Signal Detection with Quadratically Convex Orthosymmetric Constraints

This paper is concerned with signal detection in Gaussian noise under quadratically convex orthosymmetric (QCO) constraints. Specifically the null hypothesis assumes no signal, whereas the alternative considers signal which is separated in Euclidean norm from zero, and belongs to the QCO constraint. Our main result establishes the minimax rate of the separation radius between the null and alternative purely in terms of the geometry of the QCO constraint -- we argue that the Kolmogorov widths of the constraint determine the critical radius. This is similar to the estimation problem with QCO constraints, which was first established by Donoho et al. (1990); however, as expected, the critical separation radius is smaller compared to the minimax optimal estimation rate. Thus signals may be detectable even when they cannot be reliably estimated.Comment: 2 figure

Application of the D-Fullness Technique for Breakdown Point Study of the Trimmed Likelihood Estimator to a Generalized Logistic Model

2000 Mathematics Subject Classification: 62J12, 62F35A new definition for a d-fullness of a set of functions is proposed and its equivalence to the original one given by Vandev [11] is proved. The breakdown point of the WTLk estimator of Vandev and Neykov [13] for a grouped binary linear regression model with generalized logistic link is studied.Research partially supported by contracts: PRO-ENBIS: GTC1-2001-4303

Revisiting Le Cam's Equation: Exact Minimax Rates over Convex Density Classes

We study the classical problem of deriving minimax rates for density estimation over convex density classes. Building on the pioneering work of Le Cam (1973), Birge (1983, 1986), Wong and Shen (1995), Yang and Barron (1999), we determine the exact (up to constants) minimax rate over any convex density class. This work thus extends these known results by demonstrating that the local metric entropy of the density class always captures the minimax optimal rates under such settings. Our bounds provide a unifying perspective across both parametric and nonparametric convex density classes, under weaker assumptions on the richness of the density class than previously considered. Our proposed `multistage sieve' MLE applies to any such convex density class. We further demonstrate that this estimator is also adaptive to the true underlying density of interest. We apply our risk bounds to rederive known minimax rates including bounded total variation, and Holder density classes. We further illustrate the utility of the result by deriving upper bounds for less studied classes, e.g., convex mixture of densities.Comment: Total paper (46 pages, 2 figures): Main paper (17 pages, 2 figures) + Appendix (29 pages). Updated to include proof of adaptivity of estimato

Detecting Precipitation Climate Changes: An Approach Based on a Stochastic Daily Precipitation Model

2002 Mathematics Subject Classification: 62M10.We consider development of daily precipitation models based on [3] for some sites in Bulgaria. The precipitation process is modelled as a two-state first-order nonstationary Markov model. Both the probability of rainfall occurrance and the rainfall intensity are allowed depend on the intensity on the preceeding day. To investigate the existence of long-term trend and of changes in the pattern of seasonal variation we use a synthesis of the methodology presented in [3] and the idea behind the classical running windows technique for data smoothing. The resulting time series of model parameters are used to quantify changes in the precipitation process over the territory of Bulgaria

Adversarial Sign-Corrupted Isotonic Regression

Classical univariate isotonic regression involves nonparametric estimation under a monotonicity constraint of the true signal. We consider a variation of this generating process, which we term adversarial sign-corrupted isotonic (\texttt{ASCI}) regression. Under this \texttt{ASCI} setting, the adversary has full access to the true isotonic responses, and is free to sign-corrupt them. Estimating the true monotonic signal given these sign-corrupted responses is a highly challenging task. Notably, the sign-corruptions are designed to violate monotonicity, and possibly induce heavy dependence between the corrupted response terms. In this sense, \texttt{ASCI} regression may be viewed as an adversarial stress test for isotonic regression. Our motivation is driven by understanding whether efficient robust estimation of the monotone signal is feasible under this adversarial setting. We develop \texttt{ASCIFIT}, a three-step estimation procedure under the \texttt{ASCI} setting. The \texttt{ASCIFIT} procedure is conceptually simple, easy to implement with existing software, and consists of applying the \texttt{PAVA} with crucial pre- and post-processing corrections. We formalize this procedure, and demonstrate its theoretical guarantees in the form of sharp high probability upper bounds and minimax lower bounds. We illustrate our findings with detailed simulations.Comment: Total paper (52 pages, 2 figures): Main paper (13 pages, 2 figures) + Appendix (39 pages