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The University of Sydney
School of Mathematics and Statistics Assignment 1
MATH1905: Statistics (Advanced) Semester 2, 2016 Web Page: http://sydney.edu.au/science/maths/MATH1905/
Lecturer: Michael Stewart This assignment is due by 11:59pm Monday 15th August 2016, via Turnitin. A
PDF copy of your answers must be uploaded in the Learning Management System
(Blackboard) at https://elearning.sydney.edu.au. Please submit only a PDF
document (scan or convert other formats). It should include your name and SID;
your tutorial time, day, room and Tutor’s name. It is your responsibility to preview
each page of your assignment after uploading to ensure each page is included in correct
order and is legible (not sideways or upside down) before confirming your submission,
and then to check your submission receipt. The School of Mathematics and Statistics encourages some collaboration between students when working on problems, but
students must write up and submit their own version of the solutions. This assignment is worth 5% of your final assessment for this course. Your answers should be
well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any
resources used and show all working. Present your arguments clearly using words of explanation
and diagrams where relevant. After all, mathematics is about communicating your ideas. This
is a worthwhile skill which takes time and e↵ort to master. The marker will give you feedback
and allocate an overall letter grade and mark to your assignment using the following criteria:
Mark
10 Grade
A+ Criterion
Outstanding and scholarly work, answering all parts of all questions correctly, with clear accurate explanations and all relevant diagrams and working. There are at most only minor or trivial errors or omissions. 9 A Very good work, making excellent progress on both questions, but with
one or two substantial errors, misunderstandings or omissions throughout
the assignment. 7 B Good work, making good progress on 1 question and moderate progress
on the other, but making more than two distinct substantial errors, misunderstandings or omissions throughout the assignment. 6 C A reasonable attempt, making moderate progress on both questions. 4 D Some attempt, with moderate progress made on only 1 question. 2 E Some attempt, with minimal progress made on only 1 question. 0 F No credit awarded. Copyright c 2016 The University of Sydney 1 This assignment explores multiple least-squares regression. Elementary calculus
and linear algebra are needed to answer these two questions; seek assistance from a
lecturer or tutor if you need help. The two parts marked with an asterisk⇤ are quite
challenging, don’t feel bad if you find them difficult! Suppose that on each of n individuals we have 3 measurements giving ordered triples
(w1 , x1 , y1 ), . . . , (wn , xn , yn ).
1. Suppose it is desired to find constants a and b such that we may express each yi via
yi = awi + bxi + "i
where the "i ’s resemble “random errors”. It is proposed to choose a and b using the
method of least squares, that is to minimise the function
S1 (a, b) = n
X [yi (awi + bxi )]2 i=1 with respect to a and b. It suffices to solve the pair of equations
@S1
=0
@a
@S1
=0
@b
so long as a unique solution exists. Write ⌃wx = (1)
(2)
Pn i=1 wi xi , ⌃xx = Pn i=1 x2i , etc. . (a) Determine both partial derivatives and write the equations (1) and (2) in matrix
form, i.e.
✓ ◆
a
M
=v
b
for a 2-by-2 matrix M and a column vector v, writing M and v in terms of ⌃wx ,
⌃xx , etc. .
(b) Write an inequality involving ⌃wx , ⌃xx , etc. which holds if and only if the determinant of M is positive.
(c) Assuming the determinant is positive, by inverting M solve the equations and
express the least squares solutions a and b in terms of ⌃wx , ⌃xx , etc. .
(d) Show that in the special case where w1 = w2 = · · · = wn = 1, the expressions for
the solutions in the previous part reduce to
(i) b = Sxy /Sxx and
*(ii) a = y¯ b¯
x P
P
P
where as usual
Sxy = ni=1 (yi y¯)(xi x¯), Sxx = ni=1 (xi x¯)2 , y¯ = n1 ni=1 yi
P
and x¯ = n1 ni=1 xi (hint: recall the computing formulae for Sxy and Sxx ). 2 2. Suppose now that we wish to include an intercept, that is choose constants a, b and c so
that with
yi = awi + bxi + c + "i ,
the resultant "i ’s resemble “random errors”. This may be desirable if the “no intercept”
version in question 1 did not give a good fit. It is proposed to again use the method of
least squares, that is to choose a, b and c minimising
S2 (a, b, c) = n
X [yi (awi + bxi + c)]2 . i=1 (a) Use calculus to show that the value of c which minimises S2 (a, b, c) when a and b
are held fixed is c = y¯ aw¯ b¯
x and thus that this problem reduces to minimising
S3 (a, b) = n
X
i=1 {(yi y¯) [a(wi w)
¯ + b(xi x¯)]}2 over a and b which is mathematically equivalent to the problem in question 1.
*(b) The solution to this last minimisation problem is the same as that for question 1
after replacing ⌃wx , ⌃xx , etc. with Swx , Sxx , etc. respectively (Swx , Sww etc. are
defined in the same way as Sxy , Sxx ). A unique solution exists if and only if the
inequality analogous to that derived in question 1 part (b) above holds.
Derive three equivalent conditions:
• one only involving sw , the standard deviation of the wi ’s,
• one only involving sx , the standard deviation of the xi ’s and
• one only involving rwx , the correlation between them which all hold if and only if a unique solution exists. 3