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CPS2220 David Degras et al.
4. Statistical inference of model parameters
MLEs in linear Gaussian SSMs are consistent and asymptotically normal
[6]. This result can likely be extended to the switching SSM (1) under mild
assumptions on the Markov chain ( ). One can thus in principle perform

statistical inference on using the limiting distribution of the MLE. We have
found however that due to the high dimension of , common techniques to
estimate the limiting covariance matrix (i.e. the inverse of the Fisher
information matrix) are numerically unfeasible. As an alternative approach,
we propose a parametric bootstrap method that enjoys a simple and easily
parallelizable implementation.
1. Apply the EM algorithm of section 3 to the data 1: and denote by
the MLE of .
̂
∗
2. Draw a bootstrap replicate of according to the probabilities ̂.

3. For 2 ≤ ≤ , draw a bootstrap replicate of according to the
∗

̂ ̂ ).
probabilities ( ∗ −1 ,1,…, −1 ,
∗

∗
4. Draw a bootstrap replicate of from ( ̂ ∗, ∗).
̂
1
1
1
1
5. For 2 ≤ ≤ , draw a bootstrap replicate ∗ of from

( ∗ x , ∗).
̂
∗ ̂

−1
1
6. For 1 ≤ ≤ , draw a bootstrap replicate ∗ of from

( ∗x , ∗).
̂

1
̂ ∗
7. Apply the EM to the bootstrap sample ∗ 1: and denote by the
bootstrap replicate of .
̂
8. Repeat steps 2–7 a large number of times, say 50 ≤ ≤ 200, to
obtain the probability distribution of the bootstrap estimator
̂ ∗
conditional on 1: .

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estimation of jump markov linear systems. IEEE Trans. Signal Processing,
49:613–624, 2001.
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