HW 19 Stat 705  Fall 2017
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Assigned 11/28/2017, due 12/6                                 16 points 
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Assigned Tuesday 11/28/17 DUE Wednesday 12/6/17 
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Read in the dataset "anorexia" from the MASS library datasets, containing 
experimental treatment labels and pre- and post-test weights for 72 anorexic women.

(A) Plot a scatterplot of the Postwt versus Prewt for these data, using 
a different plotting character for each of the three  Treat  groups. Provide axis 
labels, a graph title, and a legend for your plotting characters.

(B) On the same graph axes, plot fits of the following curves 
y = g(x) = E(Postwt | Prewt=x)    to these data:

   (i) a polynomial regression of degree 3 (fitted by ordinary or weighted least squares),
   (ii) a cubic smoothing spline with all points as knots,
   (iii) a local polynomial regression of degree 2.

You may use the defaults in the R functions you call to do the fitting, but you should 
say what they are on these data. Also, you will have to select reasonable bandwidths 
in (ii), (iii).

Find a way to display the raw scatterplot along with the 4 fitted curves in one or two 
pictures, with appropriate labeling and legends for the nonparametric regressions.

Hand in the R code for your curve fits and the pictures.

(C) Consider the density function  f(t) = C t/(1+t^3+t^5)  for t > 0.
Code a function  Finv(x)  closely approximating the inverse of the cumulative 
distribution function for this density by the following steps.

(i) For the points  pts[i] = 1/(1-i/5001)^0.25-1, i=1,...,5000, evaluate the 
distribution function  cdf[i] = F(pts[i]) using numerical integration to accuracy 
at least 1.e-6.

(ii) Fit a smoothing spline to the points (cdf[i], pts[i]), i=1,...,5000, with high 
smoothing parameter (say, spar=0.5 or 0.6), and code Finv(x) as the (vectorized) 
function on new data x  on the interval (0,cdf[5000]) obtained by using  predict  
on the fitted smooth.spline object.

(iii) Find a way to estimate the accuracy of Finv as an approximation of the inverse 
distribution function.