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‎D/para.md‎

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---
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**Definition:** A parameter, also "statistical parameter", is any fixed quantity, i.e. [constant](/D/const) scalar, vector or matrix, that describes a parametrized [probability distribution](/D/dist).
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**Definition:** A parameter, also "statistical parameter", is any fixed quantity, i.e. [constant](/D/const) scalar, vector or matrix, that describes a parametrized [probability distribution](/D/dist) by influencing its [probability mass function](/D/pmf) or [probability density function](/D/pdf).
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Examples of parameters include the mean and variance parameters of a [normal distribution](/D/norm), covariance parameters in a [multivariate](/D/mvn) or [matrix](/D/matn)-normal distribution, shape and rate parameters of the [gamma distribution](/D/gam) or the vector of category probabilities in a [multinomial distribution](/D/mult).

‎I/DbA.md‎

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- [Sample covariance](/D/cov-samp)
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### JoramSoch (187 definitions)
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### JoramSoch (188 definitions)
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- [Akaike information criterion](/D/aic)
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- [Alternative hypothesis](/D/h1)
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- [One-tailed and two-tailed test](/D/test-tail)
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- [One-way analysis of variance](/D/anova1)
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- [p-value](/D/pval)
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- [Parameter](/D/para)
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- [Point and set hypothesis](/D/hyp-point)
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- [Poisson distribution](/D/poiss)
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- [Poisson distribution with exposure values](/D/poissexp)

‎I/DbN.md‎

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| D201 | post-pred | [Posterior predictive distribution](/D/post-pred) | aloctavodia | 2024-08-18 |
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| D202 | prior-pred | [Prior predictive distribution](/D/prior-pred) | aloctavodia | 2024-08-19 |
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| D203 | data | [Data](/D/data) | JoramSoch | 2024-09-20 |
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| D204 | para | [Parameter](/D/para) | JoramSoch | 2024-09-27 |

‎I/DbT.md‎

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### P
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- [p-value](/D/pval)
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- [Parameter](/D/para)
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- [Point and set hypothesis](/D/hyp-point)
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- [Poisson distribution](/D/poiss)
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- [Poisson distribution with exposure values](/D/poissexp)

‎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 (435 proofs)
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### JoramSoch (436 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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- [Gibbs' inequality](/P/gibbs-ineq)
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- [Independence of estimated parameters and residuals in multiple linear regression](/P/mlr-ind)
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- [Independence of products of multivariate normal random vector](/P/mvn-indprod)
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- [Independent random variables are uncorrelated](/P/corr-ind)
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- [Inflection points of the probability density function of the normal distribution](/P/norm-infl)
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- [Invariance of the covariance matrix under addition of constant vector](/P/covmat-inv)
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- [Invariance of the differential entropy under addition of a constant](/P/dent-inv)
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- [Variance of the exponential distribution](/P/exp-var)
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- [Variance of the log-normal distribution](/P/lognorm-var)
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### salbalkus (1 proof)
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### salbalkus (2 proofs)
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- [The expected value minimizes the mean squared error](/P/mean-mse)
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- [The median minimizes the mean absolute error](/P/med-mae)
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### StatProofBook (1 proof)
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‎I/PbN.md‎

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| P468 | mean-wlln | [Weak law of large numbers](/P/mean-wlln) | JoramSoch | 2024-09-13 |
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| P469 | mean-mse | [The expected value minimizes the mean squared error](/P/mean-mse) | salbalkus | 2024-09-13 |
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| P470 | ind-self | [Self-independence of random event](/P/ind-self) | JoramSoch | 2024-09-20 |
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| P471 | med-mae | [The median minimizes the mean absolute error](/P/med-mae) | salbalkus | 2024-09-23 |
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| P472 | corr-ind | [Independent random variables are uncorrelated](/P/corr-ind) | JoramSoch | 2024-09-27 |

‎I/PbT.md‎

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- [Independence of estimated parameters and residuals in multiple linear regression](/P/mlr-ind)
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- [Independence of products of multivariate normal random vector](/P/mvn-indprod)
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- [Independent random variables are uncorrelated](/P/corr-ind)
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- [Inflection points of the probability density function of the normal distribution](/P/norm-infl)
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- [Invariance of the covariance matrix under addition of constant vector](/P/covmat-inv)
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- [Invariance of the differential entropy under addition of a constant](/P/dent-inv)
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- [t-test for multiple linear regression using contrast-based inference](/P/mlr-t)
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- [The expected value minimizes the mean squared error](/P/mean-mse)
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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 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)

