Page 270 - Contributed Paper Session (CPS) - Volume 2

CPS1854 Shonosuke S. et al.
We also consider regularized estimation of regression coefficients and random
effects by putting penalties on the objective function.

2. Robust Estimation of Linear Mixed Models

2.1 Weighted estimating equations and objective functions
We consider the following linear mixed model:
= + + , = 1, . . . , , = 1, . . . , , (1)

where xij and zij are p- and q-dimensional vectors of covariates, is a
−dimensional vector of regression coefficients, bi is a vector of random
effects and εij is an error term. Here we assume that bi and εij are independent
2
2
and distributed as bi ∼ (0, ) and εij ∼ (0,σ ), where R and σ are unknown
variance-covariate matrix and variance parameter, respectively.
We first assume that both R and σ are known. Then, the maximum
2
likelihood estimators of and bi are the solution of the following estimating
equations:

1 ∑ ∑ ( − + ) = 0

2
=1 =1

1

∑ ( − + ) − −1 = 0, = 1, … , . (2)

2
=1
When there exist outliers in yij, the estimating equations may produce poor
estimates of as well as .

2
We introduce weights wij = w(yij;bi, ,σ ) and ui = ( ; ) to modify the

estimating functions in (2) to

1

= ∑ ∑ ( − + )

2
=1 =1

1

= ∑ ( − + ) − −1 , = 1, … , (3)
2

=1
Specifically, we consider the following forms of the weights:
2
( | ; , ) ( ; )

= , = ,

2
−1 ∑ ∑ ( | ; , ) −1 ∑ ( ; )
=1 =1 =1
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