Consider the simple regression model Y = a +3X + ε where ~ N(0,0²), for unknown parameters a, ß, o². Let â, ß, ô² be their MLE estima- tors, respectively, for sample (x₁, y₁), (X2; Y2), ···, (Xn, Yn). In this exercise, we look at the probability distribution of the estimator 3 as a random variable. (i) Verify that the estimator as a random variable may be expressed as (ii) Show that ß := Â(Y1, Y2, · Yn) Var (3) - = n 1 1 —— Vx n 1 0² Σ(x¡ — ñ)Yį. - i=1 Vx n (Hint: use the "variances of independent RV's add up" property from Homework 2 Exercise 6 (ii). If you are not sure how to proceed, look at the computation of Eß in lecture.)
Consider the simple regression model Y = a +3X + ε where ~ N(0,0²), for unknown parameters a, ß, o². Let â, ß, ô² be their MLE estima- tors, respectively, for sample (x₁, y₁), (X2; Y2), ···, (Xn, Yn). In this exercise, we look at the probability distribution of the estimator 3 as a random variable. (i) Verify that the estimator as a random variable may be expressed as (ii) Show that ß := Â(Y1, Y2, · Yn) Var (3) - = n 1 1 —— Vx n 1 0² Σ(x¡ — ñ)Yį. - i=1 Vx n (Hint: use the "variances of independent RV's add up" property from Homework 2 Exercise 6 (ii). If you are not sure how to proceed, look at the computation of Eß in lecture.)
A First Course in Probability (10th Edition)
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ISBN:9780134753119
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Chapter1: Combinatorial Analysis
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Can you show me the answer to (i) and (ii)? Thank you so much for your help
![Consider the simple regression model
Y = a + 3X + €
where ~ N(0,0²), for unknown parameters a, 3, o². Let â, , ² be their MLE estima-
tors, respectively, for sample (x₁, y₁), (x2, y2),, (xn, yn). In this exercise, we look at the
probability distribution of the estimator 3 as a random variable.
(i) Verify that the estimator as a random variable may be expressed as
(ii) Show that
8:8(Y₁, Y₂, Yn)
:=
;
=
Var(3) =
n
11
Vx n
10²
Σ(x₁ - π) Yi.
i=1
Vx n
(Hint: use the "variances of independent RV's add up" property from Homework 2
Exercise 6 (ii). If you are not sure how to proceed, look at the computation of E3 in
lecture.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33efa0ee-e3c4-4640-bf2a-d63d72536f00%2F11408ddb-794f-4c96-b676-ca9a0638cc5b%2Fykxww1p_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the simple regression model
Y = a + 3X + €
where ~ N(0,0²), for unknown parameters a, 3, o². Let â, , ² be their MLE estima-
tors, respectively, for sample (x₁, y₁), (x2, y2),, (xn, yn). In this exercise, we look at the
probability distribution of the estimator 3 as a random variable.
(i) Verify that the estimator as a random variable may be expressed as
(ii) Show that
8:8(Y₁, Y₂, Yn)
:=
;
=
Var(3) =
n
11
Vx n
10²
Σ(x₁ - π) Yi.
i=1
Vx n
(Hint: use the "variances of independent RV's add up" property from Homework 2
Exercise 6 (ii). If you are not sure how to proceed, look at the computation of E3 in
lecture.)
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