Consider a Gaussian linear model Y=aX+epsilon in a Bayesian view. Consider the prior pi (a)=1 for all ain mathbb {R}. Determine whether each of the following statements is true or false.
pi (a) a uniform prior.
True
False
pi (a) is a Jeffreys prior when we consider the likelihood L(Y=y|A=a, X=x) (where we assume x is known).
True
False
Consider a linear regression model mathbf{Y}=mathbb {X}{oldsymbol eta }+sigma {oldsymbol varepsilon } where
{oldsymbol varepsilon }in mathbb {R}^ n is a random vector with mathbb {E}[{oldsymbol varepsilon }]=mathbf{0}, mathbb {E}[{oldsymbol varepsilon }{oldsymbol varepsilon }^ T]=I_ n, and no further assumptions are made about {oldsymbol varepsilon }
mathbb {X} is an n by p deterministic matrix, and mathbb {X}^ Tmathbb {X} is invertible.
sigma >0 is an unknown constant.
Let hat{{oldsymbol eta }} denote the least squares estimator of eta in this context. Determine whether each of the the following statements is true or false.
hat{{oldsymbol eta }} is the maximum likelihood estimator for {oldsymbol eta }.
True
False
With the model written as mathbf{Y}=mathbb {X}{oldsymbol eta }+sigma {oldsymbol varepsilon }, hat{{oldsymbol eta }} has dimension 1 imes p (i.e. is a row vector of length p).
True
False
hat{eta } has a Gaussian distribution (even for small n).
True
False
Under the same setup and assumptions, mathbb {X}hat{{oldsymbol eta }} is...
(Check all that apply.)
Equal to (mathbb {X}^ Tmathbb {X})^{-1}mathbb {X}^{T}mathbf{Y}
An unbiased estimator of mathbb {X}{oldsymbol eta }
A vector in mathbb {R}^ p