The expected value, variance, and other details of the distribution are given in the sidebox; for , the
excess kurtosis is
While the related
beta distribution is the
conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in
odds. The distribution is a
Pearson type VI distribution.[1]
The mode of a variate X distributed as is .
Its mean is if (if the mean is infinite, in other words it has no well defined mean) and its variance is if .
Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution.
This generalization can be obtained via the following invertible transformation. If and for , then .
Compound gamma distribution
The compound gamma distribution[3] is the generalization of the beta prime when the scale parameter, q is added, but where p = 1. It is so named because it is formed by
compounding two
gamma distributions:
where is the gamma pdf with shape and inverse scale .
The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q2.
Another way to express the compounding is if and , then . (This gives one way to generate random variates with compound gamma, or beta prime distributions. Another is via the ratio of independent gamma variates, as shown below.)
Properties
If then .
If , and , then .
If then .
If and two iid variables, then with and , as the beta prime distribution is infinitely divisible.
More generally, let iid variables following the same beta prime distribution, i.e. , then the sum with and .
^Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021). "A new regression model for positive random variables with skewed and long tail". Metron. 79: 33–55.
doi:
10.1007/s40300-021-00203-y.
S2CID233534544.
Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", Metron, 79: 33–55,
doi:
10.1007/s40300-021-00203-y,
S2CID233534544
The expected value, variance, and other details of the distribution are given in the sidebox; for , the
excess kurtosis is
While the related
beta distribution is the
conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in
odds. The distribution is a
Pearson type VI distribution.[1]
The mode of a variate X distributed as is .
Its mean is if (if the mean is infinite, in other words it has no well defined mean) and its variance is if .
Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution.
This generalization can be obtained via the following invertible transformation. If and for , then .
Compound gamma distribution
The compound gamma distribution[3] is the generalization of the beta prime when the scale parameter, q is added, but where p = 1. It is so named because it is formed by
compounding two
gamma distributions:
where is the gamma pdf with shape and inverse scale .
The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q2.
Another way to express the compounding is if and , then . (This gives one way to generate random variates with compound gamma, or beta prime distributions. Another is via the ratio of independent gamma variates, as shown below.)
Properties
If then .
If , and , then .
If then .
If and two iid variables, then with and , as the beta prime distribution is infinitely divisible.
More generally, let iid variables following the same beta prime distribution, i.e. , then the sum with and .
^Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021). "A new regression model for positive random variables with skewed and long tail". Metron. 79: 33–55.
doi:
10.1007/s40300-021-00203-y.
S2CID233534544.
Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", Metron, 79: 33–55,
doi:
10.1007/s40300-021-00203-y,
S2CID233534544