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Multiplicative Censoring: Estimation of a Density and Its Derivatives

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Multiplicative Censoring: Estimation of a Density and Its Derivatives REVSTAT – Statistical Journal Volume 11, Number 3, November 2013, 255–276 MULTIPLICATIVE CENSORING: ESTIMATION OF A DENSITY AND ITS DERIVATIVES UNDER THE Lp-RISK Authors: Mohammad Abbaszadeh – Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran [email protected] Christophe Chesneau – Universit´ede Caen, LMNO, Campus II, Science 3, 14032, Caen, France [email protected] Hassan Doosti – Department of Mathematics, Kharazmi University, Tehran, Iran and Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia [email protected] Received: May 2012 Revised: January 2013 Accepted: January 2013 Abstract: We consider the problem of estimating a density and its derivatives for a sample of • multiplicatively censored random variables. The purpose of this paper is to present an approach to this problem based on wavelets methods. Two different estimators are developed: a linear based on projections and a nonlinear using a term-by-term selection of the estimated wavelet coefficients. We explore their performances under the Lp-risk with p 1 and over a wide class of functions: the Besov balls. Fast rates of convergence are≥ obtained. Finite sample properties of the estimation procedure are studied on a simulated data example. Key-Words: density estimation; multiplicative censoring; inverse problem; wavelets; Besov balls; • Lp-risk. AMS Subject Classification: 62G07, 62G20. • 256 Mohammad Abbaszadeh, Christophe Chesneau and Hassan Doosti Multiplicative Censoring: Estimation of a Density and its Derivatives... 257 1. INTRODUCTION The multiplicative censoring density model can be described as follows. We observe n i.i.d. random variables Y ,...,Y where, for any i 1,...,n , 1 n ∈ { } (1.1) Yi = UiXi , U1,...,Un are n unobserved i.i.d. random variables having the common uniform distribution on [0, 1] and X1,...,Xn are n unobserved i.i.d. random variables with common unknown density f : [0, 1] [0, ]. For any i 1,...,n , we suppose → ∞ ∈ { } that Ui and Xi are independent. Our aim is to estimate f (or a transformation of f) from Y1,...,Yn. Details, applications and results of this model can be found in, e.g., [37], [38], [2] and [1]. For recent applications in the field of signal processing, we refer to [7] and references therein for further readings. In this paper, we investigate the estimation of f (m) (including f for m = 0). This is particularly of interest to detect possible bumps, concavity or convexity properties of f. The estimation of the derivatives of a density have been investi- gated by several authors. The pioneers are [4], [35] and [36]. Recent studies can be found in [31], [9, 10], [32] and [8]. In recent years, wavelet methods in nonparametric function estimation have become a powerful technique. The major advantages of these methods are their spatial adaptivity and asymptotic optimality properties over large function spaces. We refer to, e.g., [3], [23] and [39]. These facts motivate the estimation of f (m) via wavelet methods. To the best of our knowledge, this has never been investigated before for (1.1). Combning the approaches of [1] and [31], we con- struct two different wavelet estimators: a linear one and a nonlinear adaptive one based on a hard thresholding rule introduced by [18]. The latter method has the advantage to be adaptive; it does not depend on the knowledge of the smoothness (m) of f in its construction. We explore their performances via the Lp-risk with p 1 (including the Mean Integrated Squared Error (MISE) which corresponds ≥ to p = 2) over a “standard” wide class of unknown functions: the Besov balls s Br,q(M). Our main result proves that the considered adaptive wavelet estimator achieves a fast rate of convergence. Then we show the finite sample properties of the considered estimators by a simulated data. The rest of the paper is organized as follows. Section 2 briefly describes the wavelet basis and the Besov balls. Assumptions on the model and the wavelet es- timators are presented in Section 3. The theoretical results are given in Section 4. A simulation study is done in Section 5. The proofs are gathered in Section 6. 