stat610hw3

# stat610hw3 - Statistics 610 Homework 3(1 Problems from...

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Statistics 610: Homework 3 (1) Problems from Keener’s notes: Chapter 5. 14, 15, 18, 19, 20, 21, 25, 40. (2) (a) Independence of X and s . Let X 1 ,X 2 ,...,X n be i.i.d. N ( μ,σ 2 ). Using the method of orthogonal transformations discussed in class, show that X and s 2 = n i =1 ( X i - X ) 2 / ( n - 1) are independent and determine the distributions of each of these quantities. (b) Simple linear regression with normal errors. Consider the model Y i = α + β X i + ± i where the ± i ’s are i.i.d. N (0 2 ), for i = 1 , 2 , 3 ,...,n and the X 1 ,X 2 ,...,X n are ﬁxed scalars. The parameters α,β,σ 2 are all unknown. (i) Write down the joint density function of ( Y 1 ,Y 2 ,...,Y n ) and treating this as a function of ( α,β,σ 2 ), ﬁnd the maximum likelihood estimates of these parameters: i.e. ﬁnd (ˆ α, ˆ β, ˆ σ 2 ) that maximizes the joint density. (ii) Deﬁne ( W 1 ,W 2 ,...,W n ) T = P ( Y 1 ,Y 2 ,...,Y n ) T where P is an orthogonal matrix. By choosing P

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stat610hw3 - Statistics 610 Homework 3(1 Problems from...

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