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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, November 14, 2000}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Hidetoshi Shimodaira}
\centerline{\sl The Institute of Statistical Mathematics, Tokyo}
\centerline{\sl visiting at Department of Statistics, Stanford University}
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\centerline{\bf Another calculation of the p-value for the problem}
\centerline{\bf of regions using the scaled bootstrap resamplings}
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An approximately unbiased test is considered for the null hypothesis
represented as a region with smooth boundaries.  This problem is discussed
previously in Efron and Tibshirani (1998), and our argument is based on
their results. We give another calculation of the corrected p-value without
finding the ``nearest point'' on the boundary to the observation, which is
required in the calculation of Efron, Halloran and Holmes (1996).  We
generate sets of bootstrap replicates with several sample sizes which may
differ from that of the observed data. For each set of replicates, the
frequency that the replicates fall in the region is counted. Only these
frequencies are used to estimate the signed distance and the curvature of
the boundary in the calculation of the p-value. Because of the simplicity
of the algorithm, it will be useful for applications with correlated data
structure where complicated bootstrap methods are used.  At the same time,
our calculation is asymptotically third-order accurate, while the method of
Efron et al. (1996) is second-order accurate and the naive p-value is only
first-order accurate. The method also gives third-order accurate confidence
intervals of a real parameter by inverting the significance tests.

Keywords: problem of regions; confidence interval; curvature; similar on
the boundary; unbiased test; scaled bootstrap; third-order accuracy


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