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Table 1 Results of numerical simulations

From: Constrained expectation maximisation algorithm for estimating ARMA models in state space representation

p

Nc

dim (𝜗)

iter.

MTLL (EM)

F.N. (EM)

MTLL (EM+BFGS)

AIC (EM+BFGS)

F.N. (EM+BFGS)

1

2

8

20

31345.44

0.5774

19068.02

19084.02

0.4600

2

2

14

33

27917.36

0.0331

13663.76

13691.76

0.0967

3

2

22

51

25368.71

0.2102

13343.41

13387.41

0.0110

4

2

32

74

25245.96

0.3000

13104.01

13168.01

0.0353

5

2

44

98

26517.65

0.3064

13215.47

13303.47

0.0123

6

2

58

119

28181.53

0.3045

13208.43

13324.43

0.0170

7

2

74

142

29638.16

0.3110

13262.68

13410.68

0.0440

8

2

92

187

31086.83

0.2716

16750.97

16934.97

0.4028

9

2

112

474

30385.52

0.3758

28282.57

28506.57

0.0977

10

2

134

513

30383.59

0.4259

29566.95

29834.95

0.2320

2

1

8

25

44391.86

-

37955.71

37971.71

-

2

3

20

35

28052.80

-

13529.03

13569.03

-

  1. Columns (from left to right) show model order (p), number of ARMA components (Nc), number of model parameters (dim (𝜗)), number of iterations until EM algorithm reaches a minimum (iter.), (−2)log(likelihood) at that minimum (MTLL), Frobenius norm (F.N.), (−2)log(likelihood) after additional optimisation by BFGS, corresponding value of the Akaike Information Criterion (AIC) and corresponding Frobenius norm. Likelihood values were computed omitting a transient of 20 initial samples. Note that Frobenius norm can only be computed for Nc=2