754 research outputs found
Optimal testing of equivalence hypotheses
In this paper we consider the construction of optimal tests of equivalence
hypotheses. Specifically, assume X_1,..., X_n are i.i.d. with distribution
P_{\theta}, with \theta \in R^k. Let g(\theta) be some real-valued parameter of
interest. The null hypothesis asserts g(\theta)\notin (a,b) versus the
alternative g(\theta)\in (a,b). For example, such hypotheses occur in
bioequivalence studies where one may wish to show two drugs, a brand name and a
proposed generic version, have the same therapeutic effect. Little optimal
theory is available for such testing problems, and it is the purpose of this
paper to provide an asymptotic optimality theory. Thus, we provide asymptotic
upper bounds for what is achievable, as well as asymptotically uniformly most
powerful test constructions that attain the bounds. The asymptotic theory is
based on Le Cam's notion of asymptotically normal experiments. In order to
approximate a general problem by a limiting normal problem, a UMP equivalence
test is obtained for testing the mean of a multivariate normal mean.Comment: Published at http://dx.doi.org/10.1214/009053605000000048 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Explicit nonparametric confidence intervals for the variance with guaranteed coverage
In this paper, we provide a method for constructing confidence intervals for the variance that exhibit guaranteed coverage probability for any sample size, uniformly over a wide class of probability distributions. In contrast, standard methods achieve guaranteed coverage only in the limit for a fixed distribution or for any sample size over a very restrictive (parametric) class of probability distributions. Of course, it is impossible to construct effective confidence intervals for the variance without some restriction, due to a result of Bahadur and Savage (1956). However, it is possible if the observations lie in a fixed compact set. We also consider the case of lower confidence bounds without any support restriction. Our method is based on the behavior of the variance over distributions that lie within a Kolmogorov-Smirnov confidence band for the underlying distribution. The method is a generalization of an idea of Anderson (1967), who considered only the case of the mean; it applies to very general parameters, and particularly the variance. While typically it is not clear how to compute these intervals explicitly, for the special case of the variance we provide an algorithm to do so. Asymptotically, the length of the intervals is of order n -1/2 in probability), so that, while providing guaranteed coverage, they are not overly conservative. A small simulation study examines the finite sample behavior of the proposed intervals
On stepdown control of the false discovery proportion
Consider the problem of testing multiple null hypotheses. A classical
approach to dealing with the multiplicity problem is to restrict attention to
procedures that control the familywise error rate (), the probability of
even one false rejection. However, if is large, control of the is so
stringent that the ability of a procedure which controls the to detect
false null hypotheses is limited. Consequently, it is desirable to consider
other measures of error control. We will consider methods based on control of
the false discovery proportion () defined by the number of false
rejections divided by the total number of rejections (defined to be 0 if there
are no rejections). The false discovery rate proposed by Benjamini and Hochberg
(1995) controls . Here, we construct methods such that, for any
and , . Based on -values of
individual tests, we consider stepdown procedures that control the ,
without imposing dependence assumptions on the joint distribution of the
-values. A greatly improved version of a method given in Lehmann and Romano
\citer10 is derived and generalized to provide a means by which any sequence of
nondecreasing constants can be rescaled to ensure control of the . We also
provide a stepdown procedure that controls the under a dependence
assumption.Comment: Published at http://dx.doi.org/10.1214/074921706000000383 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stepup procedures for control of generalizations of the familywise error rate
Consider the multiple testing problem of testing null hypotheses
. A classical approach to dealing with the multiplicity problem is
to restrict attention to procedures that control the familywise error rate
(), the probability of even one false rejection. But if is
large, control of the is so stringent that the ability of a
procedure that controls the to detect false null hypotheses is
limited. It is therefore desirable to consider other measures of error control.
