Consistency and asymptotic normality of the sieve estimator and an approximate maximum likelihood estimator of the drift coefficient of an interacting particles of diffusions are studied. For the sieve estimator, observations are taken on a fixed time interval [0,<i>T</i>] and asymptotics are studied as the number of interacting particles increases with the dimension of the sieve. For the approximate maximum likelihood estimator, discrete observations are taken in a time interval [0,<i>T</i>] and asymptotics are studied as the number of interacting particles increases with the number of observation time points

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Abstract. Consistency and limit distribution of the maximum likelihood estimator of a parameter in the drift coefficient of an anticipative Skorohod stochastic differential equation satisfying a boundary condition are obtained based on n independent trajectories of the corresponding Skorohod diffusion inside a time interval [0, T ] as n → ∞. The results are illustrated for the anticipative Ornstein-Uhlenbeck process.

The paper shows that the distribution of the normalized least squares estimator of the drift parameter in the fractional Ornstein-Uhlenbeck process observed over [0, T] converges to the standard normal distribution with an uniform optimal error bound of the order O(T −1/2) for 0.5 ≤ H ≤ 0.63 and of the order O(T4H-3) for 0.63 < H < 0.75 where H is the Hurst exponent of the fractional Brownian motion driving the Ornstein-Uhlenbeck process. For the normalized quasi-least squares estimator, the error bound is of the order O(T−1/4) for 0.5 ≤ H ≤ 0.69 and of the order O(T4H−3) for 0.69 < H < 0.75.

We obtain explicit form of fine large deviation theorems for the log-likelihood ratio in testing models with fractional nonergodic Ornstein-Uhlenbeck processes with Hurst parameter more than half and get explicit rates of decrease of the error probabilities of Neyman-Pearson, Bayes and minimax tests.