Page 271 - Contributed Paper Session (CPS) - Volume 2

CPS1854 Shonosuke S. et al.
2
where = ∑ , ( | ; , ) is the conditional distribution of under

=1

the model (1), that is, | ~ ( + , ), and π( ; ) is the marginal
2

distribution of , i.e. ∼ (0, ) . If is a outlier, the weight for the

information of would be very small, so that the information of such outliers

are automatically ignored in (3). The tuning parameter controls the effects
of outliers, and the larger value leads to the smaller weight for when is

an outlier. The value of is specified by the user or choose in a objective way
as discussed later. Note that if = 0, the weights are 1, so that the modified
estimating functions (3) reduce to the conventional ones. Since the estimating
functions and are not necessarily unbiased, the modified estimating

equations are − E[ ] = 0 and − E[ ] = 0, = 1, . . . , , where the

expectations are taken with respect to the conditional distribution
( ; , ) and the marginal distribution π( ; ), respectively.
2

The modified estimating equations can be derived by considering the
following objective function:

2
2 1+
log {∑ ∑ ( | ; , ) } − log {∑ ∑ (| ; , ) }
1 +
=1 =1 =1 =1 (4)

+ log {∑ ( ; ) } − log {∑ ∫ (; ) 1+ },
1 +
=1 =1
It is easy to show that the partial derivatives of (4) with respect to β and bi
are the modified estimating functions Fβ and Fbi given in (3), respectively. Note
that the objective function (4) can be seen as a weighted combination of two
-divergence (Fujisawa and Eguchi, 2008).
From the forms of f( |bi;β,σ ) and π(bi;R), we may evaluate the integrals
2

appeared in (4). By ignoring irrelevant terms, we have the following function
to be minimized:

2
(, ) = − log {∑ ∑ ( | ; , ) } − log 2

2(1 + )
=1 =1
(5)

− log {∑ ( ; ) } − log||,
2(1 + )
=1
where = ( , () , ) is a vector of unknown parameters. Then, the new
2

robust
260 | I S I W S C 2 0 1 9