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                Department of Mathematical Sciences                       
                   The Johns Hopkins University                           
                                                                        
                              SEMINAR                                
                                                                        
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Michael Woodroofe                                 March 29, 2001
Department of Statistics                          304 Whitehead Hall
University of Michigan                            Preseminar: 3:00 p.m.
                                                  Refreshments: 3:30 p.m.
                                                  Seminar: 4:00 p.m.

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                        ISOTONIC REGRESSION:
              ANOTHER LOOK AT THE CHANGE POINT PROBLEM

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                             ABSTRACT

In simple versions of the change point problem, independent random
variables  X_k  have one (marginal) distribution  F_0, say, for  k < \nu,
and another  F_1  for  k \geq \nu, where the change point  \nu  is an
unknown parameter and  1 \leq \nu \leq \infty.  Statistical questions
include testing for the existence of a change, and estimating the location
of  \nu  when it exists.  The problem arises in industrial quality
assessment but also more generally -- for example, in assessing changes in
weather patterns and disease rates.  In this talk, I will explore a
modified version of the change point problem in which the abrupt change is
replaced by a montonic, but otherwise arbitrary, sequence of changes.  In
the simplest case, suppose that  X_k = \mu_k + \epsilon_k, where  \mu_1,
..., \mu_n  are the unknown parameters and  \epsilon_1, ..., \epsilon_n
are independent normal random variables with a common variance  \sigma^2.
Suppose that  \mu_1 \leq ... \leq \mu_n  and consider testing the
hypothesis  H_0: \mu_1 = ... = \mu_n.  This is the same null hypothesis
encountered in the change point problem, but the alternative is different.
A penalized likelihood ratio test of  H_0  is developed and its asymptotic
distribution is obtained.  The asymptotic distribution is obtained under
the more general assumption that  \epsilon_1, ..., \epsilon_n  are part of
a zero-mean, square-integrable, stationary ergodic process that exhibits
suitable short-range dependence.  The test is illustrated by rainfall data
from the Tucuman Region of Argentina. 

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