Page 107 - Contributed Paper Session (CPS) - Volume 1

CPS1167 Indrani B.

∗
Here, , and denote constant factors. If : = () and = () are
3
2
1
statistics for which
((), (); ) = sup (, ; ),
,
then () is said to be the MLP of and () the predictive maximum
:
likelihood estimator (PMLE) of .

Case 1 : (1 < < − 1 and = 1,· · ·, ) The logarithm of the
1
1
predictive likelihood function (log PLF) of = , corresponding to the PLF
:
given by the first equation of (6), is given by

1 (7)
− 1 1
1
−
log = −( + 1) log + ∑ ( ) − + ( − 1) log (1 − )

=1

− 1 − 2 2

+ ( ) + ∑ ( ) −

= 1 +1

in which with = (, , ) =
1,

− 1 − 1 1 − 1
( − ) and = ; =
1
1 − 2 ln − 2 ln − 1
1, … , and = ( − + ) ; = + 1, … , .
2

1
The predictive likelihood equations (PLEs) are obtained by differentiating
log L in (7) with respect to , , and σ and these are as follows:
2
1

− 1 − 1

where = ( − ) − ( − ) 1 . The MLP : of : is obtained
1

1
1
by solving the first equation of (8) and is given by
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