Page 59 - Contributed Paper Session (CPS) - Volume 5

CPS886 Marcelo Bourguignon
In particular, the mean and the variance associated with (2) are given by
( + − 1)
[] = , > 1, [] = , > 2. (3)
− 1 ( − 2)( − 1) 2

The rest of the paper proceeds as follows. In Section 2, we introduce a new
parameterization of the BP distribution that is indexed by the mean and
precision parameters and presents the BP regression model with varying mean
and precision. In Section 3, some numerical results of the estimators are
presented with a discussion of the obtained results. In Section 4, we discuss an
application to real data that demonstrates the usefulness of the proposed
model. Concluding remarks are given in the final section.

2. A BP distribution parameterized by its mean and precision parameters
Regression models are typically constructed to model the mean of a
distribution. However, the density of the BP distribution is given in Equation
(2), where it is indexed by α and . In this context, in this section, we considered
a new parameterization of the BP distribution in terms of the mean and
precision parameters. Consider the parameterization = ∕ ( − 1) and ∅ =
− 2, i.e., = (1 + ∅) and = 2 + ∅. Under this new parameterization, it
follows from (3) that
(1 + )
[] = [] = .

From now on, we use the notation Y ∼ (, ) to indicate that Y is a
random variable following a BP distribution with mean µ and precision
parameter . Note that V() = (1 + ) is similar to the variance function of
the gamma distribution, for which the the variance has a quadratic relation
with its mean. We note that this parameterization was not proposed in the
statistical literature. Using the proposed parameterization, the BP density in
(2) can be written as
(+1)−1 (1 + ) −[(+1)++2]
(|, ) = , > 0, (4)
((1 + ), + 2)

where > 0 > 0.
Let Y1,...,Yn be independent random variables, where each = 1, . . . . , ,
,
follows the pdf given in (4) with mean and precision parameter . Suppose

the mean and the precision parameter of Yi satisfies the following functional
relations
⊺
⊺
( ) = = x ( ) = 2 = z (5)

2

1
1

T
T
where = ( , . . . , ) and = ( , . . . , ) are vectors of unknown regression

1
1

coefficients which are assumed to be functionally independent, ∈ ℝ and
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