STA 410/2102 - Topics and Study Questions for Test 1

A) Arithmetic on the computer. 

  The idea of floating-point arithmetic - representation in terms
  of mantissa and exponent.  Round-off error, underflow and overflow.
  The problems of saturation and of cancellation.  See Chapter 2 of
  the textbook.

  Exercise 1:  Suppose arithmetic is done with decimal floating-point
               numbers with two decimal digits in the mantissa and a
               two-digit exponent.  What will be the result of the 
               following operations?  (Here "e" means "times ten to the
               power".)

                     a) 0.46e10 + 0.36e9
                     b) (0.10e-60 * 0.10e-50) / (0.1e-40)
                     c) (0.99e0 + 0.3e-2) + 0.3e-2
                     d) 0.99e0 + (0.3e-2 + 0.3e-2)
                     e) (0.10e-2 + 0.16e-3) - (0.10e-2 + 0.14e-3)

B) Simulation and permutation tests.

   Generation of random variates.  Simulation to determine properties
   of statistical procedures (eg, correctness of the distribution of
   p-values under the null hypothesis, power of a test for an alternative).
   Permutation tests.

   Exercise 2:  Write a S/R function called my.rnorm that generates
                normal random variates, taking the same arguments as 
                the built-in rnorm (sample size, mean, standard deviation).
                This function should use the built-in runif and qnorm
                functions.  The arguments of qnorm are a vector of
                numbers between 0 and 1, a mean, and a standard deviation.
                It returns a vector of quantiles of the normal distribution
                with that mean and standard deviation, with the quantile
                positions being given by the first argument.

C) Least-squares regression by solving the normal equations.

   The Cholesky decomposition, forward substitution, and backward
   substitution.  See sections 3.1 to 3.3 of the text for this and
   for (D).

   Exercise 3:  Solve the following system of equation for z by hand,
                using the Cholesky decomposition and forward and
                backward substitution:

                    [ 4   2  2 ]        [ 10 ]
                    [          ]        [    ]
                    [ 2  10  1 ]  z  =  [  2 ]
                    [          ]        [    ]
                    [ 2   1  5 ]        [  1 ]
  
D) Least-squares regression using orthogonal transformations.

   Definition of an orthogonal transformation.  Effect of an orthogonal
   transformation on the regression problem.  Using orthogonal 
   transformations to produce an upper-triangular X matrix, and why this 
   helps.  Accomplishing this with Givens rotations and Householder
   reflections.  

   Exercise 4:  Find a Householder transformation which when applied to
                the vector [ 5 3 0 4 ]' will produce a vector in which
                the last two coordinates are zero, and the first coordinate
                is the same as before.  What is the full orthogonal matrix 
                corresponding to this transformation?