))(1/2). With a formula for EZ(n)(2) we prove that EZ(n)(2) >= 1 and that EZ(n)(2) = 1 if and only if the summands are symmetrically distributed. We also construct examples where Z(n) converges to the standard normal distribution as n tends to infinity while EZ(n)(2) tends to infinity (the distribution of the summands varies with n).

Keyword
Quadratic moment, Self-normalization, Symmetric distributions, Student's t-test
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-135426 (URN)10.1016/j.spl.2010.04.008 (DOI)000280916100004 ()
Available from: 2010-12-07 Created: 2010-12-06 Last updated: 2017-12-11Bibliographically approved
3. On the tails of linear combinations of Rademacher random variables through exponential tilting
Open this publication in new window or tab >>On the tails of linear combinations of Rademacher random variables through exponential tilting
2011 (English)Report (Other academic)
Place, publisher, year, edition, pages
Uppsala: , 2011. 22 p.
Series
U.U.D.M. report / Uppsala University, Department of Mathematics, ISSN 1101-3591 ; 2011:16
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:uu:diva-158199 (URN)
Available from: 2011-09-01 Created: 2011-09-01 Last updated: 2011-11-10Bibliographically approved
4. Characterizing optimality among three-decision procedures for directional conclusions
Open this publication in new window or tab >>Characterizing optimality among three-decision procedures for directional conclusions
2011 (English)Report (Other academic)
Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, Uppsala University, 2011. 13 p.
Series
U.U.D.M. report / Uppsala University, Department of Mathematics, ISSN 1101-3591 ; 2011:21
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:uu:diva-161739 (URN)
Available from: 2011-11-17 Created: 2011-11-17 Last updated: 2011-11-17Bibliographically approved
5. Applying the Benjamini–Hochberg procedure to a set of generalized p-values
Open this publication in new window or tab >>Applying the Benjamini–Hochberg procedure to a set of generalized p-values
2011 (English)Report (Other academic)
Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, Uppsala University, 2011. 14 p.
Series
U.U.D.M. report / Uppsala University, Department of Mathematics, ISSN 1101-3591 ; 2011:22
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:uu:diva-161740 (URN)
Available from: 2011-11-17 Created: 2011-11-17 Last updated: 2011-11-23Bibliographically approved

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