Testing monotonicity via local least concave majorants

Abstract

We propose a new testing procedure for detecting localized departures from monotonicity of a signal embedded in white noise. In fact, we perform simultaneously several tests that aim at detecting departures from concavity for the integrated signal over various intervals of different sizes and localizations. Each of these local tests relies on estimating the distance between the restriction of the integrated signal to some interval and its least concave majorant. Our test can be easily implemented and is proved to achieve the optimal uniform separation rate simultaneously for a wide range of H\"{o}lderian alternatives. Moreover, we show how this test can be extended to a Gaussian regression framework with unknown variance. A simulation study confirms the good performance of our procedure in practice.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ496 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Central limit theorems for the $L_{p}$-error of smooth isotonic estimators

“…Then, consider the case nb 5 → C 2 0 > 0. Note that ζ i depends only on the Brownian motion on the interval (1). Hence, the left hand side of (72), can be written as…”

“…We denote by T B the local mean test of [2] and S reg n the test proposed in [1] on the basis of the distance between the least concave majorant of Λ n and Λ n . The result of the simulations for n = 100, α = 0.05, b = 0.1, are given in Table 1.…”

“…The results of the simulation, again for n = 100, α = 0.05, b = 0.1 and various values of a and σ 2 are given in Table 2. We denote by S reg n the test of [1] and by T run the test of [18]. Note that when a = 0, the regression function is decreasing so H 0 is satisfied.…”

Testing k-monotonicity of a discrete distribution. Application to the estimation of the number of classes in a population

“…the last equality being deduced from the definition of ∆ k given at Equation (1). Therefore if we denote by p k 0 the value of p 0 under the assumption that p is a k-monotone abundance distribution, then…”

“…In order to validate the chosen model before estimating the number of classes, we propose a goodnessof-fit test for testing k-monotonicity. To the best of our knowledge, very few works are available for testing a shape constraint on a discrete density: Akakpo et al [1] proposed a procedure for testing monotonicity (k = 1), while Durot et al [15] and Balabdaoui et al [7] considered the problem of testing convexity (k = 2). The testing procedures they proposed rely on the asymptotic distribution of some distance between the empirical distribution and the estimation of the density under the shape constraint.…”