
Bayesian Semiparametric Longitudinal Data Modeling Using NI Densities 177
• a ga mma distribution in the Student-t case i.e., Gamma
ν+n
i
2
,
ν
2
+
λ
i
2σ
2
e
;
• a truncated gamma distribution in the slash ca se i.e.,
TGamma
ν +
n
i
2
,
λ
i
2σ
2
e
, (0, 1)
; and
• a discrete distribution taken one of two sta tes in the contaminated normal
case i.e.,
π(u
i
|y
i
, b
i
, θ) =
(
η
i
ξ
i
+η
i
if u
i
= ρ
ξ
i
ξ
i
+η
i
if u
i
= 1
where η
i
= νρ
n
i
/2
exp
n
−
ρλ
i
2σ
2
e
o
and ξ
i
= ν exp
n
−
λ
i
2σ
2
e
o
.
π(ν|y, b, u, θ) is:
• in the Student-t case, proportional to
(ν/2)
nν/2
(Γ(ν/2))
n
exp
−
ν
2
"
n
X
i=1
(u
i
− log(u
i
)) + γ
#!
I
{(2,∞)}
(ν)
• a Gamma distribution i.e., Gamma(a + n, b −
P
n
i=1
log u
i
) in the slash
case;
• and given by,
π(ν|y, b, u