On moments of Pitman estimators: the case of fractional Brownian motion

A. A. Novikov   ; N. E. Kordzakhia   ; T. Ling

Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 4, pp. 695-710

Bibliographie

[1] Abramovits M., Stigan I., Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979, 830 pp.

[2] Antoch J., Hušková M., “Bayesian-type estimators of change points”, J. Statist. Plann. Inference, 91:2 (2000), 195–208 | DOI | Zbl

[3] Berger J. O., Statistical Decision Theory and Bayesian Analysis, Springer-Verlag, New York, 1985, 617 pp. | Zbl

[4] Borovkov A. A., Matematicheskaya statistika, Nauka, Novosibirsk, 1997, 771 pp.

[5] Cameron R. H., Martin W. T., “Transformations of Wiener integrals under translations”, Ann. Math., 45 (1944), 386–396 | DOI | Zbl

[6] Dachian S., “Estimation of cusp location for Poisson observations”, Statist. Inference Stoch. Process., 6:1 (2003), 1–14 | DOI | Zbl

[7] Fujii T., “An extension of cusp estimation problem in ergodic diffusion processes”, Statist. Probab. Lett., 80:9–10 (2010), 779–783 | DOI | Zbl

[8] Golubev G. K., “O vychislenii effektivnosti otsenki maksimalnogo pravdopodobiya pri nablyudenii razryvnogo signala v belom shume”, Probl. peredachi inform., 15:3 (1979), 61–69 | Zbl

[9] Gushchin A. A., Küchler U., “On estimation of delay location”, Statist. Inference Stoch. Process., 14:3 (2011), 273–305 | DOI | Zbl

[10] Ibragimov I. A., Khasminskii R. Z., “Ob asimptoticheskom povedenii obobschennykh baiesovskikh otsenok”, Dokl. AN SSSR, 194:2 (1970), 257–260 | Zbl

[11] Ibragimov I. A., Khasminskii R. Z., “Otsenka parametra razryvnogo signala v belom gaussovskom shume”, Probl. peredachi inform., 11:3 (1970), 31–43

[12] Ibragimov I. A., Khasminskii R. Z., Asimptoticheskaya teoriya otsenivaniya, Nauka, M., 1979, 527 pp.

[13] Kawazu K., “A note on the Brownian motion”, From Stochastic Calculus to Mathematical Finance, Springer, Berlin, 2006, 385–392 | DOI | Zbl

[14] Kawazu K., Tanaka H., “On the maximum of a diffusion process in a drifted Brownian environment”, Lecture Notes in Math., 1557, 1993, 78–85 | DOI | Zbl

[15] Kutoyants Y. A., Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2004, 481 pp. | Zbl

[16] Ling T., Novikov A., On simulation of exponential functionals and maximim of fractional Brownian motion, Preprint, UTS, Sydney, 2013

[17] Molchan G. M., “Maximum of a fractional Brownian motion: Probabilities of small values”, Comm. Math. Phys., 205:1 (1999), 97–111 | DOI | Zbl

[18] Novikov A. A., Kordzakhiya N. E., “Otsenki Pitmana: novyi podkhod k vychisleniyu asimptoticheskoi dispersii”, Teoriya veroyatn. i ee primen., 57:3 (2012), 603–611 | DOI

[19] Pitman E. J. G., “The estimation of the location and scale parameters of a continuous population of any given form”, Biometrika, 30 (1939), 391–421

[20] Rubin H., Song K.-S., “Exact computation of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise”, Ann. Statist., 23:3 (1995), 732–739 | DOI | Zbl

[21] Terentev A. S., “Raspredelenie veroyatnostei vremennogo polozheniya absolyutnogo maksimuma na vykhode soglasovannogo filtra”, Radiotekhnika i elektronika, 13:4 (1968), 652–657

[22] Wood A., Chan G., “Simulation of stationary Gaussian processes in $[0, 1]^{d}$”, J. Comput. Graph. Statist., 3:4 (1994), 409–432

[23] Yao Y.-C., “Approximating the distribution of the maximum likelihood estimate of the change-point in a sequence of independent random variables”, Ann. Statist., 15:3 (1987), 1321–1328 | DOI | Zbl

[24] Shepp L. A., “The joint density of the maximum and its location for a Wiener process with drift”, J. Appl. Probab., 16:2 (1979), 423–427 | DOI | Zbl

[25] Shiryaev A., Probability, Springer-Verlag, New York, 1995, 609 pp. | Zbl