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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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\begin{center}
  4:15 p.m., Tuesday, February 14, 2006\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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\begin{center}
  \textsl{Yehua Li} \\
Department of Statistics\\
Texas A\&M University\\
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\begin{center}
  \textbf{Nonparametric Estimation of Correlation Functions \\
In Longitudinal And Spatial Data, With Application To \\
Colon Carcinogenesis Experiments}
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\noindent

In longitudinal and spatial studies,
observations at a particular time or location within one subject
could have complicated structures, e.g. vectors or even functions.
Measurements within the same subject usually demonstrate strong
correlations that are stationary in time or distance lags. The
times or locations of these data being sampled may not be
homogeneous. We propose a nonparametric estimator of the
correlation function in such data, using kernel methods. We show
that the proposed estimator has a pointwise asymptotic normal
distribution, when the number of subjects is fixed and the number
of vectors or functions within each subject goes to infinity.
Based on the asymptotic theory, we propose a weighted block
bootstrapping method in making inferences on the correlation
function, where the weights account for the inhomogeneity of the
distribution of the times or locations. The method is applied to a
data set from a colon carcinogenesis study, in which colonic
crypts were sampled from a piece of colon segment from each of the
12 rats in the experiment and the expression level of p27, an
important cell cycle protein, was then measured for each cell
within the sampled crypts. A simulation study is also provided to
illustrate the numerical performance of the proposed method.



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