On the Goodness-of-Fit Tests for Some Continuous Time Processes

We present a review of several results concerning the construction of the Cramer-von Mises and Kolmogorov-Smirnov type goodness-of-fit tests for continuous time processes. As the models we take a stochastic differential equation with small noise, ergodic diffusion process, Poisson process and self-exciting point processes. For every model we propose the tests which provide the asymptotic size $\alpha$ and discuss the behaviour of the power function under local alternatives. The results of numerical simulations of the tests are presented.Comment: 22 pages, 2 figure

Estimating linear functionals of a sparse family of Poisson means

Assume that we observe a sample of size n composed of p-dimensional signals, each signal having independent entries drawn from a scaled Poisson distribution with an unknown intensity. We are interested in estimating the sum of the n unknown intensity vectors, under the assumption that most of them coincide with a given 'background' signal. The number s of p-dimensional signals different from the background signal plays the role of sparsity and the goal is to leverage this sparsity assumption in order to improve the quality of estimation as compared to the naive estimator that computes the sum of the observed signals. We first introduce the group hard thresholding estimator and analyze its mean squared error measured by the squared Euclidean norm. We establish a nonasymptotic upper bound showing that the risk is at most of the order of {\sigma}^2(sp + s^2sqrt(p)) log^3/2(np). We then establish lower bounds on the minimax risk over a properly defined class of collections of s-sparse signals. These lower bounds match with the upper bound, up to logarithmic terms, when the dimension p is fixed or of larger order than s^2. In the case where the dimension p increases but remains of smaller order than s^2, our results show a gap between the lower and the upper bounds, which can be up to order sqrt(p)

Uncertainty quantification for matrix compressed sensing and quantum tomography problems

We construct minimax optimal non-asymptotic confidence sets for low rank matrix recovery algorithms such as the Matrix Lasso or Dantzig selector. These are employed to devise adaptive sequential sampling procedures that guarantee recovery of the true matrix in Frobenius norm after a data-driven stopping time n^ for the number of measurements that have to be taken. With high probability, this stopping time is minimax optimal. We detail applications to quantum tomography problems where measurements arise from Pauli observables. We also give a theoretical construction of a confidence set for the density matrix of a quantum state that has optimal diameter in nuclear norm. The non-asymptotic properties of our confidence sets are further investigated in a simulation study

Nonparametric inference with generalized likelihood ratio tests

Asymptotic null distribution, Bootstrap, Generalized likelihood ratio, Nonparametric test, Power function, Wilks’ phenomenon, 62G07, 62G10, 62J12,