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Equation Solving

Problem 1 (10 points)
Let z be a 2 ◊ 1 random vector with distribution,
z ≥ N
350
0
6
,
5 1 ≠1
2
≠1
2 1
64 .
Find a 2 ◊ 2 matrix A such that ÎAzÎ2 has a chi-square distribution, and specify the associated degrees of
freedom.
Problem 2 (15 points) 3+3+3+6 = 15
Consider the following two regression models (at the population level):
M1 : y = —0 + —1×1 + —2×2 + Á,
M2 : y = —0 + —1×1 + —2×2 + —3×2
1 + —4×2
2 + —5x1x2 + Á.
Assume that we fit the two models to the same data.
(a) Can we compare the R2 of the two models without looking at the results of the fittings? Justify your
answer.
(b) Can we say that the residual vector of one model is orthogonal to the fitted value vector of the other
model? If so specify the details, if not argue why not.
(c) Assume that the data is actually generated from the following model (at the population level)
y = —ú
0 + —ú
1×1 + —ú
2×2 + —ú
3x1x2 + Á,
with —ú
0 , —ú
1 , —ú
2 and —ú
3 all nonzero. Which of the two models M1 and M2 will produce an unbiased
LSE of —ú? Justify your answer.
(d) Now assume the data of sample size n = 100 is actually generated from model M1 with noise vector
following our standard Gaussian assumption Á ≥ N(0, ‡2In). Consider an F-test for comparing M1
and M2. What is the probability that the p-value of that test is Ø 0.05? Justify your answer.
1
Problem 3 (20 points) 10+10 = 20
Consider the following two regression models (at the population level):
M1 : y = —0 + —1×1 + —2×2 + Á,
M2 : y = —0 + —1×1 + —2×2 + —3×2
1 + —4x1x2 + Á.
We fit the two models to the following data (n = 10):
t(dat)

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]

X1 -0.4 0.6 -0.2 0.8 0.9 -0.9 0.1 0.8 0.1 -0.1

X2 0.9 -0.1 0.4 0.1 -0.8 0.8 -0.5 -0.9 -0.3 0.9

y 2.0 -0.3 0.7 -0.4 -0.6 2.0 -0.4 -1.0 -0.2 0.9

(a) Compare the two models and choose one based on an F-test at level – = 0.05.
(b) Find the variance inflation factor for —1 in model M2.
Problem 4 (20 points)
Consider the following three regression models (at the population level):
M0 : y = —0 + Á,
M1 : y = —0 + —1×1 + —2×2 + Á,
M2 : y = —0 + —1×1 + —2×2 + —3×2
1 + —4×2
2 + —5x1x2 + Á.
We fit the three models to the same data with sample size n. For i = 0, 1, 2, let SSE(Mi) œ R and µˆ(Mi) œ Rn
denote the SSE and the fitted value vector of model Mi, respectively. The following information is given:
• The sample size n = 15.
• εˆ(M0) ≠ µˆ(M1)Î2 = 4 ◊ SSE(M1).
• The F-statistic for an F-test comparing models M1 and M2 is Fˆ = 4.
What is the R2 for model M2?
2
Problem 5 (15 points) 3 + 3 + 3 + 3 + 3 = 15
Consider regression model y = —0 + q4
j=1 —jxj + Á with the data given by
t(dat)

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]

X1 0.0 -1 0.0 -1 -1 0 1.0 -1.0 1 0

X2 -1.0 0 1.0 1 0 -1 0.0 0.0 0 -1

X3 -1.0 1 1.0 -1 1 1 1.0 1.0 0 0

X4 -1.0 -1 1.0 -1 0 0 -1.0 0.0 0 0

y -0.7 -1 -0.1 1 -1 -4 -0.8 -0.3 -3 2

Under the standard assumption Á ≥ N(0, ‡2In), answer the following:
(a) Which of the 10 data points has the highest influence on the regression, based on the Cook’s distance?
(b) Which of the 10 data points has the highest leverage without having much of an influence on the
regression?
(c) Using the PRESS statistic, among the the following two models, which one should be selected:
(1) the model that includes x1 and x2 or
(2) the model that includes x3 and x4?
Both models also include the intercept.
(d) Is it reasonable to use R2 to select among the two models in part (c)? Justify your answer.
(e) Argue that the comparison based on R2 in part (d) is equivalent to comparing the p-values of two
F-tests. Specify those tests.
Problem 6 (20 points) 10+10 = 20
Consider linear regression model y = —0 + q2
j=1 —jxj + Á with the transposed design matrix XT given by

[,1] [,2] [,3] [,4] [,5]

X1 0 -1 0 -1 -1

X2 0 1 -1 1 0

and the transposed response vector yT given by

[,1] [,2] [,3] [,4] [,5]

y 3 -1 -4 -5 -3

The noise vector Á = [Á1,…, Án]
T follows the following model: (n = 5)
Á1 = z1
Ái ≠ Ái≠1 = zi, i = 2,…,n
zi ≥ N!
0, i
n

, i = 1, . . . , n.
Assume that {z1, z2,…,zn} are independent.
(a) What is the distribution of Á = [Á1, Á2,…, Án]
T ?
(b) What is the smallest possible variance of an unbiased estimate of —2?

Sample Solution

hand, is concerned with decayed emotion. An inconsistency in Gothic is that ‘Gothic novelists did not know how to release their own feelings of frustration and rebelliousness. Their fiction is both exploratory and fearful’ as Kilgour tells us. It usually results in the death of a villain. Miles has a valid point about how you cannot constrain Gothic to a particular type of text, preferring to class Gothic literature as a taste or preference. Overall, I will show that the reception of gothic writing-its institutional and commercial recognition as a kind of literature- played a fundamental role in shaping many of the ideological assumptions about high culture that we now associate with the term Romanticism. The Gothic novel was first invented almost single-handedly by Walpole as The Castle of Otranto fits most of the classifications we see in Gothic today. ‘The Gothic, like any genre, depends on a system of classification, and because genres, as Derrida argues, are never pure, and systems of classification, according to Foucault, cannot be verified, one is pressed to investigate and contest the validity of the definitions and conceptions typically attributed to the term “Gothic”, a kind of writing that is evidently heterogeneous and impure’ (Alshatti, A. (2008).). Walpole’s novel was imitated in the eighteenth century, but it was enjoyed widespread influence in the nineteenth century partly because of the era’s understanding in dark and fascinating themes. He could be said to have been influenced by Shakespearean dramas because in The Castle Of Otranto he plays around with mental disturbances, where Manfred seeks to marry the soon to be wife of his dead son Conrad to keep his genes alive throughout generations. Lady Macbeth evidently suffers from a psychotic disorder with the misfortune of hallucinations which can be induced by extreme guilt. She has the sense of heavy guilt because her and her husband killed King Duncan in cold blood. Gothic, it can be argued, was instrumental in the decisive shift towards popular fiction in its modern form, aimed at a brood readership, commercially streamlined, form
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