MAST90104: A First Course in Statistical Learning
Assignment 1, 2018
Due: 5.30pm Friday August 10. Please submit a scanned or other electronic copy of
your work via the Learning Management System - see this link for instructions
This assignment is worth 5% of your total mark. You must ll in the online plagiarism declaration
form. prior to submitting your assignment
You may use R for this assignment, but for matrix calculations only (you may not use the lm function).
If you do, include your R commands and output.
1. Suppose that A is a symmetric matrix with Ak = Ak+1 for some integer k 1. Show that A is
idempotent.
2. Let A1;A2;:::;Am be a set of symmetric k k matrices. Suppose that there exists an orthogonal
matrix P such that PTAiP is diagonal for all i. Show that AiAj = AjAi for every pair i;j =
1;2;:::;m.
3. Show that for any random vector y and compatible matrix A, we have var Ay = A(var y)AT.
4. Let y be a 3-dimensional multivariate normal random vector with mean and variance
(a) Describe the distribution of Ay.
(b) Find E[yTAy].
(c) Describe the distribution of yTAy.
(d) Find all linear combinations of y elements which are independent of yTAy.
5. The table below shows prices in US cents per pound received by shermen and vessel owners for
various species of sh and shell sh in 1970 and 1980. (Taken from Moore McCabe, Introduction
to the Practice of Statistics, 1989.)
Type of sh Price (1970) Price (1980)
Cod 13.1 27.3
Flounder 15.3 42.4
Haddock 25.8 38.7
Menhaden 1.8 4.5
Ocean perch 4.9 23.0
Salmon, chinook 55.4 166.3
Salmon, coho 39.3 109.7
Tuna, albacore 26.7 80.1
Clams, soft-shelled 47.5 150.7
Clams, blue hard-shelled 6.6 20.3
Lobsters, american 94.7 189.7
Oysters, eastern 61.1 131.3
Sea scallops 135.6 404.2
Shrimp 47.6 149.0
1
We will model this data using a linear model.
(a) The linear model is of the form. y = X +". Write down the matrices and vectors involved
in this equation.
(b) Using matrices, nd the least squares estimators of the parameters.
(c) Calculate the sample variance s2.
(d) A sherman sold ocean trout for 18c/pound in 1970. Predict the price for ocean trout in 1980.
(e) Calculate the standardised residual for sea scallops.
(f) Calculate the Cook’s distance for sea scallops.
(g) Does sea scallops t the linear model? Justify your argument.