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Q1. Suppose that for k= 1,..., K and i = 1,...,nk we observe independent Xki ~ Np(µk, Ip), where K, n₁,...,nk € N. In this question we consider the K-sample testing problem, where we want to test the null hypothesis Ho μ₁ = ... = μK. Note that the observations here have identity covariance matrix, so this is a simplification of the setting found in the notes. (a) Write down the likelihood function L(μ₁,..., μK) for this model, and prove that where Xk -1 = nk supμERÐ L(μ,..., μl) supμ1,...,KERP L(µ₁,. MK) (ii) Show that we may write K 1 kXk EXP(-¹4X - X1²), k=1 i=1 1Xki and X = N-¹ Σk_₁ nkXk with N = ₁ k. Zk=1 K k=1 = exp K 2 (b) In this part of the question we will find the distribution of Σk_₁ nkXk – Ã|² under Ho. - k=1 √nk)¹. Give, with jus- (i) Define the Kp-dimensional random vector Y (√n₁X₁, tification, the distribution of Y. and calculate q nk (√n₁(×₁ – ī)¹,..., √nk (Xk − Ī)¹)¹ = (Ikp — N−¹VV¹)Y, (iv) Using Proposition where V is the Kp × p matrix given by VT =(√√₁Ip... √nkIp). (iii) Prove that P = = IKp – N−¹VVT is an orthogonal projection matrix satisfying PV Tr(P). = 2.3.2 and the previous parts of this question, or otherwise, prove that K under Ho we have h_₁ nkXk – X² ~ x ²4. 0, X₁ = (2,0,1)¹, X2 = (0,1,2)¹, X3 = (0,0,0)¹. Using part (b), carry out a test of Ho at the 5% significance level. (c) Suppose that we have data on K = 3 groups, with n₁ = n₂ = n3 = 50, and suppose that we observe

Q1. Suppose that for k= 1,..., K and i = 1,...,nk we observe independent Xki ~ Np(µk, Ip), where K, n₁,...,nk € N. In this question we consider the K-sample testing problem, where we want to test the null hypothesis Ho μ₁ = ... = μK. Note that the observations here have identity covariance matrix, so this is a simplification of the setting found in the notes. (a) Write down the likelihood function L(μ₁,..., μK) for this model, and prove that where Xk -1 = nk supμERÐ L(μ,..., μl) supμ1,...,KERP L(µ₁,. MK) (ii) Show that we may write K 1 kXk EXP(-¹4X - X1²), k=1 i=1 1Xki and X = N-¹ Σk_₁ nkXk with N = ₁ k. Zk=1 K k=1 = exp K 2 (b) In this part of the question we will find the distribution of Σk_₁ nkXk – Ã|² under Ho. - k=1 √nk)¹. Give, with jus- (i) Define the Kp-dimensional random vector Y (√n₁X₁, tification, the distribution of Y. and calculate q nk (√n₁(×₁ – ī)¹,..., √nk (Xk − Ī)¹)¹ = (Ikp — N−¹VV¹)Y, (iv) Using Proposition where V is the Kp × p matrix given by VT =(√√₁Ip... √nkIp). (iii) Prove that P = = IKp – N−¹VVT is an orthogonal projection matrix satisfying PV Tr(P). = 2.3.2 and the previous parts of this question, or otherwise, prove that K under Ho we have h_₁ nkXk – X² ~ x ²4. 0, X₁ = (2,0,1)¹, X2 = (0,1,2)¹, X3 = (0,0,0)¹. Using part (b), carry out a test of Ho at the 5% significance level. (c) Suppose that we have data on K = 3 groups, with n₁ = n₂ = n3 = 50, and suppose that we observe

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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= Q1. Suppose that for k = 1,..., K and i 1,...,nk we observe independent Xki ~ Np(µk, Ip), where K, n₁,...,nk € N. In this question we consider the K-sample testing problem, where we want to test the null hypothesis Ho: μ₁ = μK. Note that the observations here have identity covariance matrix, so this is a simplification of the setting found in the notes. (a) Write down the likelihood function L(µ₁,…,‚µK) for this model, and prove that -X₁²), supμERP L(μ,..., μl) L(µ₁, ´μ1,...,KERP sup =... (ii) Show that we may write .., MK) = = K ink K where X₁ = n²¹ [₁ Xki and X = N-¹Σk_₁ nķīk with N = Σk 1_1 nk. -k i=1 k=1 exp 2 k=1 (b) In this part of the question we will find the distribution of K_₁7k|Xk – X² under Hō. (i) Define the Kp-dimensional random vector Y = (√n₁×¹, ..√nKX). Give, with jus- nk tification, the distribution of Y. K Σ k=1 (c) Suppose that we have data on K observe = nk Xk (√n₁(×₁ – Ī)¹,..., √nk(Xk — X)¹)¹ = (Ikp — N−¹VV¹)Y, where V is the Kp × p matrix given by VT = (√n₁Ip….. √nkIp). (iii) Prove that P IKp - N-¹VVT is an orthogonal projection matrix satisfying PV and calculate q = Tr(P). -1,..., (iv) Using Proposition 2.3.2 and the previous parts of this question, or otherwise, prove that under Ho we have Σk_1 ¹k|Xk — Ñ| ² ~ x ²² = = 3 groups, with n₁ = n₂ - n3 = = X₁ (2,0,1)T, X2 = (0,1,2)T, X3 = (0,0,0). Using part (b), carry out a test of Ho at the 5% significance level. 0, 50, and suppose that we
Transcribed Image Text:= Q1. Suppose that for k = 1,..., K and i 1,...,nk we observe independent Xki ~ Np(µk, Ip), where K, n₁,...,nk € N. In this question we consider the K-sample testing problem, where we want to test the null hypothesis Ho: μ₁ = μK. Note that the observations here have identity covariance matrix, so this is a simplification of the setting found in the notes. (a) Write down the likelihood function L(µ₁,…,‚µK) for this model, and prove that -X₁²), supμERP L(μ,..., μl) L(µ₁, ´μ1,...,KERP sup =... (ii) Show that we may write .., MK) = = K ink K where X₁ = n²¹ [₁ Xki and X = N-¹Σk_₁ nķīk with N = Σk 1_1 nk. -k i=1 k=1 exp 2 k=1 (b) In this part of the question we will find the distribution of K_₁7k|Xk – X² under Hō. (i) Define the Kp-dimensional random vector Y = (√n₁×¹, ..√nKX). Give, with jus- nk tification, the distribution of Y. K Σ k=1 (c) Suppose that we have data on K observe = nk Xk (√n₁(×₁ – Ī)¹,..., √nk(Xk — X)¹)¹ = (Ikp — N−¹VV¹)Y, where V is the Kp × p matrix given by VT = (√n₁Ip….. √nkIp). (iii) Prove that P IKp - N-¹VVT is an orthogonal projection matrix satisfying PV and calculate q = Tr(P). -1,..., (iv) Using Proposition 2.3.2 and the previous parts of this question, or otherwise, prove that under Ho we have Σk_1 ¹k|Xk — Ñ| ² ~ x ²² = = 3 groups, with n₁ = n₂ - n3 = = X₁ (2,0,1)T, X2 = (0,1,2)T, X3 = (0,0,0). Using part (b), carry out a test of Ho at the 5% significance level. 0, 50, and suppose that we
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