Page 109 - Contributed Paper Session (CPS) - Volume 1

CPS1167 Indrani B.

1 (15)
1 − 1
∑( + 1) 1 + = + 1.
1
1
=1

The linearized PMLE ̃ is given by

(16)

in which is as given in Case 1. The PMLE ̃ corresponding to the linearized
1
1
PMLE ̃ in (16) is obtained by solving (15).

Case 3 : (0 ≤ ≤ − 1 and = + 1, … , )Following similar derivations,
1

the modified MLP (MMLP) in this case is given by
if =1 (17)
= { − ̃ 2

: ̃ +̃ log [ ̃ + (, ] if 1 < ≤
2
)

3.2. Conditional Median Predictor
The median of the conditional distribution of , given , can be used as
:
a predictor of : . This predictor is called the Conditional Median Predictor

(CMP) of : . and will be denoted by : . The CMP : of : is such that

: 1
∫ (| ) = in which (| ) is the conditional density of , given

2 :
: − −

= . Since [ − | = ] where and are the location and scale

parameters respectively (in general), has the same distribution as the random
variable , the CMP of is given by
: : :
− 1
= + log [ + ]
1
: ,
Where is the -th order statistics out of units from Exp(1) and
:
Med[ ] = .
: :

Case 1 : 1 ≤ ≤ − 1 and = 1, … , The CMP : of : is given by
1
1
− ̂ 1 (18)
= ̂ + ̂ log [ ̂ + ] ,
: 1 ,
in which ̂ and ̂ can be used as the linearized uniformly minimum variance
1
unbiased estimator (UMVUE) of and to reduce the variability in the
1
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