Multiwavelet density estimation for biased data

Abstract In this article, we present our method for multiwavelet estimation for a density function based on random samples from a weighted distribution. First, we propose linear and nonlinear multiwavelet estimators for the biased data. Then, we provide and prove the upper bound of risk for the proposed nonlinear estimator over a large class of densities in the Besov space, . Finally, we provide simulation and real data experiments to illustrate the effectiveness and behavior of the proposed method and estimator.

[1]  M. C. Jones.  Kernel density estimation for length biased data , 1991 .

[2]  Zhengxing Cheng,et al.  An algorithm for constructing symmetric orthogonal multiwavelets by matrix symmetric extension , 2004, Appl. Math. Comput..

[3]  B. Alpert.  A class of bases in L 2 for the sparse representations of integral operators , 1993 .

[4]  C. Chui,et al.  A study of orthonormal multi-wavelets , 1996 .

[5]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[6]  S. L. Lee,et al.  WAVELETS OF MULTIPLICITY r , 1994 .

[7]  T. R. Downie,et al.  The discrete multiple wavelet transform and thresholding methods , 1998, IEEE Trans. Signal Process..

[8]  Xiang-Gen Xia,et al.  Design of prefilters for discrete multiwavelet transforms , 1996, IEEE Trans. Signal Process..

[9]  Weixing Wang,et al.  On a family of biorthogonal armlets with prescribed properties , 2007, Appl. Math. Comput..

[10]  I. D. Feis,et al.  Wavelet density estimation for weighted data , 2014 .

[11]  Nonlinear wavelet density estimation for biased data in Sobolev spaces , 2013 .

[12]  I. Johnstone,et al.  Density estimation by wavelet thresholding , 1996 .

[13]  Qingtang Jiang,et al.  BALANCED MULTI-WAVELETS IN R , 2005 .

[14]  Jian-ao Lian,et al.  Armlets and balanced multiwavelets: flipping filter construction , 2005, IEEE Trans. Signal Process..

[15]  Y. Cen,et al.  Explicit construction of high-pass filter sequence for orthogonal multiwavelets , 2009, Appl. Math. Comput..

[16]  Weixing Wang,et al.  Odd-length armlets with flipping property and its application in image compression , 2009, Expert Syst. Appl..

[17]  George C. Donovan,et al.  Construction of Orthogonal Wavelets Using Fractal Interpolation Functions , 1996 .

[18]  Charles K. Chui,et al.  Analysis-ready multiwavelets (armlets) for processing scalar-valued signals , 2004, IEEE Signal Processing Letters.

[19]  D. Hardin,et al.  Multiwavelet prefilters. 1. Orthogonal prefilters preserving approximation order p/spl les/2 , 1998 .

[20]  Jo Yew Tham,et al.  Some properties of symmetric-antisymmetric orthonormal multiwavelets , 2000, IEEE Trans. Signal Process..

[21]  Adrian M. Peter,et al.  Multiwavelet density estimation , 2012, Appl. Math. Comput..

[22]  Y. Vardi,et al.  Nonparametric Estimation in the Presence of Length Bias , 1982 .

[23]  Xiang-Gen Xia,et al.  A new prefilter design for discrete multiwavelet transforms , 1997, Conference Record of the Thirty-First Asilomar Conference on Signals, Systems and Computers (Cat. No.97CB36136).

[24]  Sam Efromovich.  Density estimation for biased data , 2004 .

[25]  A. Tsybakov,et al.  Wavelets, approximation, and statistical applications , 1998 .

[26]  S. L. Lee,et al.  Wavelets in wandering subspaces , 1993 .

[27]  Martin Vetterli,et al.  High-order balanced multiwavelets: theory, factorization, and design , 2001, IEEE Trans. Signal Process..

[28]  B. Vidakovic,et al.  Wavelet density estimation for stratified size-biased sample , 2010 .

[29]  L. Cui,et al.  m-Band orthogonal vector-valued multiwavelets for vector-valued signals , 2008 .

[30]  Y. Vardi.  Empirical Distributions in Selection Bias Models , 1985 .

[31]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .