MAST30025 Linear Statistical Models 2020
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Semester 1 Assessment, 2022
MAST30025 Linear Statistical Models
Question 1 (14 marks)
[3 marks] Let A be a symmetric matrix with eigenvalues all either 0 or 1. Show that A is idempotent.
[3 marks] Let B be a symmetric matrix with positive eigenvalues. Show that B is positive definite.
(c) [3 marks] Let y be a random vector with E[y] = µ and Var y = V , and let C be a matrix. Show directly that Var Cy = CVCT .
[2 marks] Let D be a matrix. Show directly that r(Dc D) = r(D).
[3 marks] Find a conditional inverse of
E = ┌ ┐
' 3 3 333 3333 ' .
Question 2 (11 marks)
Let
y = ┌ y(y)扌(|) ┐ ( MVN ╱┌ 2(1) ┐ , ┌ 1(3) 2(1) ┐、,
and
A = ┌ 2一1 一11 ┐ , V = ┌ 1(3) 2(1) ┐ .
(a) [2 marks] Calculate E[yT Ay].
[2 marks] Let z = y 一 E[y]. Describe the distribution of zT V-|z, where V = Var y.
[3 marks] Let x = (y|, y扌 , y| 一 y扌 )T . Describe the distribution of x.
[4 marks] Find a constant a such that ay| )y| 一 3y扌) |
( t| . |
Question 3 (16 marks)
For this question, we study the relationship between the compressible strength (MPA) of a concrete slump and the following seven ingredients:
❼ Cement: cement (kg/m扌 )
❼ Slag: blast furnace slag (kg/m扌 )
❼ Flyash: fly ash (kg/m扌 )
❼ Water: water (kg/m扌 )
❼ SP: superplasticizer (kg/m扌 )
❼ Coarse: coarse sand aggregate (kg/m扌 )
❼ Fine: fine sand aggregate (kg/m扌 )
> slump <- readRDS("slump.Rds")
> head(slump, n = 5)
Cement 1 273 2 163 3 162 4 162 5 154 |
Slag 82 149 148 148 112 |
Flyash 105 191 191 190 144 |
Water SP Coarse
|
Fine 680 746 743 741 658 |
MPA 34.99 41.14 41.81 42.08 26.82 |
> y = slump$MPA
> X1 = as.matrix(cbind(1, slump$SP))
> round(solve(t(X1)%*%X1),6)
[,1] [,2]
[1,] 0.100417 -0.010622
[2,] -0.010622 0.001244
> t(X1)%*%y
[,1]
[1,] 3712.06
[2,] 31615.27
> sum(y^2)
[1] 140047.1
> sum((y - mean(y))^2)
[1] 6266.664
> fullmodel <- lm(MPA ~ ., data = slump)
> summary(fullmodel)
Call:
lm(formula = MPA ~ ., data = slump)
Residuals:
Min 1Q Median 3Q Max
-5.8411 -1.7063 -0.2831 1.2986 7.9424
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 139.78150 Cement 0.06141 Slag -0.02971 Flyash 0.05053 Water -0.23270 SP 0.10315 Coarse -0.05562 Fine -0.03908 |
71.10128 0.02282 0.03176 0.02316 0.07166 0.13459 0.02744 0.02882 |
1.966 2.691 -0.935 2.182 -3.247 0.766 -2.027 -1.356 |
0.05222 . 0.00842 ** 0.35200 0.03159 * 0.00161 ** 0.44532 0.04546 * 0.17833 |
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.609 on 95 degrees of freedom Multiple R-squared: 0.8968, Adjusted R-squared: 0.8892 F-statistic: 118 on 7 and 95 DF, p-value: < 2.2e-16
> qvec = c(0.025, 0.05, 0.1, 0.9, 0.95, 0.975)
> qchisq(qvec, df = 95)
[1] 69.92487 73.51984 77.81843 113.03769 118.75161 123.85797 > qt(qvec, df = 95)
[1] -1.985251 -1.661052 -1.290527 1.290527 1.661052 1.985251 > qf(0.95, 5:10, 95)
[1] 2.310225 2.195548 2.107506 2.037370 1.979923 1.931838
(a) [2 marks] For the model MPA = βb + β|SP + ε, calculate the least squares estimate b.
[2 marks] For the model MPA = βb + β|SP + ε, calculate SSRes .
[2 marks] For the full model, find a 95% confidence interval for β扌in。.
[3 marks] For the full model, test the hypothesis Hb : βslag = 0 at the 5% significance level using an F-test. State the F-statistic, the p-value, and your conclusion in the context of the study.
[4 marks] For the full model, complete the following ANOVA table: |
Variation SS |
Regression |
Residual |
Total |
|
[3 marks] For the full model, test Hb : β廿。m。nt = βslag = β扌lyash = βwat。r = β廿oars。 = β扌in。 = 0 (i.e., all parameters except the intercept and βsP ), with significance level α = 0.05. (Hint: Use your results from part (b).) |
|
Question 4 (11 marks)
Consider the generalised least squares estimator
b = (XT V-|X)-|XT V-|y
for a full rank linear model
y = Xβ + ε ,
where X is an n × p matrix with rank p, ε ( MVN (0, V), and V is known.
[2 marks] Show directly that E[b] = β .
[3 marks] Calculate Var b.
[3 marks] For design variables x* with response y* , find a 100(1 一 α)% confidence interval
for E[y*] = (x* )T β .
[3 marks] For the null hypothesis Hb : β = 0, find the distribution of the test statistic bT XT V-|Xb, and the rejection region for significance level α .
2022-06-10