A model in the form of y = β 0 + β 1z 1 + β 2z 2 + . . . + β pzp+ ε, where each independent variable zj (for j = 1, 2, . . ., p) is a function of x 1, x 2 ,..., xk, is known as the
a. |
general curvilinear model. |
|
b. |
general linear model. |
|
c. |
pth-order z model. |
|
d. |
experimental model. |
In the first-order model E( y) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3, β 2 represents the slope of the line relating y to x 2 when β 1 and β 3 are both held fixed. True or False
Exercise5 Consider a linear model with n -2m in which yi Bo Pi^i +ei,i-1,...,m, and Here €1, ,En are 1.1.d. from N(0,ơ), β-(A ,A, β), and σ2 are unknown parameters, zı, known constants with x1 +... + Xm-Tm+1 + +xn0 , zn are 1, write the model in vector form as Y = Xß+ε describing the entries in the matrix X. 2, Determine the least squares estimator β of β.
Exercise5 Consider a linear model with n -2m in which...
How do you prove this equation?
In the full rank general linear model y Ξ ε ~ MVN(0.021). Th is given by β +e, assume 2 en the maximum likelihood estimator tor σ SSRes
In the full rank general linear model y Ξ ε ~ MVN(0.021). Th is given by β +e, assume 2 en the maximum likelihood estimator tor σ SSRes
The general solution to the second-order differential equation d2ydt2−4dydt+7y=0d2ydt2−4dydt+7y=0 is in the form y(x)=eαx(c1cosβx+c2sinβx).y(x)=eαx(c1cosβx+c2sinβx). Find the values of αα and β,β, where β>0.β>0.Answer: α=α= and β=β=
Let y = Xß + ε where ε ~ N(0, σ21). Let β = (XTX)-"XTy and let è-y-X β. (a) Show that è-(1-Pxje where Px (b) Compute Ee -e 2. X(XTX)-1x" (1 (c) Compute Varle-e.
Let y = Xß + ε where ε ~ N(0, σ21). Let β = (XTX)-"XTy and let è-y-X β. (a) Show that è-(1-Pxje where Px (b) Compute Ee -e 2. X(XTX)-1x" (1 (c) Compute Varle-e.
in a Bayesian view. Consider the prior π(a)-1 for all a e R Consider a Gaussian linear model Y = aX+ E Determine whether each of the following statements is true or false. π(a) a uniform prior. (1) (a) True (b) False L(Y=y14=a,X=x) (2) π(a) is a jeffreys prior when we consider the likelihood (where we assume xis known) (a) True (b)False Y-XB+ σε where ε E R" is a random vector with Consider a linear regression model E[ε1-0, E[eErJ-1....
1.Given the Multiple Linear regression model as Y-Po + β.X1 + β2X2 + β3Xs + which in matrix notation is written asy-xß +ε where -έ has a N(0,a21) distribution + + ßpXo +ε A. Show that the OLS estimator of the parameter vector B is given by B. Show that the OLS in A above is an unbiased estimator of β Hint: E(β)-β C. Show that the variance of the estimator is Var(B)-o(Xx)-1 D. What is the distribution o the...
2. Let the following data be given where X is the independent variable and Y is the dependent variable Find the correlation coefficient r a. a and B,onpfrom the sample for the model: b. Estimate +tx and the actual data Y where ε is the random error between the fitted model f by Y write the linear equation P=" dr-h Predict the value of P when x = 8 c. d.
2. Let the following data be given where X...
Exercise 2.6: Consider the models y Xßte and y* X"β+c" where E(e) = 0, cov(e) = σ21, y* = ГУ, X* = ГХ, e* =「ε and r is a known n x n orthogonal matrix. Show that: 1. E(e) 0, cov(e) σ21 2. b b and s2 s2, where b and b' are the least squares estimates of β and 82 and s+2 are the estimates of σ2 obtained from the two models.
The random vector Y = (Y1, ...,
Yn)T is such that Y = Xβ + ε, where X is an n
× p full-rank matrix of known constants, β is a p-length vector of
unknown parameters, and ε is an n-length vector of random
variables. A multiple linear regression model is fitted to the
data.
(a) Write down the multiple linear regression model assumptions in
matrix format.
(b) Derive the least squares estimator β^ of β.
(c) Using the data:...