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National Institute for Applied Statistics Research
Australia

University of Wollongong, Australia

Working Paper

11-19

The Multivariate Moments Problem: A Practical High Dimensional Density
Estimation Method

Bradley Wakefield, Yan-Xia Lin, Rathin Sarathy and Krishnamurty
Muralidhar

Copyright © 2019 by the National Institute for Applied Statistics Research Australia, UOW. Work in progress, no part of this paper may be reproduced without permission from the Institute.

National Institute for Applied Statistics Research Australia, University of Wollongong,
Wollongong NSW 2522, Australia Phone +61 2 4221 5435, Fax +61 2 42214998.

Email: [email protected]

The Multivariate Moments Problem:
A Practical High Dimensional Density Estimation Method

October 8, 2019

Bradley Wakefield

National Institute for Applied Statistics Research Australia
School of Mathematics and Applied Statistics University of Wollongong, NSW 2522, Australia

Email: [email protected]

Yan-Xia Lin

National Institute for Applied Statistics Research Australia
School of Mathematics and Applied Statistics University of Wollongong, NSW 2522, Australia

Email: [email protected]

Rathin Sarathy

Department of Management Science & Information Systems
Oklahoma State University, OK 74078, USA

Email: [email protected]

Krishnamurty Muralidhar

Department of Marketing & Supply Chain Management, Research Director for the Center for Business of Healthcare
University of Oklahoma, OK 73019, USA

Email: [email protected]

1

Abstract

The multivariate moments problem and it’s application to the estimation of joint density functions is often considered highly impracticable in modern day analysis. Although many results exist within the framework of this issue, it is often an undesirable method to be used in real world applications due to the computational complexity and intractability of higher order moments. This paper aims to take a new look at the application of a multivariate moment based method in estimating a joint density and demonstrate that sample moment based approximations can offer a legitimate rival to current density approximation methods.
Within this paper we use tools previously derived, such as the properties of hyper-geometric type eigenfunction polynomials, properties in the approximation of density functions and the completeness of orthogonal multivariate polynomials to obtain a sample based multivariate density approximation and associated properties of determinacy and convergence. Furthermore we attempt to overcome one of the most considerable issues in applying a moment based density approximation, the determination of the required parameters, by exploiting the analytical properties of orthogonal polynomials and applying various statistical estimation techniques.
It then follows that by deriving these various analytic results, this paper obtains the specific statistical tools required to take the multivariate moments problem from a highly specified mostly analytical idea to a practical high dimensional density estimation method.

Keywords: Density Estimation, Multivariate, Moments Problem, Orthogonal Polynomial, Sample Moments, Multivariate Density, Joint Density Estimation, Hyper-geometric Polynomial

2
1. INTRODUCTION

The construction of unique polynomial based series expansions for continuous functions is a common practice applied in a range of areas of mathematics, physics engineering and statistics (Munkhammar et al., 2017). As such, polynomial based expansions have also been applied in the study of probability density functions on both closed and infinite domains (Elderton and Johnson, 1969; Solomon and Stephens, 1978; Kendall et al., 1987; Provost, 2005) . Known more commonly as the moments problem, this area of study has been well studied and developed from as far back as the 19th century. However, practical use of such expansions are only more recently been applied due to the computational complexity required in the estimation becoming less significant with the advent of high processing computational technology. Moreover, the study of such an approach to estimating multivariate distributions is much more sporadic. As noted by Kleiber and Stoyanov (Kleiber and Stoyanov, 2013), the results for the multivariate case tend to be scattered throughout specialised journals and tend to be focused on a more analytical perspective.
Throughout the literature, researchers have tended to focus on the usefulness and effectiveness of a particular orthogonal polynomial basis in regards to estimating certain distributions, usually selecting classical polynomial families as the basis for the expansion (Gram, 1883; Edgeworth, 1905; Charlier, 1914; Szego˝, 1939; Cramer, 1946; Alexits, 1961; Kendall et al., 1987; Provost, 2005). These classical polynomials are usually chosen based on the resemblance of their weight functions to the canonical distributions. Orthogonal polynomials with which a density approximation has been derived include Hermite (Edgeworth and Gram-Charlier Expansions), Legendre, Jacobi and Laguerre Polynomials (Gram, 1883; Edgeworth, 1905; Charlier, 1914; Szego˝, 1939; Cramer, 1946; Alexits, 1961; Kendall et al., 1987; Provost, 2005) which have weight functions resembling that of the normal, uniform, beta and gamma distributions respectively. These polynomials can all be used to provide moment based expressions of certain density functions and are all eigenfunctions of a hypergeometric type second order differential equation (S´anchez-Ruiz and Dehesa, 1998).
Multivariate orthogonal polynomial expansions particularly with reference to Hermite, Jacobi and Laguerre polynomials have also been investigated (Xu, 1997; Griffiths and Spano, 2011). Joint density expressions for quasi-multivariate normal distributions have been developed by using a Hermite polynomial based expansion (Mehler, 1866; Withers and Nadarajah, 2010) and also Laguerre polynomial based expansions have been shown to be appropriate for a marginally gamma multivariate distribution (Kibble, 1940; Krishnamoorthy and Parthasarathy,

