zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Estimating the Hurst effect and its application in monitoring clinical trials. (English) Zbl 05373967
Summary: We use a direct maximum likelihood method in estimating the Hurst coefficient for fractional Brownian motion with a short length of observations. We show that the estimate is asymptotically unbiased and derive an easy to use formula for computing the variance of the estimate. We also investigate the finite sample properties via Monte Carlo simulations. The simulation studies indicate that the theoretical formulas based on large sample arguments can be used when the number of observations is relatively small. A real example for monitoring a clinical trial is used to illustrate and compare various methods.

MSC:
62-99Statistics (MSC2000)
WorldCat.org
Full Text: DOI
References:
[1] Beran, J.: Statistics for long-memory processes. (1994) · Zbl 0869.60045
[2] Beran, J.: On a class of M-estimators for Gaussian long-memory models. Biometrika 81, 755-766 (1994) · Zbl 0812.62089
[3] Beran, J.; Terrin, N.: Testing for a change of the long-memory parameter. Biometrika 83, 627-638 (1996) · Zbl 0866.62055
[4] Brockwell, P. J.; Davis, R. A.: Time series: theory and methods. (1987) · Zbl 0604.62083
[5] Dahlhaus, R.: Efficient parameters estimation for self-similar processes. Ann. statist. 17, 1749-1766 (1989) · Zbl 0703.62091
[6] Davis, B. R.; Hardy, R. J.: Upper bound for type I and type II error rates in conditional power calculation. Commun. stat. Theory methods 19, 3571-3584 (1990)
[7] Davies, R. B.; Harte, D. S.: Tests for Hurst effect. Biometrika 74, 95-101 (1987) · Zbl 0612.62123
[8] Demets, D. L.; Hardy, R. J.; Friedman, L. F.; Lan, K. K. G.: Statistical aspects of early termination in the beta-blocker heart attack trial. Control. clin. Trials 5, 362-372 (1984)
[9] Digital Equipment Corporation, 1998. DIGITAL Visual Fortran. Maynard, MA.
[10] Duncan, T. E.; Hu, Y.; Pasik-Duncan, B.: Stochastic calculus for fractional Brownian motion, I. Theor. SIAM J. Control optim. 38, 582-612 (2000) · Zbl 0947.60061
[11] Gripenberg, G.; Norros, I.: On the prediction of fractional Brownian motion. J. appl. Probab. 33, 400-410 (1996) · Zbl 0861.60049
[12] Lai, D. J.; Davis, B. R.; Hardy, R. J.: Fractional Brownian motion and clinical trials. J. appl. Statist. 27, 103-108 (2000) · Zbl 0937.62114
[13] Lan, K. K. G.; Wittes, J.: The B-valuea tool for monitoring data. Biometrics 44, 579-585 (1988)
[14] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noise and applications. SIAM rev. 10, 422-437 (1968) · Zbl 0179.47801
[15] Mannella, R.; Grigolini, P.; West, B. J.: A dynamical approach to fractional Brownian motion. Fractals 2, 81-94 (1994)
[16] Mardia, K. V.; Marshall, R. J.: Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71, 135-146 (1984) · Zbl 0542.62079
[17] Mathsoft, 1999. S-plus 2000. Seattle, WA.
[18] Mesa, O. J.; Poveda, G.: The Hurst effectthe scale of fluctuation approach. Water resour. Res. 29, 3995-4002 (1993)
[19] Molz, F. J.; Boman, G. K.: A fractal-based stochastic interpolation scheme in subsurface hydrology. Water resour. Res. 29, 3769-3774 (1993)
[20] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P.: Numerical recipes in Fortran: the art of scientific computing. (1992) · Zbl 0778.65002
[21] Schott, J. R.: Matrix analysis for statistics. (1997) · Zbl 0872.15002
[22] Seber, G. A. F.: Linear regression analysis. (1997) · Zbl 0354.62055
[23] Sweeting, T. J.: Uniform asymptotic normality of the maximum likelihood estimator. Ann. statist. 8, 1375-1381 (1980) · Zbl 0447.62041
[24] Visual Numerics, 1994. IMSL Library. Houston, TX.
[25] Wallinger, W.; Taqqu, S.; Leland, W. E.; Wilson, D. V.: Self-similarity in high-speed packet trafficanalysis and modeling of Ethernet traffic measurements. Statist. sci. 10, 67-85 (1995) · Zbl 1148.90310