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| Probability density function|
| Cumulative distribution function|
|Parameters|| shape (real)|
Probability density functionEdit
where is the shape parameter and is the scale parameter of the gamma distribution. (NOTE: this parameterization is what is used in the infobox and the plots.)
Alternatively, the gamma distribution can be parameterized in terms of a shape parameter and an inverse scale parameter , called a rate parameter:
Both parameterizations are common because they are convenient to use in certain situations and fields.
The information entropy is given by:
where is the polygamma function.
If for and then
If , then . Or, more generally, for any it holds that . That is the meaning of θ (or β) being the scale parameter.
Parameter estimation Edit
The likelihood function is
from which we calculate the log-likelihood function
Finding the maximum with respect to by taking the derivative and setting it equal to zero yields the maximum likelihood estimate of the parameter:
Substituting this into the log-likelihood function gives:
Finding the maximum with respect to by taking the derivative and setting it equal to zero yields:
where is the digamma function.
There is no closed-form solution for . The function is numerically very well behaved, so if a numerical solution is desired, it can be found using Newton's method. An initial value of can be found either using the method of moments, or using the approximation:
If we let then is approximately
which is within 1.5% of the correct value.
Generating Gamma random variables Edit
Given the scaling property above, it is enough to generate Gamma variables with as we can later convert to any value of β with simple division.
Using the fact that if , then also , and the method of generating exponential variables, we conclude that if U is uniformly distributed on (0, 1], then . Now, using the "α-addition" property of Gamma distribution, we expand this result:
where are all uniformly distributed on (0, 1 ] and independent.
All that is left now is to generate a variable distributed as for and apply the "α-addition" property once more. This is the most difficult part, however.
We provide an algorithm without proof. It is an instance of the acceptance-rejection method:
- Let m be 1.
- Generate and — independent uniformly distributed on (0, 1] variables.
- If , where , then go to step 4, else go to step 5.
- Let . Go to step 6.
- Let .
- If , then increment m and go to step 2.
- Assume to be the realization of .
Now, to summarize,
where is the integral part of α, ξ has been generating using the algorithm above with (the fractional part of α), and are distributed as explained above and are all independent.
- is an exponential distribution if .
- if for any c > 0 .
- is a gamma distribution if and if the are all independent and share the same parameter .
- is a chi-square distribution if .
- If is an integer, the gamma distribution is an Erlang distribution (so named in honor of A. K. Erlang) and is the probability distribution of the waiting time until the -th "arrival" in a one-dimensional Poisson process with intensity .
- then if , where is the inverse-gamma distribution.
- is a beta distribution if < and and are also independent.
- is a Maxwell-Boltzmann distribution if .
- is a normal distribution as where .
- The real vector follows a Dirichlet distribution if are independent, and .
- R. V. Hogg and A. T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)
See also Edit
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