258 Mohammad Abbaszadeh, Christophe Chesneau and Hassan Doosti 2. WAVELETS AND BESOV BALLS This section is devoted to bascics on wavelets and Besov balls. 2.1. Wavelets Let N be a positive integer such that N > 10(m + 1) (where m refers to the estimation of f (m)). Throughout the paper, we work within an orthonormal multiresolution analysis of L ([0, 1]) = h: [0, 1] R; 1 h(x)2 dx < , associated with the ini- 2 → 0 ∞ tial wavelet functions φ and ψ of the Daubechies wavelets db2N. The features of R these functions are to be compactly supported and m+1. C Set φ (x) = 2j/2 φ(2jx k) , ψ (x) = 2j/2ψ(2jx k) . j,k − j,k − Then, with an appropriate treatment at the boundaries, there exists an integer τ satisfying 2τ 2N such that, for any ℓ τ, the system ≥ ≥ = φ ; k 0,..., 2ℓ 1 ; ψ ; j N 0, ..., ℓ 1 , k 0,..., 2j 1 S ℓ,k ∈ { − } j,k ∈ − { − } ∈ { − } n o is an orthonormal basis of L2([0, 1]). For any integer ℓ τ, any h L ([0, 1]) can be expanded on as ≥ ∈ 2 S 2ℓ−1 ∞ 2j −1 (2.1) h(x) = c φ (x) + d ψ (x) , x [0, 1] , ℓ,k ℓ,k j,k j,k ∈ Xk=0 Xj=ℓ Xk=0 where 1 1 (2.2) cj,k = h(x) φj,k(x) dx, dj,k = h(x) ψj,k(x) dx . Z0 Z0 See, e.g., [14] and [27]. As usual in nonparametric statistics via wavelets, we will suppose that the unknown function f (m) belongs to Besov balls defined below. 2.2. Besov balls Let M>0, s>0, r 1, q 1 and L ([0,1])= h: [0,1] R; 1 h(x) rdx< . ≥ ≥ r → 0 | | ∞ Set, for every measurable function h on [0,1] and ǫ 0, ∆ǫ(h)(x)= h(x+ǫ) h(x), 2 N ≥ N−1 R − ∆ǫ (h)(x) = ∆ǫ(∆ǫ(h))(x) and, identically, ∆ǫ (h)(x) = ∆ǫ (∆ǫ(h))(x). Multiplicative Censoring: Estimation of a Density and its Derivatives... 259 Let 1 1/r ρN (t,h,r) = sup ∆N (h)(x) r dx . | ǫ | |ǫ|≤t Z0 Then, for s [0,N), we define the Besov ball Bs (M) by ∈ r,q q 1/q 1 ρN (t,h,r) dt Bs (M) = h L ([0, 1]); M , r,q ∈ r ts t ≤ ( Z0 ) with the usual modifications if r = or q = . ∞ ∞ We have the equivalence: h Bs (M) if and only if there exists a constant ∈ r,q M ∗ > 0 (depending on M) such that (2.2) satisfy 1/r q 1/q 2τ −1 1/r ∞ 2j −1 c r + 2j(s+1/2−1/r) d r M ∗ , | τ,k| | j,k| ≤ k=0 ! j=τ k=0 X X X with the usual modifications if r = or q = . ∞ ∞ In this expression, s is a smoothness parameter and r and q are norm parameters. Details on Besov balls can be found in [28] and [23, Chapter 9]. 3. ESTIMATORS This section describes our wavelet estimation approach. 3.1. Wavelet methodology Suppose that, for any v 0,...,m , f (v) L ([0, 1]). Then we have the ∈ { } ∈ 2 wavelet series expansion: 2ℓ−1 ∞ 2j −1 f (m)(x) = c(m)φ (x) + d(m)ψ (x) , x [0, 1] , ℓ,k ℓ,k j,k j,k ∈ Xk=0 Xj=ℓ Xk=0 (m) 1 (m) (m) 1 (m) where cj,k = 0 f (x)φj,k(x) dx and dj,k = 0 f (x)ψj,k(x) dx and m is the order of the density derivative to be estimated. R R We now aim to construct natural estimators for these unknown wavelet coefficients. Combining the approaches of [1] and [31], let us investigate a more (m) (m) arranging expression for cj,k (the same development holds for dj,k ). 260 Mohammad Abbaszadeh, Christophe Chesneau and Hassan Doosti Suppose that, for any v 0,...,m , f (v)(0) = f (v)(1) = 0. It follows from ∈ { } m-fold integration by parts that 1 c(m) = ( 1)m f(x)(φ )(m)(x) dx . j,k − j,k Z0 Note that, since U ([0, 1]) and U and X are independent, the density of Y 1 ∼U 1 1 1 is 1 f(y) (3.1) g(x) = dy, x [0, 1] . y ∈ Zx Hence f(x)= xg′(x), x [0, 1]. − ∈ One integration by parts yields 1 c(m) = ( 1)m g′(x) x(φ )(m)(x) dx j,k − − j,k Z0 1 m (m) (m+1) = ( 1) (φj,k) (x)+ x(φj,k) (x) g(x) dx − 0 Z = E ( 1)m (φ )(m)(Y )+ Y (φ )(m+1)(Y ) . − j,k 1 1 j,k 1 (m) The method of moments gives the following unbiased estimator for cj,k : n ( 1)m (3.2) cˆ(m) = − (φ )(m)(Y )+ Y (φ )(m+1)(Y ) j,k n j,k i i j,k i Xi=1 (m) and, similarly, an unbiased estimator for dj,k is n ( 1)m (3.3) dˆ(m) = − (ψ )(m)(Y )+ Y (ψ )(m+1)(Y ) . j,k n j,k i i j,k i Xi=1 Further properties of these wavelet coefficients estimators are explored in Propo- sitions 6.1 and 6.2 below. We are now in the position to present the considered estimators for f (m). 3.2. Main estimators ˆ(m) We define the linear estimator flin by 2j0 −1 (m) (m) (3.4) fˆ (x) = cˆ φj ,k(x) , x [0, 1] , lin j0,k 0 ∈ Xk=0 (m) wherec ˆj,k is defined by (3.2) and j0 is an integer which will be properly chosen later.
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