Efficiently Implementing the Maximum Likelihood Estimator for Hurst Exponent

Table 2

Efficiency comparison for four cases covering six approaches, each value representing either the average time (in seconds) of 13 Hurst exponents spent by each approach in two executing procedures (with and without considering the computational efficiency) or their corresponding time ratio.

Cases	𝑁=128	𝑁=256	𝑁=512	𝑁=1024	𝑁=2048	𝑁=4096
C1	1.27𝐸−01	3.98𝐸−01	1.64𝐸+00	1.06𝐸+01	1.11𝐸+02	8.78𝐸+02
C1^b	1.02𝐸−01	2.03𝐸−01	4.40𝐸−01	1.05𝐸+00	2.88𝐸+00	1.04𝐸+01
Ratio	1.25𝐸+00	1.97𝐸+00	3.73𝐸+00	1.01𝐸+01	3.86𝐸+01	8.46𝐸+01
C2	1.19𝐸−01	3.63𝐸−01	1.61𝐸+00	1.03𝐸+01	1.11𝐸+02	8.70𝐸+02
C2^b	9.83𝐸−02	2.02𝐸−01	4.37𝐸−01	1.16𝐸+00	2.88𝐸+00	1.03𝐸+01
Ratio	1.21𝐸+00	1.80𝐸+00	3.68𝐸+00	8.94𝐸+00	3.85𝐸+01	8.47𝐸+01
C3-A1^a	1.29𝐸−01	3.62𝐸−01	1.65𝐸+00	1.05𝐸+01	1.09𝐸+02	8.80𝐸+02
C3-A1^a,b	1.09𝐸−01	2.14𝐸−01	4.57𝐸−01	1.09𝐸+00	2.93𝐸+00	1.09𝐸+01
Ratio	1.18𝐸+00	1.69𝐸+00	3.60𝐸+00	9.71𝐸+00	3.73𝐸+01	8.08𝐸+01
C3-A2^a	1.27𝐸−01	3.50𝐸−01	1.59𝐸+00	1.06𝐸+01	1.10𝐸+02	8.70𝐸+02
C3-A2^a,b	1.00𝐸−01	2.05𝐸−01	4.36𝐸−01	1.05𝐸+00	2.88𝐸+00	1.03𝐸+01
Ratio	1.27𝐸+00	1.71𝐸+00	3.64𝐸+00	1.01𝐸+01	3.82𝐸+01	8.42𝐸+01
C4-A1^a	1.16𝐸−01	3.74𝐸−01	1.68𝐸+00	1.06𝐸+01	1.10𝐸+02	8.71𝐸+02
C4-A1^a,b	1.01𝐸−01	2.14𝐸−01	4.52𝐸−01	1.08𝐸+00	2.93𝐸+00	1.06𝐸+01
Ratio	1.15𝐸+00	1.74𝐸+00	3.71𝐸+00	9.78𝐸+00	3.76𝐸+01	8.18𝐸+01
C4-A2^a	1.17𝐸−01	3.53𝐸−01	1.65𝐸+00	1.03𝐸+01	1.10𝐸+02	8.78𝐸+02
C4-A2^a,b	9.44𝐸−02	2.02𝐸−01	4.42𝐸−01	1.04𝐸+00	2.88𝐸+00	1.07𝐸+01
Ratio	1.24𝐸+00	1.75𝐸+00	3.75𝐸+00	9.88𝐸+00	3.83𝐸+01	8.21𝐸+01

aA1 denotes Approach 1 and A2 denotes Approach 2.
^bEstimation implemented by the two newly proposed procedures.