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Merge pull request #158 from StatProofBook/master
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D/anova2.md

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$$
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where the weights are $w_{ij} = n_{ij}/n$ and the total sample size is $n = \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij}$.
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where the weights are $w_{ij} = n_{ij}/n$ and the total [sample size](/D/samp-size) is $n = \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij}$.

D/hyp-simp.md

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**Definition:** Let $H$ be a [statistical hypothesis](/D/hyp). Then,
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* $H$ 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.
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* $H$ 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.
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* $H$ 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.

I/DbA.md

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- [Sample covariance](/D/cov-samp)
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### JoramSoch (193 definitions)
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### JoramSoch (194 definitions)
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- [Akaike information criterion](/D/aic)
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- [Alternative hypothesis](/D/h1)
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- [Residual sum of squares](/D/rss)
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- [Residual variance](/D/resvar)
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- [Residual-forming matrix](/D/rfmat)
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- [Sample](/D/samp)
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- [Sample correlation coefficient](/D/corr-samp)
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- [Sample correlation matrix](/D/corrmat-samp)
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- [Sample covariance matrix](/D/covmat-samp)

I/DbN.md

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| D207 | dist-uni | [Unimodal and multimodal probability distribution](/D/dist-uni) | JoramSoch | 2024-10-25 |
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| D208 | est | [Estimand, estimator and estimate](/D/est) | JoramSoch | 2024-11-01 |
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| D209 | est-bias | [Biased vs. unbiased estimator](/D/est-bias) | JoramSoch | 2024-11-08 |
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| D210 | samp | [Sample](/D/samp) | JoramSoch | 2024-11-15 |

I/DbT.md

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### S
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- [Sample](/D/samp)
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- [Sample correlation coefficient](/D/corr-samp)
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- [Sample correlation matrix](/D/corrmat-samp)
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- [Sample covariance](/D/cov-samp)

I/PbA.md

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- [Posterior predictive distribution is a marginal distribution of the joint likelihood](/P/postpred-jl)
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### JoramSoch (442 proofs)
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### JoramSoch (443 proofs)
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- [Accuracy and complexity for Bayesian linear regression](/P/blr-anc)
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- [Accuracy and complexity for Bayesian linear regression with known covariance](/P/blrkc-anc)
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- [t-distribution is a special case of multivariate t-distribution](/P/t-mvt)
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- [t-test for multiple linear regression using contrast-based inference](/P/mlr-t)
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- [The p-value follows a uniform distribution under the null hypothesis](/P/pval-h0)
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- [The product of independent log-normal random variables is a log-normal random variable](/P/lognorm-prodind)
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- [The regression line goes through the center of mass point](/P/slr-comp)
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- [The residuals and the covariate are uncorrelated in simple linear regression](/P/slr-rescorr)
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- [The sum of residuals is zero in simple linear regression](/P/slr-ressum)

I/PbN.md

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| P476 | bvn-mi | [Mutual information of the bivariate normal distribution](/P/bvn-mi) | JoramSoch | 2024-11-01 |
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| P477 | mvn-mi | [Mutual information of the multivariate normal distribution](/P/mvn-mi) | JoramSoch | 2024-11-01 |
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| P478 | mgf-sumind | [Moment-generating function of a sum of independent random variables](/P/mgf-sumind) | JoramSoch | 2024-11-08 |
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| P479 | lognorm-prodind | [The product of independent log-normal random variables is a log-normal random variable](/P/lognorm-prodind) | JoramSoch | 2024-11-15 |

I/PbT.md

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- [The log probability scoring rule is a strictly proper scoring rule](/P/lpsr-spsr)
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- [The median minimizes the mean absolute error](/P/med-mae)
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- [The p-value follows a uniform distribution under the null hypothesis](/P/pval-h0)
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- [The product of independent log-normal random variables is a log-normal random variable](/P/lognorm-prodind)
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- [The regression line goes through the center of mass point](/P/slr-comp)
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- [The residuals and the covariate are uncorrelated in simple linear regression](/P/slr-rescorr)
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- [The sum of residuals is zero in simple linear regression](/P/slr-ressum)

P/anova2-ols.md

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$$
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with the sample size numbers
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with the [sample size](/D/samp-size) numbers
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$$ \label{eq:samp-size}
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P/ci-wilks.md

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\log \Lambda(\phi) = \log p(y|\phi,\hat{\lambda}) - \log p(y|\hat{\phi},\hat{\lambda}) \; .
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[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:
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[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:
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$$ \label{eq:wilks}
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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

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