Asymptotic Normality of the Estimators for Fractional Brownian Motions with Discrete Data

Table 1

The estimation results of comparative methods for sample size 𝑁=50.

Estimation method	Statistic	True values of 𝐻
		0.5600	0.6200	0.8500	0.9200
R/S method	Mean	0.50506	0.59232	0.82667	0.87692
	S.D.	0.62322	0.43601	0.48195	0.51479
	√MSE	0.62564	0.43689	0.48251	0.51659
	CPU time	13	15	18	20
Variations estimators	Mean	0.59451	0.65676	0.89267	0.92050
	S.D.	0.12365	0.23050	0.23739	0.15369
	√MSE	0.12838	0.23341	0.24119	0.15369
	CPU time	3	5	4	6
Wavelet method	Mean	0.58500	0.63520	0.87150	0.92963
	S.D.	0.04265	0.02518	0.06251	0.09654
	√MSE	0.04944	0.02941	0.06610	0.09702
	CPU time	22	27	28	28
Our method	Mean	0.56667	0.62905	0.85507	0.92102
	S.D.	0.03319	0.03092	0.01350	0.01325
	√MSE	0.03385	0.03222	0.01442	0.01329
	CPU time	3	3	5	5