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ANALYSIS OF MULTI-UNIT VARIANCE COMPONENTS MODELS

WITH STATE SPACE PROFILES

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JOHN V. TSIMIKAS^{1} AND JOHANNES LEDOLTER^{2}

^{1} *Department of Health Care Policy, Harvard Medical School,*

25 Shattuck Str, Boston MA 02115, U.S.A.

^{2} *Department of Statistics and Actuarial Science, University of Iowa,*

Iowa City, IA 52242, U.S.A.
(Received April 11, 1996; revised March 27, 1997)

**Abstract.**
We apply the Kalman Filter to the
analysis of multi-unit variance components models where
each unit's response profile follows a state space model.
We use mixed model results to obtain estimates of
unit-specific random effects, state disturbance terms and
residual noise terms. We use the signal extraction
approach to smooth individual profiles. We show how to
utilize the Kalman Filter to efficiently compute the
restricted loglikelihood of the model. For the important
special case where each unit's response profile follows a
continuous structural time series model with known
transition matrix we derive an EM algorithm for the
restricted maximum likelihood (REML) estimation of the
variance components. We present details for the case where
individual profiles are modeled as local polynomial trends
or polynomial smoothing splines.

*Key words and phrases*:
Continuous-time
stochastic models, EM algorithm, Kalman Filter, mixed
model prediction, restricted maximum likelihood,
smoothing splines, unequally spaced observations, variance
components.

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