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On Confidence Intervals and Two-Sided Hypothesis Testing
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary and six papers, dealing with confidence intervals and two-sided tests of point-null hypotheses.

In Paper I, we study Bayesian point-null hypothesis tests based on credible sets. A decision-theoretic justification for tests based on central credible intervals is presented.

Paper II is concerned with a new two-sample test for the difference of mean vectors, in the high-dimensional setting where the number of variables is greater than the sample size. A simulation study indicates that the proposed test yields higher power when the variables are correlated. Computational aspects of the test are discussed.

In Paper III, we discuss randomized confidence intervals for a binomial proportion. How some classical intervals fare is compared to how a recently proposed interval fares, in terms of coverage, length and sensitivity to the randomization.

In Paper IV, a level-adjustment of the Clopper-Pearson interval for a binomial proportion is proposed. The adjusted interval is shown to have good coverage properties and short expected length.

In Paper V we study the cost of using the exact Clopper-Pearson interval rather than shorter approximate intervals, in terms of the increase in expected length and the increase in sample size required to obtain a given length. Comparisons are made using asymptotic expansions.

Paper VI deals with exact confidence intervals and point-null hypothesis tests for parameters of a class of discrete distributions. A large class of intervals are shown to lack strict nestedness and to have bounds that are not strictly monotone and typically also discontinuous. The p-values of the corresponding hypothesis test are shown to lack desirable continuity properties, and to typically also lack certain monotonicity properties.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2014. , 47 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 85
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:uu:diva-229399ISBN: 978-91-506-2408-3 (print)OAI: oai:DiVA.org:uu-229399DiVA: diva2:736874
Public defence
2014-09-26, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2014-09-04 Created: 2014-08-06 Last updated: 2014-09-04
List of papers
1. Decision-theoretic justifications for Bayesian hypothesis testing using credible sets
Open this publication in new window or tab >>Decision-theoretic justifications for Bayesian hypothesis testing using credible sets
2014 (English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 146, 133-138 p.Article in journal (Refereed) Published
Abstract [en]

In Bayesian statistics the precise point-null hypothesis theta=theta(0) can be tested by checking whether theta(0) is contained in a credible set. This permits testing of theta=theta(0) without having to put prior probabilities on the hypotheses. While such inversions of credible sets have a long history in Bayesian inference, they have been criticized for lacking decision-theoretic justification. We argue that these tests have many advantages over the standard Bayesian tests that use point-mass probabilities on the null hypothesis. We present a decision-theoretic justification for the inversion of central credible intervals, and in special case HPD sets, by studying a three-decision problem with directional conclusions. Interpreting the loss function used in the justification, we discuss when tests based on credible sets are applicable. We then give some justifications for using credible sets when testing composite hypotheses, showing that tests based on credible sets coincide with standard tests in this setting. 

Keyword
Bayesian inference, Credible set, Decision theory, Directional conclusion, Hypothesis testing, Three-decision problem
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-216025 (URN)10.1016/j.jspi.2013.09.014 (DOI)000328593200012 ()
Available from: 2014-01-20 Created: 2014-01-17 Last updated: 2017-12-06Bibliographically approved
2. A high-dimensional two-sample test for the mean using random subspaces
Open this publication in new window or tab >>A high-dimensional two-sample test for the mean using random subspaces
2014 (English)In: Computational Statistics & Data Analysis, ISSN 0167-9473, E-ISSN 1872-7352, Vol. 74, 26-38 p.Article in journal (Refereed) Published
Abstract [en]

A common problem in genetics is that of testing whether a set of highly dependent gene expressions differ between two populations, typically in a high-dimensional setting where the data dimension is larger than the sample size. Most high-dimensional tests for the equality of two mean vectors rely on naive diagonal or trace estimators of the covariance matrix, ignoring dependences between variables. A test using random subspaces is proposed, which offers higher power when the variables are dependent and is invariant under linear transformations of the marginal distributions. The p-values for the test are obtained using permutations. The test does not rely on assumptions about normality or the structure of the covariance matrix. It is shown by simulation that the new test has higher power than competing tests in realistic settings motivated by microarray gene expression data. Computational aspects of high-dimensional permutation tests are also discussed and an efficient R implementation of the proposed test is provided.

