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On large deviations in testing Ornstein-Uhlenbeck-type models. (English) Zbl 1204.62144

Summary: We obtain exact large deviation rates for the log-likelihood ratio in testing models with observed Ornstein-Uhlenbeck processes and get explicit rates of decrease for the error probabilities of Neyman-Pearson, Bayes, and minimax tests. Moreover, we give expressions for the rates of decrease for the error probabilities of Neyman-Pearson tests in models with observed processes solving affine stochastic delay differential equations.

MSC:

62M02 Markov processes: hypothesis testing
60F10 Large deviations
62F05 Asymptotic properties of parametric tests
62F15 Bayesian inference
62C20 Minimax procedures in statistical decision theory
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[1] Bahadur RR (1960). Asymptotic efficiency of tests and estimates. Sankhya 22: 229–252 · Zbl 0109.12503
[2] Bahadur RR (1967). Rates of convergence of estimates and test statistics. Annal Math Stat 38: 303–324 · Zbl 0201.52106
[3] Bahadur RR (1971). Some limit theorems in statistics. SIAM, Philadelphia · Zbl 0257.62015
[4] Bahadur RR, Zabell SL, Gupta JC (1980) Large deviations, tests and estimates. In: Chakravarti IM (ed) Asymptotic theory of statistical tests and estimation. Volume in Honor of Wassily Hoeffding. Academic Press, New York, pp. 33–64
[5] Björk T (1997) Interest rate theory. In: Rungaldier W (ed) Financial mathematics Lecture notes in mathematics 1656 Springer, Berlin · Zbl 0904.90007
[6] Borodin AN and Salminen P (1996). Handbook of Brownian Motion – Facts and Formulae. Birkhäuser, Basel · Zbl 0859.60001
[7] Borovkov AA (1984) Mathematical Statistics. Moscow, Nauka (in Russian); English translation: (1998) Gordon and Breach, Amsterdam
[8] Birgé L (1981). Vitesses maximales de décroissance des erreurs et tests optimaux associés. Zeits Wahrscheinlichkeitstheorie und verwandte Gebiete 55: 261–273 · Zbl 0486.62029
[9] Chernoff H (1952). A measure of asymptotic efficiency for tests of hypothesis based on the sum of observations. Annal Math Stat 23: 493–507 · Zbl 0048.11804
[10] Dacunha-Castelle D and Duflo M (1986). Probability and Statistics I. Springer, New York · Zbl 0586.62003
[11] Dietz HM (1992). A non-Markovian relative of the Ornstein–Uhlenbeck process and some of its local statistical properties. Scand J of Stat 19: 363–379 · Zbl 0770.60056
[12] Gushchin AA and Küchler U (1999). Asymptotic inference for a linear stochastic differential equation with time delay. Bernoulli 5(3): 483–493 · Zbl 0943.60016
[13] Gushchin AA and Küchler U (2000). On stationary solutions of delay differential equations driven by a Lévy process. Stoch Process Appl 88: 195–211 · Zbl 1045.60057
[14] Gushchin AA and Küchler U (2001). Addendum to ’Asymptotic inference for a linear stochastic differential equation with time delay’. Bernoulli 7: 629–632 · Zbl 1008.62080
[15] Gushchin AA and Küchler U (2003). On parametric statistical models for stationary solutions of affine stochastic delay differential equations. Math Meth Stat 12(1): 31–61
[16] Jacod J and Shiryaev AN (1987). Limit Theorems for Stochastic Processes. Springer, Berlin · Zbl 0635.60021
[17] Küchler U and Kutoyants YA (2000). Delay estimation for some stationary diffusion-type processes. Scand J Stat 27(3): 405–414 · Zbl 0976.62083
[18] Küchler U, Vasil’ev VA (2001) On guaranteed parameter estimation of stochastic differential equations with time delay by noisy observations. Discussion Paper 14 of Sonderforschungsbereich 373, Humboldt University Berlin, 25 pp
[19] Kutoyants YA (2004). Statistical Inference for Ergodic Diffusion Processes. Springer Series in Statistics, New York · Zbl 1038.62073
[20] Kutoyants YA (2005). On delay estimation for stochastic differential equations. Stoch Dynamics 5(2): 333–342 · Zbl 1070.62066
[21] Lin’kov YN (1993) Asymptotic Statistical Methods for Stochastic Processes. Naukova Dumka, Kiev (in Russian); English translation: (2001) American Mathematical Society, Providence, R:I
[22] Lin’kov YN (1999). Large deviation theorems for extended random variables and some applications. J Math Sci 93(4): 563–573 · Zbl 0943.60014
[23] Lin’kov YN (2002) Large deviations in testing of models with fractional Brownian motion. University of Helsinki, Preprint, 14 pp
[24] Liptser RS and Shiryaev AN (1977). Statistics of Random Processes I. Springer, Berlin
[25] Putschke U (2000) Affine stochastische Funktionaldifferentialgleichungen und lokal asymptotische Eigenschaften ihrer Parameterschätzungen. Doctoral Dissertation, Humboldt University of Berlin, Institute of Mathematics
[26] Rockafellar RT (1970). Convex Analysis. Princeton University Press, Princeton · Zbl 0193.18401
[27] Szimayer A and Maller R (2004). Testing for mean reversion in processes of Ornstein–Uhlenbeck type. Stat Inf Stoch Process 7(2): 95–113 · Zbl 1056.62090
[28] Vajda I (1990). Generalization of discrimination-rate theorems of Chernoff and Stein. Cybernetics 26(4): 273–288 · Zbl 0727.62026
[29] Vasiček OA (1977). An equilibrium characterization of the term structure. J Finan Econ 5: 177–188 · Zbl 1372.91113
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