Transformed Gamma Distribution

The transformed gamma distribution, also known as the generalized gamma distribution, is a continuous probability distribution that generalizes the gamma distribution. It is used to model the time until an event occurs a specified number of times.

The transformed gamma distribution has two shape parameters $a$ and $b$, and a scale parameter $\theta$. The probability density function (PDF) is given by:

$$f(x;a,b,\theta )=\frac{b{x}^{a-1}{e}^{-(x/\theta {)}^b}}{{\theta }^a\mathrm{\Gamma }(a/b)}$$

where $x\ge 0$, $a>0$, $b>0$, and $\theta >0$. The cumulative distribution function (CDF) is:

$$F(x;a,b,\theta )=\gamma (\frac{a}{b},{\left(\frac{x}{\theta }\right)}^b)/\mathrm{\Gamma }\left(\frac{a}{b}\right)$$

where $\gamma$ is the lower incomplete gamma function.

• Reliability engineering uses it for modeling failure times.

• Survival analysis employs it in medical research.

• Environmental science uses it for modeling extreme events.

• Material science applies it to strength and lifetime modeling.

Notable properties include:

• The hazard rate can be increasing, decreasing, or bathtub-shaped depending on the parameter values.

• The distribution exhibits great flexibility in shape due to its three parameters.

• The distribution is unimodal for $a>1$.

• When $b=1$, it reduces to the gamma distribution.

• When $a=b$, it becomes the Weibull distribution.

• When $b=2$, it is related to the generalized normal distribution.

• When $a=1$, it becomes the stretched exponential distribution.

The transformed gamma distribution is implemented by the TransformedGammaDistribution class. It has one constructor that takes three arguments. The first and second are the two shape parameters. The third is the scale parameter.

The following constructs a transformed gamma distribution with shape parameters 4.2 and 3, and scale parameter 1:

var trgamma1 = new TransformedGammaDistribution(4.2, 3.0, 1.0);

Dim trgamma1 = New TransformedGammaDistribution(4.2, 3.0, 1.0)

No code example is currently available or this language may not be supported.

let trgamma1 = TransformedGammaDistribution(4.2, 3.0, 1.0)

The TransformedGammaDistribution class has three specific properties, ShapeParameter1, ShapeParameter2, and ScaleParameter, which return the shape and scale parameters of the distribution.

TransformedGammaDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. It takes fourth parameters. The first argument is the random number generator. The second to fourth parameters are the two shape parameters and the scale parameter of the distribution.

var random = new Pcg32();
double sample = TransformedGammaDistribution.Sample(random, 4.2, 3.0, 1.0);

Dim random = New Pcg32()
Dim sample = TransformedGammaDistribution.Sample(random, 4.2, 3.0, 1.0)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = TransformedGammaDistribution.Sample(random, 4.2, 3.0, 1.0)

The above example uses the Pcg32 to generate uniform random numbers.

For more information on the transformed gamma distribution, refer to the following sources:

• Venter, G. (1994). "Transformed beta and gamma distributions and aggregate losses"

  Proceedings of the Casualty Actuarial Society 70 (133), 289-308

• Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

• Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Other Resources