corrected some pages

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

Author: JoramSoch

D/chi2.md

@@ -22,7 +22,7 @@ sources:
     url: "https://en.wikipedia.org/wiki/Chi-square_distribution#Definitions"
   - authors: "Robert V. Hogg, Joseph W. McKean, Allen T. Craig"
     year: 2018
-    title: "The χ2-Distribution"
+    title: "The Chi-Squared-Distribution"
     in: "Introduction to Mathematical Statistics"
     pages: "Pearson, Boston, 2019, p. 178, eq. 3.3.7"
     url: "https://www.pearson.com/store/p/introduction-to-mathematical-statistics/P100000843744"

P/ci-wilks.md

@@ -74,7 +74,7 @@ $$ \label{eq:llr}
 \log \Lambda(\phi) = \log p(y|\phi,\hat{\lambda}) - \log p(y|\hat{\phi},\hat{\lambda}) \; .
 $$
 
-[Wilks' theorem](llr-wilks) states that, when comparing two statistical models with parameter spaces $\Theta_1$ and $\Theta_0 \subset \Theta_1$, as the sample size approaches infinity, the quantity calculated as $-2$ times the log-ratio of maximum likelihoods follows a [chi-squared distribution](/D/chi2), if the null hypothesis is true:
+[Wilks' theorem](/P/llr-wilks) states that, when comparing two statistical models with parameter spaces $\Theta_1$ and $\Theta_0 \subset \Theta_1$, as the sample size approaches infinity, the quantity calculated as $-2$ times the log-ratio of maximum likelihoods follows a [chi-squared distribution](/D/chi2), if the null hypothesis is true:
 
 $$ \label{eq:wilks}
 H_0: \theta \in \Theta_0 \quad \Rightarrow \quad -2 \log \frac{\operatorname*{max}_{\theta \in \Theta_0} p(y|\theta)}{\operatorname*{max}_{\theta \in \Theta_1} p(y|\theta)} \sim \chi^2_{\Delta k}

P/kl-nonsymm.md

@@ -39,10 +39,10 @@ $$ \label{eq:KL-nonsymm}
 \mathrm{KL}[P||Q] \neq \mathrm{KL}[Q||P]
 $$
 
-for some [probability distributions](dist) $P$ and $Q$.
+for some [probability distributions](/D/dist) $P$ and $Q$.
 
 
-**Proof:** Let $X \in \mathcal{X} = \left\lbrace 0, 1, 2 \right\rbrace$ be a discrete [random variable](/D/rvar) and consider the two [probability distributions](dist)
+**Proof:** Let $X \in \mathcal{X} = \left\lbrace 0, 1, 2 \right\rbrace$ be a discrete [random variable](/D/rvar) and consider the two [probability distributions](/D/dist)
 
 $$ \label{eq:P-Q}
 \begin{split}

P/norm-gi.md

@@ -22,7 +22,7 @@ sources:
     url: "https://proofwiki.org/wiki/Gaussian_Integral"
   - authors: "ProofWiki"
     year: 2020
-    title: "Integral to Infinity of Exponential of -t^2"
+    title: "Integral to Infinity of Exponential of minus t squared"
     in: "ProofWiki"
     pages: "retrieved on 2020-11-25"
     url: "https://proofwiki.org/wiki/Integral_to_Infinity_of_Exponential_of_-t%5E2"

P/wald-mgf.md

@@ -72,13 +72,13 @@ $$ \label{eq:bessel-de}
 x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx}-(x^2+p^2)y=0 \; .
 $$
 
-The first of these [identities](https://dlmf.nist.gov/10.39.2) gives an explicit solution for $K_{-1/2}$:
+The [first of these identities](https://dlmf.nist.gov/10.39.2) gives an explicit solution for $K_{-1/2}$:
 
 $$ \label{eq:bessel-fact1}
 K_{-1/2}(x) = \sqrt{\frac{\pi}{2x}} e^{-x} \; .
 $$
 
-The second of these [identities](https://dlmf.nist.gov/10.32.10) gives an integral representation of $K_p$:
+The [second of these identities](https://dlmf.nist.gov/10.32.10) gives an integral representation of $K_p$:
 
 $$ \label{eq:bessel-fact2}
 K_p(\sqrt{ab}) = \frac{1}{2}\left(\frac{a}{b}\right)^{p/2} \int_0^{\infty}x^{p-1}\cdot \exp\left[-\frac{1}{2}\left(ax + \frac{b}{x}\right)\right]dx \; .
@@ -104,10 +104,10 @@ Combining with \eqref{eq:bessel-fact1} and simplifying gives
 
 $$ \label{eq:wald-mgf-s4}
 \begin{split}
-  M_X(t) &= \frac{\alpha}{\sqrt{2\pi}}\cdot e^{\alpha \gamma}\cdot 2\left(\frac{\gamma^2-2t}{\alpha^2}\right)^{1/4} \cdot \sqrt{\frac{\pi}{2\sqrt{\alpha^2(\gamma^2-2t)}}}\cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
-         &= \frac{\alpha}{\sqrt{2}\cdot \sqrt{\pi}}\cdot e^{\alpha \gamma}\cdot 2 \cdot \frac{(\gamma^2-2t)^{1/4}}{\sqrt{\alpha}}\cdot \frac{\sqrt{\pi}}{\sqrt{2}\cdot \sqrt{\alpha}\cdot (\gamma^2-2t)^{1/4}}\cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
-         &= e^{\alpha \gamma} \cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
-         &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
+M_X(t) &= \frac{\alpha}{\sqrt{2\pi}}\cdot e^{\alpha \gamma}\cdot 2\left(\frac{\gamma^2-2t}{\alpha^2}\right)^{1/4} \cdot \sqrt{\frac{\pi}{2\sqrt{\alpha^2(\gamma^2-2t)}}}\cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+       &= \frac{\alpha}{\sqrt{2}\cdot \sqrt{\pi}}\cdot e^{\alpha \gamma}\cdot 2 \cdot \frac{(\gamma^2-2t)^{1/4}}{\sqrt{\alpha}}\cdot \frac{\sqrt{\pi}}{\sqrt{2}\cdot \sqrt{\alpha}\cdot (\gamma^2-2t)^{1/4}}\cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+       &= e^{\alpha \gamma} \cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+       &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
 \end{split}
 $$