Statistics 4M03/6M03 Assignment #1

Statistics 4M03/6M03 - Assignment #1

2003-10-12

Due: 2003-10-30 17:00

Part A

Problems from Srivastava, Methods of Multivariate Statistics.

2.9.9, 2.9.16, 2.9.20, 2.9.27, 2.9.28, 2.9.29, 2.9.33, 2.9.34

Part B

Draw a perspective plot and a contour plot for the bivariate distribution

f(x, y) = x + y, 0 < x < 1, 0 < y < 1, and 0 elsewhere.

Derive and graph the marginal and conditional distributions. Find the mean vector and the covariance matrix. Find the regression of x on y and the regression of y on x and add these lines to the contour plot.

Consider a bivariate normal distribution having the same first and second moments as the distribution above. Add its 95% ellipse of concentration and both regression lines to the contour plot.

Part C

The "Mineral Content in Bones" data below show mineral content measured by photon absorptiometry in 6 different bones (dominant radius, radius, dominant humerus, humerus, dominant ulna, ulna) in a study involving 25 women. Give a scatterplot matrix and a stars plot. Assess the univariate normality of each variable with a normal probability plot, using qqnorm() and qqline(). Draw a 3-D frequency polygon for the bivariate distribution of the variables "ulna and "dominant ulna". Compute the sample mean, S and R arrays. Make a scatterplot of "ulna" versus "dominant ulna" with the bivariate normal 50%, 90% and 95% ellipses of concentration superimposed. Explain what you have learned from each plot. Submit your answers as a brief report, pasting the graphs and analyses into a word processor document and adding comments and discussion.

Part D

How many independent normal observations are required before the histogram looks like a normal distribution? Try hist(rnorm(n)) a few times with different values of n.

Write a function rnormp(n, p = 2, mu = rep(0, p), sigma = diag(p)) to generate n independent observations from a p-variate Normal distribution with mean vector mu and covariance matrix sigma. The function should return an n by p data frame. The function should work in both R and in Splus.

Use your function rnormp() with the function hist3d() developed in class to see how many independent bivariate normal observations are required before the 3-D frequency polygon looks like a bivariate normal distribution. Does your answer depend on the correlation coefficient?

Data for mineral content in bones

 raddom radius humdom humerus ulnadom   ulna
  1.103  1.052  2.139   2.238   0.873  0.872
  0.842  0.859  1.873   1.741   0.590  0.744
  0.925  0.873  1.887   1.809   0.767  0.713
  0.857  0.744  1.739   1.547   0.706  0.674
  0.795  0.809  1.734   1.715   0.549  0.654
  0.787  0.779  1.509   1.474   0.782  0.571
  0.933  0.880  1.695   1.656   0.737  0.803
  0.799  0.851  1.740   1.777   0.618  0.682
  0.945  0.876  1.811   1.759   0.853  0.777
  0.921  0.906  1.954   2.009   0.823  0.765
  0.792  0.825  1.624   1.657   0.686  0.668
  0.815  0.751  2.204   1.846   0.678  0.546
  0.755  0.724  1.508   1.458   0.662  0.595
  0.880  0.866  1.786   1.811   0.810  0.819
  0.900  0.838  1.902   1.606   0.723  0.677
  0.764  0.757  1.743   1.794   0.586  0.541
  0.733  0.748  1.863   1.869   0.672  0.752
  0.932  0.898  2.028   2.032   0.836  0.805
  0.856  0.786  1.390   1.324   0.578  0.610
  0.890  0.950  2.187   2.087   0.758  0.718
  0.688  0.532  1.650   1.378   0.533  0.482
  0.940  0.850  2.334   2.225   0.757  0.731
  0.493  0.616  1.037   1.268   0.546  0.615
  0.835  0.752  1.509   1.422   0.618  0.664
  0.915  0.936  1.971   1.869   0.869  0.868