Multiscale testing of qualitative hypotheses

Suppose that one observes a process Y on the unit interval, where dY(t) = n 1/ 2f(t)dt + dW(t) with an unknown function parameter f, given scale parameter n N > 1 (sample size) and standard Brownian motion W. We propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity. These tests are asymptotically optimal and adaptive in a certain sense. They are constructed via a new class of multiscale statistics and an extension of Levy's modulus of continuity of Brownian Motion.

[1]  B. Silverman,et al.  Using Kernel Density Estimates to Investigate Multimodality , 1981 .

[2]  J. Rice.  Bandwidth Choice for Nonparametric Regression , 1984 .

[3]  J. Hartigan,et al.  The Dip Test of Unimodality , 1985 .

[4]  T. Gasser,et al.  Residual variance and residual pattern in nonlinear regression , 1986 .

[5]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[6]  Yu. I. Ingster.  Minimax Testing of Nonparametric Hypotheses on a Distribution Density in the $L_p$ Metrics , 1987 .

[7]  M. Ermakov.  Minimax Detection of a Signal In a Gaussian White Noise , 1991 .

[8]  G. Sawitzki,et al.  Excess Mass Estimates and Tests for Multimodality , 1991 .

[9]  D. Donoho,et al.  Renormalization Exponents and Optimal Pointwise Rates of Convergence , 1992 .

[10]  James Stephen Marron,et al.  Some asymptotics for multimodality tests based on kernel density estimates , 1992 .

[11]  R. L. Eubank,et al.  Testing Goodness-of-Fit in Regression Via Order Selection Criteria , 1992 .

[12]  Wilbert C.M. Kallenberg,et al.  Consistency and Monte Carlo Simulation of a Data Driven Version of smooth Goodness-of-Fit Tests , 1995 .

[13]  T. Ledwina.  Data-Driven Version of Neyman's Smooth Test of Fit , 1994 .

[14]  D. Donoho.  Asymptotic minimax risk for sup-norm loss: Solution via optimal recovery , 1994 .

[15]  D. Donoho.  Statistical Estimation and Optimal Recovery , 1994 .

[16]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[17]  Jianqing Fan.  Test of Significance Based on Wavelet Thresholding and Neyman's Truncation , 1996 .

[18]  L. Brown,et al.  Asymptotic equivalence of nonparametric regression and white noise , 1996 .

[19]  V. Spokoiny.  Adaptive hypothesis testing using wavelets , 1996 .

[20]  M. Nussbaum.  Asymptotic Equivalence of Density Estimation and Gaussian White Noise , 1996 .

[21]  Anton Schick,et al.  Testing for the equality of two nonparametric regression curves , 1997 .

[22]  Jeffrey D. Hart,et al.  Nonparametric Smoothing and Lack-Of-Fit Tests , 1997 .

[23]  Ion Grama,et al.  Asymptotic equivalence for nonparametric generalized linear models , 1998 .

[24]  Sergei Leonov.  Remarks on extremal problems in nonparametric curve estimation , 1999 .

[25]  J. Marron,et al.  SiZer for Exploration of Structures in Curves , 1999 .

[26]  P. Hall,et al.  Mode testing in difficult cases , 1999 .

[27]  A. Tsybakov,et al.  Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point , 2000 .

[28]  P. Hall,et al.  Linear functions , 2018, Algebra and Geometry.

[29]  Lutz Dümbgen,et al.  Application of local rank tests to nonparametric regression , 2002 .