Merge pull request #158 from StatProofBook/master

JoramSoch · web-flow · commit df28e5d0dca3 · 2024-11-22T11:48:51.000+01:00
update to master
diff --git a/D/anova2.md b/D/anova2.md
@@ -83,4 +83,4 @@ $$ \label{eq:anova2-cons}
 \end{split}
 $$
 
-where the weights are $w_{ij} = n_{ij}/n$ and the total sample size is $n = \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij}$.
+where the weights are $w_{ij} = n_{ij}/n$ and the total [sample size](/D/samp-size) is $n = \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij}$.
diff --git a/D/hyp-simp.md b/D/hyp-simp.md
@@ -29,6 +29,6 @@ username: "JoramSoch"
 
 **Definition:** Let $H$ be a [statistical hypothesis](/D/hyp). Then,
 
-* $H$ is called a simple hypothesis, if it completely specifies the population distribution; in this case, the [sampling distribution](/D/dist-samp) of the [test statistic](/D/tstat) is a function of sample size alone.
+* $H$ is called a simple hypothesis, if it completely specifies the population distribution; in this case, the [sampling distribution](/D/dist-samp) of the [test statistic](/D/tstat) is a function of [sample size](/D/samp-size) alone.
 
 * $H$ is called a composite hypothesis, if it does not completely specify the population distribution; for example, the hypothesis may only specify one parameter of the distribution and leave others unspecified.
diff --git a/I/DbA.md b/I/DbA.md
@@ -13,7 +13,7 @@ title: "Definition by Author"
 
 - [Sample covariance](/D/cov-samp)
 
-### JoramSoch (193 definitions)
+### JoramSoch (194 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Alternative hypothesis](/D/h1)
@@ -169,6 +169,7 @@ title: "Definition by Author"
 - [Residual sum of squares](/D/rss)
 - [Residual variance](/D/resvar)
 - [Residual-forming matrix](/D/rfmat)
+- [Sample](/D/samp)
 - [Sample correlation coefficient](/D/corr-samp)
 - [Sample correlation matrix](/D/corrmat-samp)
 - [Sample covariance matrix](/D/covmat-samp)
diff --git a/I/DbN.md b/I/DbN.md
@@ -216,3 +216,4 @@ title: "Definition by Number"
 | D207 | dist-uni | [Unimodal and multimodal probability distribution](/D/dist-uni) | JoramSoch | 2024-10-25 |
 | D208 | est | [Estimand, estimator and estimate](/D/est) | JoramSoch | 2024-11-01 |
 | D209 | est-bias | [Biased vs. unbiased estimator](/D/est-bias) | JoramSoch | 2024-11-08 |
+| D210 | samp | [Sample](/D/samp) | JoramSoch | 2024-11-15 |
diff --git a/I/DbT.md b/I/DbT.md
@@ -222,6 +222,7 @@ title: "Definition by Topic"
 
 ### S
 
+- [Sample](/D/samp)
 - [Sample correlation coefficient](/D/corr-samp)
 - [Sample correlation matrix](/D/corrmat-samp)
 - [Sample covariance](/D/cov-samp)
diff --git a/I/PbA.md b/I/PbA.md
@@ -12,7 +12,7 @@ title: "Proof by Author"
 
 - [Posterior predictive distribution is a marginal distribution of the joint likelihood](/P/postpred-jl)
 
-### JoramSoch (442 proofs)
+### JoramSoch (443 proofs)
 
