Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case. (English) Zbl 1105.62004
Summary: Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourhoods can be attained for any dimension, provided the regularity of the drift is sufficiently large. In addition, a heteroskedastic Gaussian regression experiment is given, which is also locally asymptotically equivalent and which does not depend on the centre of localisation. For one direction of the equivalence an explicit Markov kernel is constructed.
MSC:
| 62B15 | Theory of statistical experiments |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62M05 | Markov processes: estimation; hidden Markov models |
| 62G07 | Density estimation |
| 62G08 | Nonparametric regression and quantile regression |
Keywords:
Asymptotic equivalence; Statistical experiment; Le Cam distance; Ergodic diffusion; Gaussian shift; Heteroskedastic regressionReferences:
| [1] | Aït-Sahalia, Y.: Closed-form likelihood expansions for multivariate diffusions. Preprint (2004), available under http://www.princeton.edu/ yacine/research.htm. · Zbl 1246.62180 |
| [2] | Bandi, F. M., Moloche, G.: On the functional estimation of multivariate diffusion processes. Preprint (2001), available under http://gsbwww.uchicago.edu/fac/federico.bandi/research. · Zbl 1393.62036 |
| [3] | Bhattacharya, R. N., Criteria for recurrence and existence of invariant measures for multidimensional diffusions, Ann. Probab., 6, 4, 541-553 (1978) · Zbl 0386.60056 |
| [4] | Brown, L. D.; Cai, T. T.; Low, M. G.; Zhang, C.-H., Asymptotic equivalence theory for nonparametric regression with random design. Dedicated to the memory of Lucien Le Cam, Ann. Statist., 30, 3, 688-707 (2002) · Zbl 1029.62044 |
| [5] | Brown, L. D.; Low, M. G., Asymptotic equivalence of nonparametric regression and white noise, Ann. Stat., 24, 6, 2384-2398 (1996) · Zbl 0867.62022 |
| [6] | Brown, L. D.; Zhang, C., Asymptotic nonequivalence of nonparametric experiments when the smoothness index is 1/2, Ann. Stat., 26, 1, 279-287 (1998) · Zbl 0932.62061 |
| [7] | Carter, A. V., A continuous Gaussian process approximation to a nonparametric regression in two dimensions, Bernoulli, 12, 1, 143-156 (2006) · Zbl 1098.62042 |
| [8] | Castellana, J.; Leadbetter, M., On smoothed probability density estimation for stationary processes, Stochastic Process. Appl., 21, 179-193 (1986) · Zbl 0588.62156 |
| [9] | Chen, M.-F.; Wang, F.-Y., Estimation of spectral gap for elliptic operators, Trans. Amer. Math. Soc., 349, 3, 1239-1267 (1997) · Zbl 0872.35072 |
| [10] | Dalalyan, A. S.; Reiß, M., Asymptotic statistical equivalence for scalar ergodic diffusions, Probab. Theory Rel. Fields, 134, 2, 248-282 (2006) · Zbl 1081.62002 |
| [11] | Delattre, S.; Hoffmann, M., Asymptotic equivalence for a null recurrent diffusion, Bernoulli, 8, 2, 139-174 (2002) · Zbl 1040.60067 |
| [12] | Efromovich, S.: Nonparametric curve estimation. Methods, theory, and applications. Springer Series in Statistics. Springer-Verlag, New York, 1999 · Zbl 0935.62039 |
| [13] | Giné, E.; Koltchinskii, V.; Zinn, J., Weighted uniform consistency of kernel density estimators, Ann. Prob., 32, 3, 2570-2605 (2004) · Zbl 1052.62034 |
| [14] | Grama, I., Neumann, M. Asymptotic equivalence of nonparametric autoregression and nonparametric regression. To appear in Ann. Statist. · Zbl 1246.62105 |
| [15] | Korostelev, A.; Nussbaum, M., The asymptotic minimax constant for sup-norm loss in nonparametric density estimation, Bernoulli, 6, 6, 1099-1118 (1999) · Zbl 0955.62037 |
| [16] | Kutoyants, Y. A.: Statistical inference for ergodic diffusion processes. Springer Series in Statistics, London, 2004 · Zbl 1038.62073 |
| [17] | Le Cam, L., Yang, G. L.: Asymptotics in statistics. Some basic concepts. Springer Series in Statistics. Springer-Verlag, New York, 1990 · Zbl 0719.62003 |
| [18] | Liptser, R. S., Shiryaev, A. N.: Statistics of random processes. 1: General theory. 2nd edition, Applications of Mathematics 5. Springer, Berlin, 2001 · Zbl 1008.62072 |
| [19] | Nussbaum, M., Asymptotic equivalence of density estimation and Gaussian white noise, Ann. Stat., 24, 6, 2399-2430 (1996) · Zbl 0867.62035 |
| [20] | Qian, Z.; Russo, F.; Zheng, W., Comparison theorem and estimates for transition probability densities of diffusion processes, Probab. Theory Related Fields, 127, 3, 388-406 (2003) · Zbl 1050.60061 |
| [21] | Qian, Z.; Zheng, W., A representation formula for transition probability densities of diffusions and applications, Stochastic Process. Appl., 111, 1, 57-76 (2004) · Zbl 1070.60072 |
| [22] | Reiß, M.: On Le Cam’s distance between Gaussian nonparametric regression and white noise. Manuscript (2006) |
| [23] | Revuz, D.,Yor, M.: Continuous martingales and Brownian motion. 3rd ed. Graduate Texts in Mathematics 293, Springer, Berlin, 1999 · Zbl 0917.60006 |
| [24] | Rogers, L. C. G., Williams, D.: Diffusions, Markov processes, and martingales. Vol. 2. Ito calculus. Wiley Series in Probability and Mathematical Statistics. New York, 1987 · Zbl 0627.60001 |
| [25] | Strasser, H.: Mathematical theory of statistics. Statistical experiments and asymptotic decision theory. De Gruyter Studies in Mathematics, 7. Walter de Gruyter, Berlin, 1985 · Zbl 0594.62017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
