corrected some pages

Several small mistakes/errors were corrected in several proofs/definitions.

Author: JoramSoch

P/mlr-mle.md

@@ -32,7 +32,7 @@ the [maximum likelihood estimates](/D/mle) of $\beta$ and $\sigma^2$ are given
 $$ \label{eq:MLE-MLE}
 \begin{split}
 \hat{\beta} &= (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} y \\
-\hat{\sigma}^2 &= \frac{1}{n} (y-X\hat{\beta})^\mathrm{T} (y-X\hat{\beta}) \; .
+\hat{\sigma}^2 &= \frac{1}{n} (y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta}) \; .
 \end{split}
 $$
 

P/mlr-wls.md

@@ -20,6 +20,12 @@ sources:
     in: "Methods and models for fMRI data analysis in neuroeconomics"
     pages: "Lecture 3, Slides 20/23"
     url: "http://www.socialbehavior.uzh.ch/teaching/methodsspring10.html"
+  - authors: "Wikipedia"
+    year: 2021
+    title: "Weighted least squares"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2021-11-17"
+    url: "https://en.wikipedia.org/wiki/Weighted_least_squares#Motivation"
 
 proof_id: "P77"
 shortcut: "mlr-wls"

P/slr-mle.md

@@ -24,7 +24,7 @@ username: "JoramSoch"
 **Theorem:** Given a [simple linear regression model](/D/mlr) with independent observations
 
 $$ \label{eq:slr}
-y = \beta_0 + \beta_1 x + \varepsilon, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; ,
+y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; ,
 $$
 
 the [maximum likelihood estimates](/D/mle) of $\beta_0$, $\beta_1$ and $\sigma^2$ are given by
@@ -90,7 +90,7 @@ and setting this derivative to zero gives the MLE for $\beta_1$:
 
 $$ \label{eq:beta1-mle}
 \begin{split}
-\frac{\mathrm{d}\mathrm{LL}(\hat{\beta}_0,\hat{\beta}_1,\hat{\sigma}^2)}{\mathrm{d}\beta_0} &= 0 \\
+\frac{\mathrm{d}\mathrm{LL}(\hat{\beta}_0,\hat{\beta}_1,\hat{\sigma}^2)}{\mathrm{d}\beta_1} &= 0 \\
 0 &= \frac{1}{\hat{\sigma}^2} \sum_{i=1}^n (x_i y_i - \hat{\beta}_0 x_i - \hat{\beta}_1 x_i^2) \\
 0 &= \sum_{i=1}^n x_i y_i - \hat{\beta}_0 \sum_{i=1}^n x_i - \hat{\beta}_1 \sum_{i=1}^n x_i^2) \\
 0 &\overset{\eqref{eq:beta0-mle}}{=} \sum_{i=1}^n x_i y_i - (\bar{y} - \hat{\beta}_1 \bar{x}) \sum_{i=1}^n x_i - \hat{\beta}_1 \sum_{i=1}^n x_i^2 \\

P/slr-mle2.md

@@ -24,7 +24,7 @@ username: "JoramSoch"
 **Theorem:** Given a [simple linear regression model](/D/mlr) with independent observations
 
 $$ \label{eq:slr}
-y = \beta_0 + \beta_1 x + \varepsilon, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; ,
+y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; ,
 $$
 
 the [maximum likelihood estimates](/D/mle) of $\beta_0$, $\beta_1$ and $\sigma^2$ are given by
@@ -70,7 +70,7 @@ $$ \label{eq:slr-mle-b}
 \end{split}
 $$
 
-which [is equal to the ordinary least squares solution for simple linear regression](/P/slr-ols):
+which [is equal to the ordinary least squares solution for simple linear regression](/P/slr-ols2):
 
 $$ \label{eq:slr-mle-b-qed}
 \begin{split}

P/slr-ols2.md

@@ -24,7 +24,7 @@ username: "JoramSoch"
 **Theorem:** Given a [simple linear regression model](/D/slr) with independent observations
 
 $$ \label{eq:slr}
-y = \beta_0 + \beta_1 x + \varepsilon, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; ,
+y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; ,
 $$
 
 the parameters minimizing the [residual sum of squares](/D/rss) are given by