 - [Accuracy and complexity for Bayesian linear regression](/P/blr-anc)
 - [Accuracy and complexity for Bayesian linear regression with known covariance](/P/blrkc-anc)
@@ -429,6 +429,7 @@ title: "Proof by Author"
 - [t-distribution is a special case of multivariate t-distribution](/P/t-mvt)
 - [t-test for multiple linear regression using contrast-based inference](/P/mlr-t)
 - [The p-value follows a uniform distribution under the null hypothesis](/P/pval-h0)
+- [The product of independent log-normal random variables is a log-normal random variable](/P/lognorm-prodind)
 - [The regression line goes through the center of mass point](/P/slr-comp)
 - [The residuals and the covariate are uncorrelated in simple linear regression](/P/slr-rescorr)
 - [The sum of residuals is zero in simple linear regression](/P/slr-ressum)
diff --git a/I/PbN.md b/I/PbN.md
@@ -484,3 +484,4 @@ title: "Proof by Number"
 | P476 | bvn-mi | [Mutual information of the bivariate normal distribution](/P/bvn-mi) | JoramSoch | 2024-11-01 |
 | P477 | mvn-mi | [Mutual information of the multivariate normal distribution](/P/mvn-mi) | JoramSoch | 2024-11-01 |
 | P478 | mgf-sumind | [Moment-generating function of a sum of independent random variables](/P/mgf-sumind) | JoramSoch | 2024-11-08 |
+| P479 | lognorm-prodind | [The product of independent log-normal random variables is a log-normal random variable](/P/lognorm-prodind) | JoramSoch | 2024-11-15 |
diff --git a/I/PbT.md b/I/PbT.md
@@ -506,6 +506,7 @@ title: "Proof by Topic"
 - [The log probability scoring rule is a strictly proper scoring rule](/P/lpsr-spsr)
 - [The median minimizes the mean absolute error](/P/med-mae)
 - [The p-value follows a uniform distribution under the null hypothesis](/P/pval-h0)
+- [The product of independent log-normal random variables is a log-normal random variable](/P/lognorm-prodind)
 - [The regression line goes through the center of mass point](/P/slr-comp)
 - [The residuals and the covariate are uncorrelated in simple linear regression](/P/slr-rescorr)
 - [The sum of residuals is zero in simple linear regression](/P/slr-ressum)
diff --git a/P/anova2-ols.md b/P/anova2-ols.md
@@ -58,7 +58,7 @@ $$ \label{eq:mean-samp}
 \end{split}
 $$
 
-with the sample size numbers
+with the [sample size](/D/samp-size) numbers
 
 $$ \label{eq:samp-size}
 \begin{split}
diff --git a/P/ci-wilks.md b/P/ci-wilks.md
@@ -74,7 +74,7 @@ $$ \label{eq:llr}
 \log \Lambda(\phi) = \log p(y|\phi,\hat{\lambda}) - \log p(y|\hat{\phi},\hat{\lambda}) \; .
 $$
 
-[Wilks' theorem](/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:
+[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](/D/samp-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:
 
 $$ \label{eq:wilks}
 H_0: \theta \in \Theta_0 \quad \Rightarrow \quad -2 \log \frac{\operatorname*{max}_{\theta \in \Theta_0} p(y|\theta)}{\operatorname*{max}_{\theta \in \Theta_1} p(y|\theta)} \sim \chi^2_{\Delta k}  \quad \text{as} \quad n \rightarrow \infty
diff --git a/P/lognorm-prodind.md b/P/lognorm-prodind.md
@@ -33,7 +33,7 @@ $$ \label{eq:X-lognorm}
 X_i \sim \ln \mathcal{N}(\mu_i, \sigma_i^2), \; i = 1, \ldots, n \; .
 $$
 
-Then, the product of these random variables also follows a [log-normal distribution](/D/lognorm):
+Then, the product of these random variables also follows a [log-normal distribution](/D/lognorm)
 
 $$ \label{eq:Z-lognorm}
 Z = \prod_{i=1}^n X_i \sim \mathcal{N}(\mu, \sigma^2)
@@ -76,7 +76,7 @@ This means that the logarithm of the product of independent [log-normal](/D/logn
 $$ \label{eq:ln-Z-norm}
      \ln Z
 =    \sum_{i=1}^n Y_i
-\sim \mathcal{N}\left( \sum_{i=1}^n \mu_i, \sum_{i=1}^n \sigma_i^2 \right) \; .
+\sim \mathcal{N}\left( \sum_{i=1}^n \mu_i, \, \sum_{i=1}^n \sigma_i^2 \right) \; .
 $$
 
 If a random variable [follows a normal distribution, then its exponential follows a log-normal distribution with the same parameters]:
@@ -92,7 +92,7 @@ Thus, from \eqref{eq:ln-Z-norm}, we have
 $$ \label{eq:Z-lognorm-qed}
      Z
 =    \exp(\ln Z) 
-\sim \ln \mathcal{N}\left( \sum_{i=1}^n \mu_i, \sum_{i=1}^n \sigma_i^2 \right)
+\sim \ln \mathcal{N}\left( \sum_{i=1}^n \mu_i, \, \sum_{i=1}^n \sigma_i^2 \right)
 $$
 
 which is equivalent to \eqref{eq:Z-lognorm} and \eqref{eq:Z-lognorm-para}.
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