Statistical Modelling Exam 27/06/2024 Exercise 1 Assume that 1,...,200 are realizations of independent Gaussian random variables with variance equal to 1 and and mean B₁ + 32 exp{z} for i = 1,..., 120, and mean ẞ₁ + ẞ3 exp{z}} for i = 121,...,200; where the z; are known constants and (31, 32, 33) are unknown real parameters. a) Are the assumptions of a Gaussian linear model satisfied in the above formulation? Moti- vate the answer. b) State the parameter space and sample space. c) Express the model in matrix form: Y = XB+E, explicitly stating how Y, X, B, and are defined and their dimensions. Write the distribution of Y and ɛ. d) Obtain the expression of the matrix XTX and the vector XTy; state how these elements should be used to obtain the maximum likelihood estimate B. e) Write the distribution of the maximum likelihood estimator (Y). f) Let e = y-Xẞ be the vector of the residuals. State which of the following identities are satisfied and motivate the answer: 200 200 Σe₁ =0 i=1 Σe; exp{z} = 0 i=1 (hint: read the indices in the sum!) 200 200 Σe₁z₁ = 0 i=1 Σe exp{z}}=0 i=1 ei 200 Σ? = 0 i=1 120 Σe exp{z} = 0 i=1

Statistical Modelling Exam 27/06/2024 Exercise 1 Assume that 1,...,200 are realizations of independent Gaussian random variables with variance equal to 1 and and mean B₁ + 32 exp{z} for i = 1,..., 120, and mean ẞ₁ + ẞ3 exp{z}} for i = 121,...,200; where the z; are known constants and (31, 32, 33) are unknown real parameters. a) Are the assumptions of a Gaussian linear model satisfied in the above formulation? Moti- vate the answer. b) State the parameter space and sample space. c) Express the model in matrix form: Y = XB+E, explicitly stating how Y, X, B, and are defined and their dimensions. Write the distribution of Y and ɛ. d) Obtain the expression of the matrix XTX and the vector XTy; state how these elements should be used to obtain the maximum likelihood estimate B. e) Write the distribution of the maximum likelihood estimator (Y). f) Let e = y-Xẞ be the vector of the residuals. State which of the following identities are satisfied and motivate the answer: 200 200 Σe₁ =0 i=1 Σe; exp{z} = 0 i=1 (hint: read the indices in the sum!) 200 200 Σe₁z₁ = 0 i=1 Σe exp{z}}=0 i=1 ei 200 Σ? = 0 i=1 120 Σe exp{z} = 0 i=1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 22E

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