situation. Suppose X1,..., Xn d N(μ₁, σ²) and Y₁, . 22d ~ , Xm 2 N(μ2, 02) are all independent, and we'd like to test Ho M1 = μ2. Let A(0; D) denote the likelihood ratio-here, evaluated on the set o D= {X1, Xn, Y₁,..., Ym}. = {(μ1, M2) μ1 = M2} CR2 and for (a) Briefly state the answers to the following questions: (i) What are the MLEs of μ1, μ2, σ² under the current model assumption? (ii) What are the MLEs of μ1, 2, σ² while being restricted to the set no? (b) Show that 2 log A(0; D) = (n+m) log 1+ nm(XY)²] (n+m)S² where X Ŷ = Y₁ + ··· + Ym n m n m and S² = Σ(X; – X)² + Σ(¥¿ − Ÿ)². = [(X;- i=1 j=1 [Hint: You can use the fact that, given a set of numbers {1,..., zn} and their average z = identity ±1(zi — a)² = Σ²²±1(²i − z)² + n(z − a)² holds for any a. - (21++zn)/n, the
situation. Suppose X1,..., Xn d N(μ₁, σ²) and Y₁, . 22d ~ , Xm 2 N(μ2, 02) are all independent, and we'd like to test Ho M1 = μ2. Let A(0; D) denote the likelihood ratio-here, evaluated on the set o D= {X1, Xn, Y₁,..., Ym}. = {(μ1, M2) μ1 = M2} CR2 and for (a) Briefly state the answers to the following questions: (i) What are the MLEs of μ1, μ2, σ² under the current model assumption? (ii) What are the MLEs of μ1, 2, σ² while being restricted to the set no? (b) Show that 2 log A(0; D) = (n+m) log 1+ nm(XY)²] (n+m)S² where X Ŷ = Y₁ + ··· + Ym n m n m and S² = Σ(X; – X)² + Σ(¥¿ − Ÿ)². = [(X;- i=1 j=1 [Hint: You can use the fact that, given a set of numbers {1,..., zn} and their average z = identity ±1(zi — a)² = Σ²²±1(²i − z)² + n(z − a)² holds for any a. - (21++zn)/n, the
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
Related questions
Question
![iid
~
N(μ2,02) are all independent, and we'd like to test
iid
situation. Suppose X1, ..., Xn
~
N(μ1,02) and Y₁,..., Xm
Ho:
M1 =μ2.
Let A(0; D) denote the likelihood ratio-here, evaluated on the set o
D = {X1, … … …, Xn, Y₁, . . ., Ym }.
=
{(M1, M2) μ1
:
=
2} CR2 and for
(a) Briefly state the answers to the following questions: (i) What are the MLEs of μ1, μ2, σ² under the current
model assumption? (ii) What are the MLES of μ1, #2, σ² while being restricted to the set no?
(b) Show that
2 log A(No; D) = (n + m) log |1+
nm(XY)²]
(n+m)S²
where
X1+
+ Xn
Y₁ +
+ Ym
Y
n
M
n
m
and S² = Σ(X; – Ñ)² + Σ(Ÿ¿ − Ÿ)².
= (x −
+
i=1
[Hint: You can use the fact that, given a set of numbers {1,..., Zn} and their average z
(z - a)² = Σ² ²±1 (zi — ž)² + n(ž − a)² holds for any a.
identity
=
(21+
+ Zn)/n, the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1e46184-5bb3-4339-852b-bb3ef2e0784b%2F76aa5f22-b9bd-4f97-86ec-10084f6e3002%2F380x6am_processed.png&w=3840&q=75)
Transcribed Image Text:iid
~
N(μ2,02) are all independent, and we'd like to test
iid
situation. Suppose X1, ..., Xn
~
N(μ1,02) and Y₁,..., Xm
Ho:
M1 =μ2.
Let A(0; D) denote the likelihood ratio-here, evaluated on the set o
D = {X1, … … …, Xn, Y₁, . . ., Ym }.
=
{(M1, M2) μ1
:
=
2} CR2 and for
(a) Briefly state the answers to the following questions: (i) What are the MLEs of μ1, μ2, σ² under the current
model assumption? (ii) What are the MLES of μ1, #2, σ² while being restricted to the set no?
(b) Show that
2 log A(No; D) = (n + m) log |1+
nm(XY)²]
(n+m)S²
where
X1+
+ Xn
Y₁ +
+ Ym
Y
n
M
n
m
and S² = Σ(X; – Ñ)² + Σ(Ÿ¿ − Ÿ)².
= (x −
+
i=1
[Hint: You can use the fact that, given a set of numbers {1,..., Zn} and their average z
(z - a)² = Σ² ²±1 (zi — ž)² + n(ž − a)² holds for any a.
identity
=
(21+
+ Zn)/n, the
Expert Solution
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Step 1: Write the given information.
VIEWStep 2: Compute the MLEs of μ1, μ2, σ² under the current model assumptions.
VIEWStep 3: Compute the MLEs of μ1, μ2, σ² while being restricted to the set Ω_0.
VIEWStep 4: Prove the given equation using the given identity.
VIEWStep 5: Substitute the likelihood ratios to get the final equation.
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