Cramér-type moderate deviations for the likelihood ratio process of Ornstein–Uhlenbeck process with shift

For the Ornstein–Uhlenbeck process in stationary and explosive cases, this paper studies Cramér-type moderate deviations for the log-likelihood ratio process. As an application, we give the negative regions of drift testing problem, and also obtain the decay rates of the error probabilities. The main methods of this paper consist of mod-[Formula: see text] convergence approach, deviation inequalities for multiple Wiener–Itô integrals and asymptotic analysis techniques.

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Large Deviations in Testing Squared Radial Ornstein-Uhleneck Model

Journal of Probability and Statistics ◽

10.1155/2011/741701 ◽

2011 ◽

Vol 2011 ◽

pp. 1-11

Author(s):

Shoujiang Zhao ◽

Qiaojing Liu

Keyword(s):

Large Deviations ◽

Hypothesis Testing ◽

Likelihood Ratio ◽

Large Deviation ◽

Decay Rates ◽

Moderate Deviations ◽

Large Deviation Principles ◽

Log Likelihood ◽

Log Likelihood Ratio ◽

Error Probabilities

We study the large deviations and moderate deviations of hypothesis testing for squared radial Ornstein-Uhleneck model. Large deviation principles for the log-likelihood ratio are obtained, by which we give negative regions in testing squared radial Ornstein-Uhleneck model and get the decay rates of the error probabilities.

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Self-normalized asymptotic properties for the parameter estimation in fractional Ornstein–Uhlenbeck process

Stochastics and Dynamics ◽

10.1142/s0219493719500187 ◽

2019 ◽

Vol 19 (03) ◽

pp. 1950018 ◽

Cited By ~ 2

Author(s):

Hui Jiang ◽

Junfeng Liu ◽

Shaochen Wang

Keyword(s):

Large Deviations ◽

Asymptotic Properties ◽

Moderate Deviations ◽

Delta Method ◽

Fourth Moment ◽

Test Statistics ◽

Uhlenbeck Process ◽

Ornstein Uhlenbeck Process ◽

Deviation Inequalities ◽

Type Ii Errors

In this paper, we consider the self-normalized asymptotic properties of the parameter estimators in the fractional Ornstein–Uhlenbeck process. The deviation inequalities, Cramér-type moderate deviations and Berry–Esseen bounds are obtained. The main methods include the deviation inequalities and moderate deviations for multiple Wiener–Itô integrals [P. Major, Tail behavior of multiple integrals and U-statistics, Probab. Surv. 2 (2005) 448–505; On a multivariate version of Bernsteins inequality, Electron. J. Probab. 12 (2007) 966–988; M. Schulte and C. Thäle, Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem, J. Funct. Anal. 270 (2016) 2223–2248], as well as the Delta methods in large deviations [F. Q. Gao and X. Q. Zhao, Delta method in large deviations and moderate deviations for estimators, Ann. Statist. 39 (2011) 1211–1240]. For applications, we propose two test statistics which can be used to construct confidence intervals and rejection regions in the hypothesis testing for the drift coefficient. It is shown that the Type II errors tend to be zero exponential when using the proposed test statistics.

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