Table 1 Probability distributions considered in this study. The probability density function f(·) is shown in the second column, and the parameter vector θ defining each distribution in the third column
From: Automatic detection of discordant outliers via the Ueda’s method
Distribution | f(·) | θ |
|---|---|---|
Normal | \(\frac {1}{\sqrt {2\pi }\sigma }e^{-\frac {(x-\mu)^{2}}{2\sigma ^{2}}}\) | μ,σ |
Student’s t | \(\frac {\Gamma (\frac {\nu +1}{2})}{\Gamma (\nu /2)\sqrt {\nu \pi }} \left (1+\frac {x^{2}}{\nu }\right)^{-\frac {\nu +1}{2}}\) | ν |
Tukey λ | See text in §2 | λ |
Beta | \(\frac {\Gamma (\alpha + \beta)}{\Gamma (\alpha) \Gamma (\beta)} x^{\alpha -1} (1-x)^{\beta -1}\) | α,β |
Ex-Gaussian | \(\frac {1}{\nu \sqrt {2\pi }}e^{\frac {\sigma ^{2}}{2\nu ^{2}} - \frac {x-\mu }{\nu }}\cdot \int _{-\infty }^{[(x-\mu)/\sigma ] - \sigma /\nu } e^{-\frac {y^{2}}{2}}dy\) | μ,σ,ν |
Gamma | \(\frac {1}{\Gamma (\alpha)\beta ^{\alpha }}x^{\alpha -1}e^{-\frac {x}{\beta }}\) | α,β |
Weibull | \(\frac {\beta }{\alpha ^{\beta }}x^{\beta -1} e^{-\left (\frac {x}{\alpha }\right)^{\beta }}\) | α,β |
Lognormal | \(\frac {1}{x\sqrt {2\pi }\sigma }e^{- \frac {\left (\ln x - \mu \right)^{2}}{2\sigma ^{2}}}\) | μ,σ |