# Stat 432 Homework 5 solution

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## Question 1 (linear regression)

[2 points] On page 23 of lecture note “LinearReg”, what are the irreducible error, bias and variance? Provide
[2 points] For Ridge regression, how does the tuning parameter trades bias and varaince of the prediction
error? Provide a brief and non-technical explanation (within 100 words).

## Question 2 (model selection criteria)

The Boston Housing data is a classical dataset that models the median house values medv of different areas
of Boston. Because a lot of variables exhibit an asymmetry, we will use some transformations.
data(Boston, package=”MASS”)

## crim zn indus chas nox rm age dis rad tax ptratio black lstat medv
## 1 0.00632 18 2.31 0 0.538 6.575 65.2 4.0900 1 296 15.3 396.90 4.98 24.0
## 2 0.02731 0 7.07 0 0.469 6.421 78.9 4.9671 2 242 17.8 396.90 9.14 21.6
## 3 0.02729 0 7.07 0 0.469 7.185 61.1 4.9671 2 242 17.8 392.83 4.03 34.7
## 4 0.03237 0 2.18 0 0.458 6.998 45.8 6.0622 3 222 18.7 394.63 2.94 33.4
## 5 0.06905 0 2.18 0 0.458 7.147 54.2 6.0622 3 222 18.7 396.90 5.33 36.2
## 6 0.02985 0 2.18 0 0.458 6.430 58.7 6.0622 3 222 18.7 394.12 5.21 28.7
useLog = c(1,3,5,6,8,9,10,14)

Boston[,useLog] = log(Boston[,useLog])
Boston[,2] = Boston[,2] / 10
Boston[,7] = Boston[,7]^2.5 / 10^4
Boston[,11] = exp(0.4 * Boston[,11])/1000
Boston[,12] = Boston[,12] / 100
Boston[,13] = sqrt(Boston[,13])

part a)
[1 point] Fit a linear regression that models medv using all other covariates, including an intercept term.
part b)

[3 points] You cannot use existing statistical functions, e.g. AIC(), for the first two questions.

• Calculate the Mallow’s Cp statistic of this model fitting.
• Based on the parameter estiamtes, if we assume that the errors follow i.i.d. Normal distribution,
calculate the −2 log-likelihood of this model fitting based on the maximum likelihood estimators of
σ
2
. Count σ
2
in the Normal density function as one additional parameter, calculate the AIC and BIC
statistics of this model fitting.

• Select the best models based on Mallow’s Cp, AIC and BIC respectively. Are they the same?

## Question 3 (ridge regression)

[2 points] Use the ridge regression to fit this dataset. You should consider a range of penalty levels and use
the generalized cross-validation criteria to select the best tuning. Report sufficient infromation of your final
model fitting results, such as parameter estiamtes and the best penalty level.