National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics; Statistics
Identifiers
urn:nbn:se:uu:diva-224136 (URN)10.1016/j.csda.2013.12.003 (DOI)000333781500003 ()
Available from: 2014-05-05 Created: 2014-05-05 Last updated: 2017-12-05Bibliographically approved
3. On split sample and randomized confidence intervals for binomial proportions
Open this publication in new window or tab >>On split sample and randomized confidence intervals for binomial proportions
2014 (English)In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 92, 65-71 p.Article in journal (Refereed) Published
Abstract [en]

We study randomized confidence intervals for binomial proportions, comparing coverage, length and the impact of the randomization. It is seen that the recently proposed split sample intervals can be improved upon in various ways. Criticisms of randomized intervals are discussed.

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-226502 (URN)10.1016/j.spl.2014.05.005 (DOI)000340313200010 ()
Available from: 2014-06-17 Created: 2014-06-17 Last updated: 2017-12-05Bibliographically approved
4. Coverage-adjusted confidence intervals for a binomial proportion
Open this publication in new window or tab >>Coverage-adjusted confidence intervals for a binomial proportion
2014 (English)In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 41, no 2, 291-300 p.Article in journal (Refereed) Published
Abstract [en]

We consider the classic problem of interval estimation of a proportion p based on binomial sampling. The ‘exact’ Clopper–Pearson confidence interval for p is known to be unnecessarily conservative. We propose coverage adjustments of the Clopper–Pearson interval that incorporate prior or posterior beliefs into the interval. Using heatmap-type plots for comparing confidence intervals, we show that the coverage-adjusted intervals have satisfying coverage and shorter expected lengths than competing intervals found in the literature.

National Category
Probability Theory and Statistics
Research subject
Statistics; Mathematical Statistics
Identifiers
urn:nbn:se:uu:diva-224408 (URN)10.1111/sjos.12021 (DOI)000335388400002 ()
Available from: 2014-05-12 Created: 2014-05-12 Last updated: 2017-12-05Bibliographically approved
5. The cost of using exact confidence intervals for a binomial proportion
Open this publication in new window or tab >>The cost of using exact confidence intervals for a binomial proportion
2014 (English)In: Electronic Journal of Statistics, ISSN 1935-7524, E-ISSN 1935-7524, Vol. 8, 817-840 p.Article in journal (Refereed) Published
Abstract [en]

When computing a confidence interval for a binomial proportion p one must choose between using an exact interval, which has a coverage probability of at least 1 a for all values of p, and a shorter approximate interval, which may have lower coverage for some p but that on average has coverage equal to 1 a. We investigate the cost of using the exact one and two-sided Clopper-Pearson confidence intervals rat her than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper-Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper-Pearson interval and Bayesian intervals based on noninformative priors.

Keyword
Asymptotic expansion, binomial distribution, confidence interval, expected length, sample size determination, proportion
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-226501 (URN)10.1214/14-EJS909 (DOI)000338327100001 ()
Available from: 2014-06-17 Created: 2014-06-17 Last updated: 2017-12-05Bibliographically approved
6. Exact confidence intervals and hypothesis tests for parameters of discrete distributions
Open this publication in new window or tab >>Exact confidence intervals and hypothesis tests for parameters of discrete distributions
2017 (English)In: Bernoulli, ISSN 1350-7265, E-ISSN 1573-9759, Vol. 23, no 1, 479-502 p.Article in journal (Refereed) Published
Abstract [en]

We study exact confidence intervals and two-sided hypothesis tests for univariate parameters of stochastically increasing discrete distributions, such as the binomial and Poisson distributions. It is shown that several popular methods for constructing short intervals lack strict nestedness, meaning that accepting a lower confidence level not always will lead to a shorter confidence interval. These intervals correspond to a class of tests that are shown to assign differing p-values to indistinguishable models. Finally, we show that among strictly nested intervals, fiducial intervals, including the Clopper-Pearson interval for a binomial proportion and the Garwood interval for a Poisson mean, are optimal.

Keyword
binomial distribution, confidence interval, expected length, fiducial interval, hypothesis test, Poisson distribution
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-229398 (URN)10.3150/15-BEJ750 (DOI)000389565500017 ()
Available from: 2014-08-06 Created: 2014-08-06 Last updated: 2017-12-05Bibliographically approved

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