Under regularity assumptions, we establish a sharp large deviation principle for Hermitian quadratic forms of stationary Gaussian processes. Our result is similar to the well-known Bahadur-Rao theorem [2] on the sample mean. We also provide several examples of application such as the sharp large deviation properties of the Neyman-Pearson likelihood ratio test, of the sum of squares, of the Yule-Walker estimator of the parameter of a stable autoregressive Gaussian process, and finally of the empirical spectral repartition function.

We study the LRT statistic for testing a single population i.i.d. model against a mixture of two populations with Markov regime. We prove that the LRT statistic converges to infinity in probability as the number of observations tends to infinity. This is a consequence of a convergence result of the LRT statistic for a subproblem where the parameters are restricted to a subset of the whole parameter set.

We observe an infinitely dimensional Gaussian random vector x = ξ + v where ξ is a sequence of standard Gaussian variables and v ∈ l2 is an unknown mean. We consider the hypothesis testing problem H0 : v = 0versus alternatives $H_{\varepsilon,\tau}:v\in V_{\varepsilon}$ for the sets $V_{\varepsilon}=V_{\varepsilon}(\tau,\rho_{\varepsilon})\subset l_2$. The sets Vε are lq-ellipsoids of semi-axes ai = i-s R/ε with lp-ellipsoid of semi-axes bi = i-r pε/ε removed or similar Besov bodies Bq,t;s (R/ε) with Besov bodies Bp,h;r (pε/ε) removed. Here $\tau =(\kappa,R)$ or $\tau =(\kappa,h,t,R);\ \ \kappa=(p,q,r,s)$ are the parameters which define the sets Vε for given radii pε → 0, 0 < p,q,h,t ≤ ∞, -∞ ≤ r,s ≤ ∞, R > 0; ε → 0 is the asymptotical parameter. We study the asymptotics of minimax second kind errors $\beta_{\varepsilon}(\alpha)=\beta(\alpha, V_{\varepsilon}(\tau,\rho_{\varepsilon}))$ and construct asymptotically minimax or minimax consistent families of tests $\psi_{\alpha;\varepsilon,\tau,\rho_{\varepsilon}}$, if it is possible. We describe the partition of the set of parameters κ into regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of various types). Analogous rates have been obtained in a signal detection problem for continuous variant of white noise model: alternatives correspond to Besov or Sobolev balls with Besov or Sobolev balls removed. The study is based on an extension of methods of constructions of asymptotically least favorable priors. These methods are applicable to wide class of “convex separable symmetrical" infinite-dimensional hypothesis testing problems in white Gaussian noise model. Under some assumptions these methods are based on the reduction of hypothesis testing problem to convex extreme problem: to minimize specially defined Hilbert norm over convex sets of sequences $\bar{\pi}$ of measures πi on the real line. The study of this extreme problem allows to obtain different types of Gaussian asymptotics. If necessary assumptions do not hold, then we obtain other types of asymptotics.

We give limit theorems specifying weak and strong rates of convergence associated to a quadratic extension of the martingale almost-sure central limit theorem. Some typical examples are discussed to illustrate how to make use of them in statistic.

The p-values are often implicitly used as a measure of evidence for the hypotheses of the tests. This practice has been analyzed with different approaches. It is generally accepted for the one-sided hypothesis problem, but it is often criticized for the two-sided hypothesis problem. We analyze this practice with a new approach to statistical inference. First we select good decision rules without using a loss function, we call them experts. Then we define a probability distribution on the space of experts. The measure of evidence for a hypothesis is the inductive probability of experts that decide this hypothesis.

Let (Xt) be a diffusion on the interval (l,r) and Δn a sequence of positive numbers tending to zero. We define Ji as the integral between iΔn and (i + 1)Δn of Xs. We give an approximation of the law of (J0,...,Jn-1) by means of a Euler scheme expansion for the process (Ji). In some special cases, an approximation by an explicit Gaussian ARMA(1,1) process is obtained. When Δn = n-1 we deduce from this expansion estimators of the diffusion coefficient of X based on (Ji). These estimators are shown to be asymptotically mixed normal as n tends to infinity.

We define a notion of delta-variance maximization and show it implies epsilon-proximity in expactations.

Let YT = (Yt)t∈[0,T] be a real ergodic diffusion process which drift depends on an unkown parameter $\theta_{0}\in \mathbb{R}^{p}$. Our aim is to estimate θ0 from a discrete observation of the process YT, (Ykδ)k=0,n, for a fixed and small δ, as T = nδ goes to infinity. For that purpose, we adapt the Generalized Method of Moments (see Hansen) to the anticipative and approximate discrete-time trapezoidal scheme, and then to Simpson's. Under some general assumptions, the trapezoidal scheme (respectively Simpson's scheme) provides an estimation of θ0 with a bias of order δ2 (resp. δ4). Moreover, this estimator is asymptotically normal. These results generalize Bergstrom's [1], which were obtained for a Gaussian diffusion process, which drift is linear in θ.