Page 58 - Contributed Paper Session (CPS) - Volume 5

CPS886 Marcelo Bourguignon
We summarize below the main contributions and advantages of the
proposed BP model over the popular gamma model. With these contributions
below, we provide a complete tool for modelling asymmetric data based on
our BP regression.
• We allow a regression structure on the precision parameter; in a manner
similar to the way the generalized linear models with dispersion
covariates extend the generalized linear models.
• The variance function of proposed model assumes a quadratic form
similar to the gamma distribution. However, the variance function of
proposed model is larger than the variance function of gamma
distribution, which may be more appropriate in certain practical
situations.
• The BP hazard rate function can have an upside-down bathtub or
increasing depending on the parameter values. Most classical two-
parameter distributions such as Weibull and gamma distributions have
monotone hazard rate functions.
• The skewness and kurtosis of the BP distribution can be much larger
than the of the gamma distribution.

The BP distribution (Keeping, 1962; McDonald, 1984) is also known as
inverted beta distribution or beta distribution of the second kind. However,
only a few works have studied the BP distribution. McDonald (1987) discussed
its properties and obtained the maximum likelihood estimates of the model
parameters. Tulupyev et al. (2013) used the BP distribution while discussing
regression models for positive random variables. However, these works have
considered the usual parameterization of the BP distribution.
A random variable Y follows the BP distribution with shape parameters α
> 0 and β > 0, denoted by Y ∼ BP(α,β), if its cumulative distribution function
(cdf) is given by
(|, ) = /(1+) (, ), > 0, (1)

Where (, ) = (, )/(, ) is the incomplete beta function

ratio, (, ) = ∫ −1 (1 − ) −1 is the incomplete function, (, ) =

0
∞
()()/( + ) is the beta function and () = ∫ −1 − is the

0
gamma function. The probability density function (pdf) associated with (1) is
−1 (1 + ) −(+)
(|, ) = , > 0. (2)
(, )

The rth moment about zero of Y is given by

( + , − )

[ ] = , − < < .
(, )
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