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Recovery of Distributions Via Moments 4 5 5 6 Robert M IMS Collections Vol. 0 (2009) 251{265 c Institute of Mathematical Statistics, 2009 1 1 arXiv: math.PR/0000011 2 2 3 3 4 Recovery of Distributions via Moments 4 5 5 6 Robert M. Mnatsakanov1,∗ and Artak S. Hakobyan2,∗ 6 7 7 West Virginia University 8 8 9 Abstract: The problem of recovering a cumulative distribution function (cdf) 9 10 and corresponding density function from its moments is studied. This problem 10 is a special case of the classical moment problem. The results obtained within 11 11 the moment problem can be applied in many indirect models, e.g., those based 12 on convolutions, mixtures, multiplicative censoring, and right-censoring, where 12 13 the moments of unobserved distribution of actual interest can be easily esti- 13 14 mated from the transformed moments of the observed distributions. Nonpara- 14 metric estimation of a quantile function via moments of a target distribution 15 15 represents another very interesting area where the moment problem arises. In 16 all such models one can use the procedure, which recovers a function via its 16 17 moments. In this article some properties of the proposed constructions are de- 17 18 rived. The uniform rates of convergence of the approximants of cdf, its density 18 function, and quantile function are obtained as well. 19 19 20 20 21 Contents 21 22 22 23 1 Introduction................................... 251 23 24 2 Some Notations and Assumptions....................... 253 24 25 3 Asymptotic Properties of Fα,ν and fα,ν ................... 254 25 26 4 Some Applications and Examples....................... 258 26 27 Acknowledgments................................. 264 27 28 References...................................... 264 28 29 29 30 30 1. Introduction 31 31 32 The probabilistic Stielties moment problem can be described as follows: let a se- 32 33 33 quence ν = fµj; j = 0; 1;::: g of real numbers be given. Find a probability distribu- 34 R j 34 tion on the non-negative real line R+ = [0; 1), such that µj = t d F (t) for j 2 35 35 N = f0; 1;::: g: The classical Stieltjes moment problem was introduced first by 36 Stieltjes [21]. When the support of the distribution F is a compact, say, suppfF g = 36 37 [0;T ] with T < 1, then the corresponding problem in known as a Hausdorff mo- 37 38 ment problem. 38 39 There are two important questions related to the Stieltjes (or Hausdorff) moment 39 40 problem: 40 41 41 (i) If the distribution F exists, is it uniquely determined by the moments fµjg? 42 (ii) How is this uniquely defined distribution F reconstructed? 42 43 43 44 44 1 45 Department of Statistics, West Virginia University, Morgantown, WV 26506, email: rmnatsak@ 45 stat.wvu.edu 46 ∗Work supported by the National Institute for Occupational Safety and Health, contract no. 46 47 212-2005-M-12857. 47 2 48 Department of Industrial and Management Systems Engineering, West Virginia University, 48 Morgantown, WV 26506, email: [email protected] 49 49 AMS 2000 subject classifications: Primary 62G05; secondary 62G20 50 Keywords and phrases: Probabilistic moment problem, M-determinate distribution, Moment- 50 51 recovered distribution, Uniform rate of convergence 51 251 imsart-coll ver. 2008/08/29 file: Mnatsak.tex date: January 31, 2009 252 Mnatsakanov and Hakobyan 1 If there is a positive answer to the question (i) we say that a distribution F is 1 2 moment-determinate (M-determinate), otherwise it is M-indeterminate. 2 3 In this paper we mainly address the question of recovering the M-determinate 3 4 distribution (density and quantile functions) via its moments in the Hausdorff mo- 4 5 ment problem, i.e., we study the question (ii). Another question we focus on here 5 6 is the estimation problem of unknown distribution and its quantile function, given 6 7 the estimated moments of the target distribution. 7 8 It is known from the probabilistic moment problem that under suitable condi- 8 9 tions M-determinate distribution is uniquely defined by its moments. There are 9 10 many articles that investigated the conditions (for example, the Carlemann's and 10 11 the Krein's conditions), under which the distributions are either M-determinate 11 12 or M-indeterminate, see, Akhiezer [2], Feller [6], Lin [10-11], and Stoyanov [22-24] 12 13 among others. However, there are very few works dealing with the reconstructions of 13 14 distributions via their moments. Several inversion formulas were obtained by invert- 14 15 ing the moment generating function and Laplace transform (Shohat and Tamarkin 15 16 [20], Widder [27], Feller [6], Chauveau et al. [4], and Tagliani and Velasquez [25]). 