This article considers two generalizations of the . The first is
the , in which one is willing to tolerate or more false
rejections for some fixed . The second is based on the false discovery
proportion (), defined to be the number of false rejections
divided by the total number of rejections (and defined to be 0 if there are no
rejections). Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995)
289--300] proposed control of the false discovery rate (), by
which they meant that, for fixed , . Here,
we consider control of the in the sense that, for fixed
and , . Beginning with any
nondecreasing sequence of constants and -values for the individual tests, we
derive stepup procedures that control each of these two measures of error
control without imposing any assumptions on the dependence structure of the
-values. We use our results to point out a few interesting connections with
some closely related stepdown procedures. We then compare and contrast two
-controlling procedures obtained using our results with the
stepup procedure for control of the of Benjamini and Yekutieli
[Ann. Statist. 29 (2001) 1165--1188].Comment: Published at http://dx.doi.org/10.1214/009053606000000461 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Generalizations of the Familywise Error Rate
Consider the problem of simultaneously testing null hypotheses H_1,...,H_s.
The usual approach to dealing with the multiplicity problem is to restrict
attention to procedures that control the familywise error rate (FWER), the
probability of even one false rejection. In many applications, particularly if
s is large, one might be willing to tolerate more than one false rejection
provided the number of such cases is controlled, thereby increasing the ability
of the procedure to detect false null hypotheses. This suggests replacing
control of the FWER by controlling the probability of k or more false
rejections, which we call the k-FWER. We derive both single-step and stepdown
procedures that control the k-FWER, without making any assumptions concerning
the dependence structure of the p-values of the individual tests. In
particular, we derive a stepdown procedure that is quite simple to apply, and
prove that it cannot be improved without violation of control of the k-FWER. We
also consider the false discovery proportion (FDP) defined by the number of
false rejections divided by the total number of rejections (defined to be 0 if
there are no rejections). The false discovery rate proposed by Benjamini and
Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] controls E(FDP).
Here, we construct methods such that, for any \gamma and \alpha,
P{FDP>\gamma}\le\alpha. Two stepdown methods are proposed. The first holds
under mild conditions on the dependence structure of p-values, while the second
is more conservative but holds without any dependence assumptions.Comment: Published at http://dx.doi.org/10.1214/009053605000000084 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the uniform asymptotic validity of subsampling and the bootstrap
This paper provides conditions under which subsampling and the bootstrap can
be used to construct estimators of the quantiles of the distribution of a root
that behave well uniformly over a large class of distributions .
These results are then applied (i) to construct confidence regions that behave
well uniformly over in the sense that the coverage probability
tends to at least the nominal level uniformly over and (ii) to
construct tests that behave well uniformly over in the sense that
the size tends to no greater than the nominal level uniformly over
. Without these stronger notions of convergence, the asymptotic
approximations to the coverage probability or size may be poor, even in very
large samples. Specific applications include the multivariate mean, testing
moment inequalities, multiple testing, the empirical process and U-statistics.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1051 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
EXPLICIT NONPARAMETRIC CONFIDENCE INTERVALS FOR THE VARIANCE WITH GUARANTEED COVERAGE
In this paper, we provide a method for constructing confidence intervals for the variance that exhibit guaranteed coverage probability for any sample size, uniformly over a wide class of probability distributions. In contrast, standard methods achieve guaranteed coverage only in the limit for a fixed distribution or for any sample size over a very restrictive (parametric) class of probability distributions. Of course, it is impossible to construct effective confidence intervals for the variance without some restriction, due to a result of Bahadur and Savage (1956). However, it is possible if the observations lie in a fixed compact set. We also consider the case of lower confidence bounds without any support restriction. Our method is based on the behavior of the variance over distributions that lie within a Kolmogorov-Smirnov confidence band for the underlying distribution. The method is a generalization of an idea of Anderson (1967), who considered only the case of the mean; it applies to very general parameters, and particularly the variance. While typically it is not clear how to compute these intervals explicitly, for the special case of the variance we provide an algorithm to do so. Asymptotically, the length of the intervals is of order n -1/2 in probability), so that, while providing guaranteed coverage, they are not overly conservative. A small simulation study examines the finite sample behavior of the proposed intervals.
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