Splus Homework Set 7, Due Wednesday May 14, 2003
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Density Estimation. Download the artificially generated dataset
HW7data.gz which you can find in the Data link on the course web-page:
it  consists of 500 scalar values. The homework assignment is to fit
density estimates and find data summaries by several specific methods:

(a) Fit the density by a kernel-density method, explaining the choice
of kernel-function and bandwidth parameter. (If you use the Splus
default density estimate, make sure to tell what the kernel function
and (optimized) bandwith parameter are.

(b) Find and plot a density-estimator for the same data, by applying a
smoothing-spline method (with higer-order smoothness) to the empirical 
distribution function.

(c) Find and plot a density-estimator for the same data, by fitting
via Maximumn Likelihood a mixture of 2 or 3 normals.

(d) Use each of the fitted density functions to estimate the upper
quartile for the distribution.

(e) Compare your estimates of the upper quartile with a bootrap
estimate. Find a `theoretical' empirical confidence interval for the
upper quartile, and find a corresponding Bootstrap estimated of
confidence interval.