16 17 These methods are too restrictive, since there are many distributions for which the 17 18 moment generating functions do not exist even though all the moments are finite. 18 19 The reconstruction of an M-determinate cdf by means of mixtures having the 19 20 same assigned moments as the target distribution have been proposed in Lindsay 20 21 et al. [12]. Note that this procedure requires calculations of high-order Hankel de- 21 22 terminants, and due to ill-conditioning of the Hankel matrices this method is not 22 23 useful when the number of assigned moments is large. The reconstruction of an un- 23 24 known density functions using the Maximum Entropy principle with the specified 24 25 ordinary and fractional moments has been studied in Kevasan and Kapur [9] and 25 26 Novi Inverardi et al. [18], among others. 26 27 In Mnatsakanov and Ruymgaart [17] the constructions (2.2) and (3.13) (see 27 28 Sections 2 and 3 below) have been introduced, and only their convergence in a 28 29 weak sense have been established. 29 30 Different types of convergence of maximum entropy approximants have been 30 31 studied by Borwein and Lewis [3], Frontini and Tagliani [7], and Novi Inverardi et 31 32 al. [18], but the rates of approximations have not been established yet. Our con- 32 33 struction enables us to derive the uniform rate of convergence for moment-recovered 33 34 cdfs Fα,ν , corresponding quantile function approximate Qα, and the uniform con- 34 35 vergence of moment-recovered density approximant fα,ν , as the parameter α ! 1. 35 36 Another constructions of moment-recovered cdfs and pdfs (see, (3.13) and (3.14) in 36 37 Remark 3.3) were proposed in Mnatsakanov [13-14], where the uniform and L1-rates 37 38 of approximations have been established for them. 38 39 The paper is organized as follows: in Section 2 we introduce some notations and 39 40 assumptions, while in Section 3 we study the properties of Fα,ν and fα,ν . Note that 40 41 our construction also gives a possibility to recover different distributions through 41 42 the simple transformations of moment sequences of given distributions (Theorem 42 43 3.1 and Corollary 3.1). In Theorem 3.2 we state the uniform rate of convergence for 43 44 moment-recovered cdfs. In Corollaries 3.2 and 3.3, and in Theorem 3.3 we applied 44 45 the constructions (2.2) and (3.11) to recover pdf f, the quantile function Q, and 45 46 corresponding quantile density function q of F given the moments of F . In Section 46 47 4 some other applications of the constructions (2.2) and (3.11) are discussed: the 47 48 uniform convergence of the empirical counterparts of (2.2) and (3.11), the rate of 48 49 approximation of moment-recovered quantile function (see (4.4) in Section 4) along 49 50 with the demixing and deconvolution problems in several particular models. 50 51 Note that our approach is particularly applicable in situations where other es- 51 imsart-coll ver. 2008/08/29 file: Mnatsak.tex date: January 31, 2009 Recovery of Distributions via Moments 253 1 timators cannot be used, e.g., in situations where only moments (empirical) are 1 2 available. The results obtained in this paper will not be compared with similar 2 3 results derived by other methods. We only carry out the calculations of moment- 3 4 recovered cdfs, pdfs, and quantile functions, and compare them with the target 4 5 distributions via graphs in several simple examples. We also compare the perfor- 5 6 mances of Fα,ν and fα,ν with the similar constructions studied in Mnatsakanov 6 7 [13-14] (see, Figures 1 (b) and 3 (b)). The moment-estimated quantile function Q^α 7 8 and well known Harrell-Davis quantile function estimator Q^HD (Sheater and Mar- 8 9 ron [19]) defined in (4.6) and (4.7), respectively, are compared as well (see, Figure 9 10 2 (b)). 10 11 11 12 12 2. Some Notations and Assumptions 13 13 14 14 Suppose that M-determinate cdf F is absolute continuous with respect to the 15 15 Lebesgue measure and has support [0;T ]; T < 1. Denote the corresponding den- 16 16 sity function by f. Our method of recovering cdf F (x); 0 x T , is based on an 17 6 6 17 inverse transformation that yields a solution of the Hausdorff moment problem. 18 18 Let us denote the ordinary moments of F by 19 19 20 Z 20 (2.1) µ = tjd F (t) = (KF )(j); j 2 ; 21 j;F N 21 22 22 23 and assume that the moment sequence ν = (µ0;F ; µ1;F ;::: ) determines F uniquely. 23 24 An approximate inverse of the operator K from (2.1) constructed according to 24 25 25 [αx] 1 j−k k 26 −1 X X (−α) α 26 (2.2) Kα ν (x) = µj;F ; 0 6 x 6 T ; α 2 R+; 27 (j − k)! k! 27 k=0 j=k 28 28 29 −1 29 is such that Kα KF !w F; as α ! 1 (see, Mnatsakanov and Ruymgaart [17]).
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