1
vote
0answers
9 views

Proving $Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{-1}{n_i-1}$ for $ j \neq k$

For the model of a single factor experiment: $y_{ij}= \mu + \alpha_i + e_{ij}$, $(1 \leq i \leq a, 1 \leq j \leq n_i)$, where a = the number of treatments, $n_i$ = the number of experimental units ...
1
vote
0answers
11 views

Principal Components vs Principal Directions

I'm trying to do statistical downscaling of some climate data and there is a module of principal component analysis by regression method required. I am confused with the different terms here. What is ...
1
vote
1answer
50 views

Prove $\operatorname{Var}(\hat{e}_{ij}) = \sigma^2 \left(\frac{n_i-1}{n_i}\right)$

$\newcommand{\Var}{\operatorname{Var}}$ Let $y_{ij}$ denote the observed response of the $j$th experimental unit in the $i$th treatment group, and the $e_{ij}$ are i.i.d. $N(0,\sigma^2)$ experimental ...
1
vote
1answer
33 views

Linear Regression with independent but non-identical noise

If I have this linear regression equation: $$y=X\beta+\epsilon $$ ($x$ and $\beta$ are vectors) The likelihood function can be written as $$L= \prod_{n=1}^N N(y_n ;x_n ,\beta ,\sigma^2)=(2\pi ...
0
votes
0answers
24 views

Identification of linear regression function under $\ell_1$-norm criterion.

Consider a linear system \begin{align*} y = \theta^Tx + e, \end{align*} where $x\in\mathbb{R}^n$ and $e\in\mathbb{R}$ are independent Gaussian random variables with distribution $\mathcal{N}(0,I_n)$ ...
0
votes
1answer
3k views

Can someone explain what plim is?

In my Introductory Econometrics class we discussed a concept of "plim" or "probability limit. I'm not sure what this means though and my professor doesn't explain it well at all. Can someone tell me ...
11
votes
3answers
1k views

Best fit ellipsoid

Given a collection of points $P \subset \mathbb R^3$, a crude characterization of the "shape" of $P$ is sometimes given by the principal components. We construct a covariance matrix, e.g., if $P$ is ...