IMS

Session Slot: 2:00- 3:50 Wednesday

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*Session Title: Advances in Hypothesis Testing
*

Theme Session: No

Applied Session: Yes

Session Organizer: **Brown, Larry**
University of Pennsylvania

Address: Statistics Department, University of Pennsylvania, Philadelphia, PA, 19104

Phone: 215-898-4753

Fax: 215-898-1280

Email: lbrown@compstat.wharton.upenn.edu

Session Timing: 110 minutes total (Sorry about format):

Session Chair: **Brown, Larry**
University of Pennsylvania

Address: Statistics Department, University of Pennsylvania, Philadelphia, PA, 19104

Phone: 215-898-4753

Fax: 215-898-1280

Email: lbrown@compstat.wharton.upenn.edu

*1. On Testing of Individual Bioequivalence*

**Wang, Weizhen**,
Wright State University

Address:

Phone:

Fax:

Email: wwang@euler.math.wright.edu

Abstract: The hypothesis of individual bioequivalence typically involves several parameters, such as the mean and variance of two populations: the generic drug and the brand name drug. The hypothesis spaces may be complicated regions in a plane or even in a 3-dimensional space. How can we construct tests involving these kinds of hypothesis regions when the data are normally distributed? In this talk, a reparametrization is introduced through a linear regression approach. A general class of hypotheses, which includes individual bioequivalence, is discussed and the exact alpha-level tests are proposed. When there is no interaction between formulation and subject, a two-by-two crossover design is sufficient to assess individual bioequivalence, while a two-by-three crossover design should be used if interaction is present.

Key words: Linear regression; Noncentral t-distribution; Power; Reparametrization.

*2. Mode Testing in Difficult Cases*

**Cheng, Ming-Yen**,
National ChungCheng University, (Taiwan)

Address: Institute of Mathematical Statistics, National Chung Cheng University, Taiwan

Phone:

Fax:

Email: mycheng@math.ccu.edu.tw

**Hall, Peter**,
Australian National University, Australia

Address: Centre for Mathematics and its Applications, Australian National University, Australia

Abstract: Usually, when testing the null hypothesis that a distribution has one mode, against the alternative that it has two, the null hypothesis is interpreted as entailing that the density of the distribution has a unique point of zero slope, which is a local maximum. We argue that a more appropriate null hypothesis is that the density has two points of zero slope, of which one is a local maximum and the other is a shoulder. The choice of null hypothesis is important when calibrating the test, and there are at least two reasons for favouring the mode-with-shoulder form. First, a unimodal density with a shoulder is the more difficult case to distinguish from bimodality, and in general a test should be constructed in such a way that it works best in difficult cases. (Here, we use the word ``difficult'' to refer to the difficulty of distinguishing between null and alternative hypotheses.) Secondly, when a test for a mode-with-shoulder is properly calibrated, so that it has asymptotically correct level, it is generally conservative when applied to the case of a mode-without-shoulder. Doing things the other way around would produce a liberal, rather than conservative, test in the important, difficult case of a mode with shoulder. The calibration method, for the bandwidth and excess mass tests, that we suggest involves the bootstrap. In contrast to other approaches it has very good adaptivity properties.

*3. Optimal nonparametric testing of qualitative
assumptions such as monotonicity or convexity*

**Duembgen, Lutz**,
Medical University of Luebeck (Germany)

Address: Medizinische Universitat zu Luebeck, Institut fur Mathematik

Phone:

Fax:

Email: duembgen@bilbo.math.mu-luebeck.de

Abstract: Suppose one observes a processYon the unit interval, wheredV=f_{o}+n^{-1/2}d Wwith an unknown function parameterf_{o}, given scale parameter (sample size) and standard Brownian motionW. We propose two types of tests of qualitative nonparametric hypotheses aboutf_{o}such as monotonicity or convexity. These tests are asymptotically optimal and adaptive with respect to two different criteria. As a by-product we obtain an extension of Lévy's modulus of continuity of Brownian motion which is of independent interest. It has potential applications to simultaneous confidence intervals in nonparametric curve estimation. (L. Duembgen and V. G. Spokoiny)

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