GammaDistribution Class

The gamma distribution is a two-parameter family of continuous probability distributions. There are three different parameterizations in common use:

• With a Shape parameter k and a Scale parameter θ.
• With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a Rate parameter.
• With a shape parameter k and a Mean parameter μ = k/β.

In each of these three forms, both parameters are positive real numbers. The parameterization with k and θ appears to be more common in econometrics and certain other applied fields, where e.g. the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution. This is the default construction method for this class.

The parameterization with α and β is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (aka rate) parameters, such as the λ of an exponential distribution or a Poisson distribution – or for that matter, the β of the gamma distribution itself. (The closely related inverse gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution.) In order to create a Gamma distribution using the Bayesian parameterization, you can use FromBayesian(Double, Double).

If k is an integer, then the distribution represents an Erlang distribution; i.e., the sum of k independent exponentially distributed random variables, each of which has a mean of θ (which is equivalent to a rate parameter of 1/θ).

The gamma distribution is the maximum entropy probability distribution for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).

References:

• Wikipedia, The Free Encyclopedia. Gamma distribution. Available on: http://en.wikipedia.org/wiki/Gamma_distribution

The following example shows how to create, test and compute the main functions of a Gamma distribution given parameters θ = 4 and k = 2:

// Create a Γ-distribution with k = 2 and θ = 4
var gamma = new GammaDistribution(theta: 4, k: 2);

// Common measures
double mean = gamma.Mean;     // 8.0
double median = gamma.Median; // 6.7133878418421506
double var = gamma.Variance;  // 32.0
double mode = gamma.Mode;     // 4.0

// Cumulative distribution functions
double cdf = gamma.DistributionFunction(x: 0.27); // 0.002178158242390601
double ccdf = gamma.ComplementaryDistributionFunction(x: 0.27); // 0.99782184175760935
double icdf = gamma.InverseDistributionFunction(p: cdf); // 0.26999998689819171

// Probability density functions
double pdf = gamma.ProbabilityDensityFunction(x: 0.27); // 0.015773530285395465
double lpdf = gamma.LogProbabilityDensityFunction(x: 0.27); // -4.1494220422235433

// Hazard (failure rate) functions
double hf = gamma.HazardFunction(x: 0.27); // 0.015807962529274005
double chf = gamma.CumulativeHazardFunction(x: 0.27); // 0.0021805338793574793

// String representation
string str = gamma.ToString(CultureInfo.InvariantCulture); // "Γ(x; k = 2, θ = 4)"

Reference