| Property . | Equation . |
|---|---|
| |$\frac{|Q|k^{k}}{x\sigma\Gamma(k)} \exp\left[ k(Qw - e^{Qw}) \right]$| |$\text{where } w = \frac{log(x) - \mu}{\sigma}$| | |
| PDF collapses to gamma when | |$Q = \sigma$| |
| PDF collapses to lognormal as | |$Q \to 0$| |
| Mean | |$\Gamma \left(\frac{k\beta + 1}{\beta }\right) \cdot \frac{1}{\Gamma (k)} \cdot e^{\mu - \frac{\log (k)}{\beta }}, \quad \text{for } Q > 0$| |
| Variance | |$\left(e^{\mu -\frac{\log (k)}{\beta }}\right)^{\!2} \cdot \left[ \frac{\Gamma \left(\frac{k\beta +2}{\beta }\right)}{\Gamma (k)} - \left(\frac{\Gamma \left(\frac{k\beta +1}{\beta }\right)}{\Gamma (k)} \!\right)^{\!2} \right], \, \, \text{for } Q > 0$| |
| CV | |$\sqrt{ \Gamma(k) \cdot \Gamma\left( \frac{k\beta + 2}{\beta} \right) \cdot \left[ \! \Gamma\left( \frac{k\beta + 1}{\beta}\! \right) \right]^{\!-2}- 1}, \, \,\text{for } Q > 0$| |
| Property . | Equation . |
|---|---|
| |$\frac{|Q|k^{k}}{x\sigma\Gamma(k)} \exp\left[ k(Qw - e^{Qw}) \right]$| |$\text{where } w = \frac{log(x) - \mu}{\sigma}$| | |
| PDF collapses to gamma when | |$Q = \sigma$| |
| PDF collapses to lognormal as | |$Q \to 0$| |
| Mean | |$\Gamma \left(\frac{k\beta + 1}{\beta }\right) \cdot \frac{1}{\Gamma (k)} \cdot e^{\mu - \frac{\log (k)}{\beta }}, \quad \text{for } Q > 0$| |
| Variance | |$\left(e^{\mu -\frac{\log (k)}{\beta }}\right)^{\!2} \cdot \left[ \frac{\Gamma \left(\frac{k\beta +2}{\beta }\right)}{\Gamma (k)} - \left(\frac{\Gamma \left(\frac{k\beta +1}{\beta }\right)}{\Gamma (k)} \!\right)^{\!2} \right], \, \, \text{for } Q > 0$| |
| CV | |$\sqrt{ \Gamma(k) \cdot \Gamma\left( \frac{k\beta + 2}{\beta} \right) \cdot \left[ \! \Gamma\left( \frac{k\beta + 1}{\beta}\! \right) \right]^{\!-2}- 1}, \, \,\text{for } Q > 0$| |
We define μ as the location parameter, σ as the scale parameter, and Q as the family or shape parameter. In this form of the PDF, μ > 0 and σ > 0, and Q ∈ (−∞, +∞). We further define two temporary variables to simplify notation: β = Q/σ and k = Q −2. The symbol Γ is the gamma function. Our parameterization is based on Prentice (1974), Lawless (1980), and Jackson (2016). Moments for |$Q > 0$| are derived in Stacy and Mihram (1965).
| Property . | Equation . |
|---|---|
| |$\frac{|Q|k^{k}}{x\sigma\Gamma(k)} \exp\left[ k(Qw - e^{Qw}) \right]$| |$\text{where } w = \frac{log(x) - \mu}{\sigma}$| | |
| PDF collapses to gamma when | |$Q = \sigma$| |
| PDF collapses to lognormal as | |$Q \to 0$| |
| Mean | |$\Gamma \left(\frac{k\beta + 1}{\beta }\right) \cdot \frac{1}{\Gamma (k)} \cdot e^{\mu - \frac{\log (k)}{\beta }}, \quad \text{for } Q > 0$| |
| Variance | |$\left(e^{\mu -\frac{\log (k)}{\beta }}\right)^{\!2} \cdot \left[ \frac{\Gamma \left(\frac{k\beta +2}{\beta }\right)}{\Gamma (k)} - \left(\frac{\Gamma \left(\frac{k\beta +1}{\beta }\right)}{\Gamma (k)} \!\right)^{\!2} \right], \, \, \text{for } Q > 0$| |
| CV | |$\sqrt{ \Gamma(k) \cdot \Gamma\left( \frac{k\beta + 2}{\beta} \right) \cdot \left[ \! \Gamma\left( \frac{k\beta + 1}{\beta}\! \right) \right]^{\!-2}- 1}, \, \,\text{for } Q > 0$| |
| Property . | Equation . |
|---|---|
| |$\frac{|Q|k^{k}}{x\sigma\Gamma(k)} \exp\left[ k(Qw - e^{Qw}) \right]$| |$\text{where } w = \frac{log(x) - \mu}{\sigma}$| | |
| PDF collapses to gamma when | |$Q = \sigma$| |
| PDF collapses to lognormal as | |$Q \to 0$| |
| Mean | |$\Gamma \left(\frac{k\beta + 1}{\beta }\right) \cdot \frac{1}{\Gamma (k)} \cdot e^{\mu - \frac{\log (k)}{\beta }}, \quad \text{for } Q > 0$| |
| Variance | |$\left(e^{\mu -\frac{\log (k)}{\beta }}\right)^{\!2} \cdot \left[ \frac{\Gamma \left(\frac{k\beta +2}{\beta }\right)}{\Gamma (k)} - \left(\frac{\Gamma \left(\frac{k\beta +1}{\beta }\right)}{\Gamma (k)} \!\right)^{\!2} \right], \, \, \text{for } Q > 0$| |
| CV | |$\sqrt{ \Gamma(k) \cdot \Gamma\left( \frac{k\beta + 2}{\beta} \right) \cdot \left[ \! \Gamma\left( \frac{k\beta + 1}{\beta}\! \right) \right]^{\!-2}- 1}, \, \,\text{for } Q > 0$| |
We define μ as the location parameter, σ as the scale parameter, and Q as the family or shape parameter. In this form of the PDF, μ > 0 and σ > 0, and Q ∈ (−∞, +∞). We further define two temporary variables to simplify notation: β = Q/σ and k = Q −2. The symbol Γ is the gamma function. Our parameterization is based on Prentice (1974), Lawless (1980), and Jackson (2016). Moments for |$Q > 0$| are derived in Stacy and Mihram (1965).
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