of School Mathematics MATH3713: Regression and Smoothing
Worksheet 3
Q1 For an vector where
and an symmetric matrix
Var
, prove the result
Hint: Show that the and elements of
where is the respectively, and express in terms of
,
tr
element of .
and
are
Q2 Express the residual sum of squares
as a quadratic form and deduce that it has degrees of freedom. Prove that
is an unbiased estimator of
. .
Q3 Consider the partitioned matrix and is of order Show that
. .
where
is of order
is of order ,
leads to prove that
and
is idempotent.
. Use these results to
Q4 For the multiple linear regression model
where we have
is of order
partitioned into . . . is of order has length and has length ,
Obtain an expression for and show that
is an unbiased estimator of
when
is true.
Q5 The table overleaf gives , the systolic blood pressure in mm Hg, , a of measure body size given by 100 , age , and smoking history, , where means non-smoker and means a current or previous smoker, for 32 white males from a local town. Carry out a regression analysis of the full three-variable model in R, and then investigate the possibility of tting a reduced model to the data. You can read in the data by typing:
smoke=read.table("http://www.maths.leeds.ac.uk/christin/math3713/smoke.dat",header=T)
The data are: > smoke blood 1 135 2 122 3 130 4 148 5 146 6 129 7 162 8 160 9 144 10 180 11 166 12 138 13 152 14 138 15 140 16 134 17 145 18 142 19 135 20 142 21 150 22 144 23 137 24 ...