, n and θ s normally distributed. Consider the linear regression model Si= α。+ α1P +...
Consider the regression model where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . . . , n. (a) (4 points) Show βˆ is normally distributed with mean β and variance σ2 . 1 1SXX Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
this is from Linear model and the book is Linear regression analysis by Seber & Lee 3. Consider the straight line model without the intercept parameter known constants. a. Show that b. Show that the RSs-{Σ4x2-β/Ld;xf} and S n-RSS is an unbiased estimate of ơ2. c. Show, that a 1-α CI for A is given by Sd where S is given in part b.
(Do this problem without using R) Consider the simple linear regression model y =β0 + β1x + ε, where the errors are independent and normally distributed, with mean zero and constant variance σ2. Suppose we observe 4 observations x = (1, 1, −1, −1) and y = (5, 3, 4, 0). (a) Fit the simple linear regression model to this data and report the fitted regression line. (b) Carry out a test of hypotheses using α = 0.05 to determine...
3. Consider the linear model: Yİ , n where E(Ei)-0. Further α +Ari + Ei for i 1, assume that Σ.r.-0 and Σ r-n. (a) Show that the least square estimates (LSEs) of α and ß are given by à--Ỹ and (b) Show that the LSEs in (a) are unbiased. (c) Assume that E(e-σ2 Yi and E(49)-0 for all i where σ2 > 0. Show that V(β)--and (d) Use (b) and (c) above to show that the LSEs are consistent...
Please help with question 4 Consider the simple linear regression model: with σ2 is known. Assume x's are fixed and known, and only y's are random. Recall Ex 3.5.22 in Homework 1. Here the design matrix is 1 T2 and the regression coefficielt is β = (α, β)T, 3. Derive the MLE of a and ß and show that it is independent of σ2· Is your MLE sane as the least square estimation in Ex 3.5.22? 4. Drive the mean...
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
Consider the simple linear regression model where Bo is known. (a) Find the least squares estimator bi of β1- (b) Is this estimator unbiased? Prove your result
Q4.. [40 points] Consider the multiple linear regression model given by y - XB -+ s, where y and e are vectors of size 8 × 1, X ls a matrix of size 8 x 3 and Disa vector of sze 3 × 1. Also, the following information are available e = 22 y -2 and XTy 3 1. [10 points) Estimate the regression coefficients in the model given above? 2. [4 points] Estimate the variance of the error term...
Consider the Bayesian linear regression model with K regressors where v) Now suppose that we have an uninformative prior such that Show that the posterior verifies: N/2 where VĮß-σ2 (XX)-1. Consider the Bayesian linear regression model with K regressors where v) Now suppose that we have an uninformative prior such that Show that the posterior verifies: N/2 where VĮß-σ2 (XX)-1.
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose that we would like to estimate the mean response at x = x*, that is we want to estimate lyx=* = Bo + B1 x*. The least squares estimator for /uyx* is = bo bi x*, where bo, b1 are the least squares estimators for Bo, Bi. ayx= (a) Show that the least squares estimator for...