The eigenvector associated with the smallest eigenvalue of the covariance matrix [S] for the January 1987 temperature data referred to in Exercise 11.2 is e 4 T = [.665, .014, .738, –.115]. Assess the...

The eigenvector associated with the smallest eigenvalue of the covariance matrix [S] for the January 1987 temperature data referred to in Exercise 11.2 is e₄
^T
= [.665, .014, .738, –.115]. Assess the normality of the linear combination e4^Tx,

a. Graphically, with a Q–Q plot. For computational convenience, evaluate F(z) using Equation 4.29.

b. Formally, with the Filliben test (see Table 5.3), assuming no autocorrelation.

Exercise 11.2

Assume that the four temperature variables in Table A.1 are MVN-distributed, with the ordering of the variables in x being [MaxIth, MinIth, MaxCan, MinCan] T . The respective means are also given in Table A.1, and the covariance matrix [S] is given in the answer to Exercise 10.7a. Assuming the true mean and covariance are the same as the sample values,

a. Specify the conditional distribution of [MaxIth, MinIth]^T, given that [Max_Can, Min_Can]
^T
= [31.77, 20.23]^T
(i.e., the average values for Canandaigua).

b. Consider the linear combinations b₁
= [1, 0, –1, 0], expressing the difference between the maximum temperatures, and b₂
= [1, –1 –1, 1], expressing the difference between the diurnal ranges, as rows of a transformation matrix [B] T . Specify the distribution of the transformed variables [B]
^T
x.