‎P/anova1-repara.md‎

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\end{split}
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$$
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2) Equation \eqref{eq:anova1-repara} is a special case of the [two-way analysis of variance](/D/anova2) with (i) just one factor $A$ and (ii) no interaction term. Thus, OLS estimates are identical to [that of two-way ANOVA](/P/anova2-ols), i.e. given by
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2) The [residual sum of squares](/D/rss) for the reparametrized model is
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$$ \label{eq:anova1-repara-c2-qed}
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$$ \label{eq:anova1-repara-rss}
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\mathrm{RSS}(\mu,\delta) = \sum_{i=1}^{k} \sum_{j=1}^{n_i} \varepsilon_{ijk}^2 = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (y_{ij} - \mu - \delta_i)^2
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$$
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and the derivatives of $\mathrm{RSS}$ with respect to $\mu$, $\delta$ are
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$$ \label{eq:anova1-repara-rss-der-mu}
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\begin{split}
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\frac{\mathrm{d}\mathrm{RSS}}{\mathrm{d}\mu}
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&= \sum_{i=1}^{k} \sum_{j=1}^{n_i} \frac{\mathrm{d}}{\mathrm{d}\mu} (y_{ij} - \mu - \delta_i)^2 \\
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&= \sum_{i=1}^{k} \sum_{j=1}^{n_i} -2 (y_{ij} - \mu - \delta_i) \\
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&= \sum_{i=1}^{k} \left( 2 n_i \mu + 2 n_i \delta_i - 2 \sum_{j=1}^{n_i} y_{ij} \right) \\
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&= 2 n \mu + 2 \sum_{i=1}^{k} n_i \delta_i - 2 \sum_{i=1}^{k} \sum_{j=1}^{n_i} y_{ij}
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\end{split}
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$$
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$$ \label{eq:anova1-repara-rss-der-delta}
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\begin{split}
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\frac{\mathrm{d}\mathrm{RSS}}{\mathrm{d}\delta_i}
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&= \sum_{j=1}^{n_i} \frac{\mathrm{d}}{\mathrm{d}\delta_i} (y_{ij} - \mu - \delta_i)^2 \\
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&= \sum_{j=1}^{n_i} -2 (y_{ij} - \mu - \delta_i) \\
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&= 2 n_i \mu + 2 n_i \delta_i - 2 \sum_{j=1}^{n_i} y_{ij} \; .
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\end{split}
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$$
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Setting these derivatives to zero, we obtain the estimates of $\mu$ and $\delta_i$:
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$$ \label{eq:anova1-repara-rss-der-mu-zero}
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\begin{split}
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0 &= 2 n \hat{\mu} + 2 \sum_{i=1}^{k} n_i \delta_i - 2 \sum_{i=1}^{k} \sum_{j=1}^{n_i} y_{ij} \\
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\hat{\mu} &= \frac{1}{n} \sum_{i=1}^{k} \sum_{j=1}^{n_i} y_{ij} - \sum_{i=1}^{k} \frac{n_i}{n} \delta_i \\
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&\overset{\eqref{eq:anova1-constr}}{=} \frac{1}{n} \sum_{i=1}^{k} \sum_{j=1}^{n_i} y_{ij} \\
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&= \bar{y}
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\end{split}
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$$
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$$ \label{eq:anova1-repara-rss-der-delta-zero}
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\begin{split}
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\hat{\mu} &= \bar{y}_{\bullet \bullet} = \frac{1}{n} \sum_{i=1}^{k} \sum_{j=1}^{n_i} y_{ij} \\
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\hat{\delta}_i &= \bar{y}_{i \bullet} - \bar{y}_{\bullet \bullet} = \frac{1}{n_i} \sum_{j=1}^{n_i} y_{ij} - \frac{1}{n} \sum_{i=1}^{k} \sum_{j=1}^{n_i} y_{ij} \; .
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0 &= 2 n_i \hat{\mu} + 2 n_i \hat{\delta}_i - 2 \sum_{j=1}^{n_i} y_{ij} \\
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\hat{\delta}_i &= \frac{1}{n_i} \sum_{j=1}^{n_i} y_{ij} - \hat{\mu} \\
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&\overset{\eqref{eq:anova1-repara-rss-der-mu-zero}}{=} \frac{1}{n_i} \sum_{j=1}^{n_i} y_{ij} - \frac{1}{n} \sum_{i=1}^{k} \sum_{j=1}^{n_i} y_{ij} \\
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&= \bar{y}_i - \bar{y} \; .
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\end{split}
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$$
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‎P/anova2-ols.md‎

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\end{split}
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$$
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Setting these derivatives to zero, we obtain the estimates of $\mu_i$, $\alpha_i$, $\beta_j$ and $\gamma_{ij}$:
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Setting these derivatives to zero, we obtain the estimates of $\mu$, $\alpha_i$, $\beta_j$ and $\gamma_{ij}$:
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$$ \label{eq:rss-der-mu-zero}
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\begin{split}

‎P/med-mae.md‎

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$$ \label{eq:med-mae-s2}
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\begin{split}
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& \frac{\partial}{\partial a} \left( \int_{-\infty}^a (a - x) f(x) \, \mathrm{d}x + \int_{a}^\infty (x - a) f(x) \, \mathrm{d}x \right) \\
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= & (a - x) f(x) + \int_{-\infty}^a f(x) \, \mathrm{d}x - (x - a) f(x) - \int_{a}^\infty f(x) \, \mathrm{d}x \; .
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&\frac{\partial}{\partial a} \left( \int_{-\infty}^a (a - x) f(x) \, \mathrm{d}x + \int_{a}^\infty (x - a) f(x) \, \mathrm{d}x \right) \\
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=\; &(a - x) f(x) + \int_{-\infty}^a f(x) \, \mathrm{d}x - (x - a) f(x) - \int_{a}^\infty f(x) \, \mathrm{d}x \; .
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\end{split}
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$$
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