Beta distribution

Source: vignettes/Distributions-Beta.Rmd

Probability density function:

f(x)=xα−1(1−x)β−1ℬ(α,β)f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathcal{B}(\alpha,\beta)} with α\alpha and β\beta two shape parameters and ℬ\mathcal B beta function.

Cumulative distribution function:

F(x)=∫0xyα−1(1−y)β−1dyℬ(α,β)=ℬ(x;α,β)F(x) = \frac{\int_{0}^{x} y^{\alpha-1}(1-y)^{\beta-1}dy} {\mathcal{B}(\alpha,\beta)} =\mathcal{B}(x; \alpha,\beta) with ℬ(x;α,β)\mathcal B (x; \alpha,\beta) incomplete beta function.

Log-likelihood function:

L(α,β;X)=∑i[(α−1)ln(x)+(β−1)ln(1−x)−lnℬ(α,β)]L(\alpha,\beta;X)=\sum_i\left[ (\alpha-1)\ln(x)+(\beta-1)\ln(1-x)-\ln \mathcal{B}(\alpha,\beta) \right]

Score function vector:

V(μ,σ;X)=(∂L∂α∂L∂β)=∑i(ψ(0)(α+β)−ψ(0)(α)+ln⁡(x)ψ(0)(α+β)−ψ(0)(β)+ln⁡(x))V(\mu,\sigma;X) =\left( \begin{array}{c} \frac{\partial L}{\partial \alpha} \\ \frac{\partial L}{\partial \beta} \end{array} \right) =\sum_i \left( \begin{array}{c} \psi^{(0)}(\alpha+\beta)-\psi^{(0)}(\alpha)+\ln(x) \\ \psi^{(0)}(\alpha+\beta)-\psi^{(0)}(\beta)+\ln(x) \end{array} \right) with ψ(0)\psi^{(0)} being log-gamma function.

Observed information matrix:

𝒥(μ,σ;X)=(ψ(1)(α)−ψ(1)(α+β)−ψ(1)(α+β)−ψ(1)(α+β)ψ(1)(β)−ψ(1)(α+β))\mathcal J (\mu,\sigma;X)= \left( \begin{array}{cc} \psi^{(1)}(\alpha)-\psi^{(1)}(\alpha+\beta) & -\psi^{(1)}(\alpha+\beta) \\ -\psi^{(1)}(\alpha+\beta) & \psi^{(1)}(\beta)-\psi^{(1)}(\alpha+\beta) \end{array} \right) with ψ(1)\psi^{(1)} being digamma function.