3

1951; Mustapha and Dimitrakopoulos, 2010). However, when considering Gaussian expansions using Hermite polynomials, such as the Edgeworth expansions, considerable issues regarding the behavior of the tail of the distribution are not well-defined (Kendall et al., 1987; Welling, 1999) . Although some Saddlepoint approximation methods have been proposed to overcome these issues (Daniels, 1983), these approximations tend to be restrictive and prone to failure (Provost, 2005). This is also of true of Laguerre polynomial based expansions especially when attempting to use sample moments in the approximation where the parameters can suffer from considerable sampling variance problems.
One of the key differences between the moment-based density expressions relates to the support of the varying orthogonal bases used in the approximation. It is imperative that the support of these bases are consistent with that of the sample space of the density function we are expressing in the expansion (Kleiber and Stoyanov, 2013). This inherently leads us to three distinct cases of problems: Hausdorff (density functions defined on a compact space), Stieltjes (density functions defined on a half bounded (lower or upper) space) (Stieltjes, 1894; Stoyanov and Tolmatz, 2005) and Hamburger (density functions defined on a completely unbounded space) (Shohat and Tamarkin, 1943; Akhiezer and Kemmer, 1965) and therefore there are varying cases we must consider in order to ensure the determinacy of the expansion. Determinacy refers to the existence of an unique moment based expansion of the density function and in terms of multivariate moments problem, this notion of determinacy is extensively reviewed in Kleiber and Stoyanov (Kleiber and Stoyanov, 2013). Interestingly, the paper investigates the relationship between the marginal determinacy and the determinacy of the multivariate distribution as a whole. It raises the Peterson result (Petersen, 1982), that marginal determinacy implies multivariate determinacy and conjectures that for absolutely continuous function, multivariate determinacy implies marginal. This issue of determinacy becomes fairly trivial for multivariate distribution functions defined on a compact spaces as the multivariate analogue of the Weierstrass Stone Approximation theorem (Sz¨okefalvi-Nagy, 1965) confirms the uniqueness of the polynomial expansion.
In many practical situations, no additional information about the joint density is available beyond that attached to a random sample of observations from the distribution. It is therefore difficult to decided exactly which type of expansion would be more representative of the original distribution especially when attempting to apply only one sort of expansion to a multitude of varying distribution functions where domains may vary. When enforcing various assumptions about the underlying marginal distributions, we can also misrepresent the original statistical

4

information contained in the random sample and hence fail to properly approximate the true joint density.
It is for this reason we may prefer a more uninformative approach by using a constant reference density on a compact space. This is the approach undertaken by a Legendre expansion (Provost, 2005) and allows for a more robust approach to density estimation by imputing less prior information into the construction of the density function, especially when trying to protect the original statistical information of the sample. Regardless, by taking a broader perspective towards the basis used, we can obtain results for all of these approaches and allow for easier comparison to select the more appropriate basis.
In this paper we use tools previously derived, such as the properties of hyper-geometric type eigenfunction polynomials (Yanez et al., 1994; S´anchez-Ruiz and Dehesa, 1998), properties in the approximation of density functions (Sz¨okefalvi-Nagy, 1965; Provost, 2005) and the completeness of orthogonal multivariate polynomials (Xu, 2014) to obtain a sample based multivariate density approximation and associated properties of determinacy (Putinar and Schmu¨dgen, 2008; Kleiber and Stoyanov, 2013) and convergence. We focus on distributions with bounded marginal domains for each variable in order to ensure determinacy and makes no specific choice about the associated reference joint density (weight) used. However, this restriction to a bounded domain is certainly not necessary if moment determinacy can be assured and as such, all results presented in this paper can apply on unbounded domains in certain situations. Furthermore, as demonstrated in (Lin, 2014), which applies a univariate form of moment-based density estimation for the purposes of recovering information from perturbed data, provided the sample size is sufficiently large enough, assuming a compact domain can result in a good estimate for density functions which exist on an unbounded domain.
Part of the most considerable issues in applying a moment based density approximation is determining how to best select the required parameters in the estimation. This is an area unexplored in the previous literature and by deriving various analytic results, this paper derives the specific statistical tools required to take the multivariate moments problem from a highly specified mostly analytical idea to a practical high dimensional density estimation method comparable to that of other multivariate density approximation methods. This paper is structured into seven main sections. The second section relates to the formula and results pertaining to using a generalised hyper-geometric type polynomial base for density estimation extensively studied in previous literature. The third section looks at applying these formula to obtain the multivariate density approximation with the fourth section then deriving

5

properties and techniques for applying this approximation to a sample. The fifth section contains a simulation study using these methods to assess the ability of this methods to adequately approximate the joint density. The sixth section extends these results to the cumulative distribution function, the conditional cumulative distribution function and proposes a strategy for simulating observations from such an approximation. We will conclude this paper with remarks about the tools that were derived as well as a direction for further research.

2. BASIC FORMULA AND RESULTS

This paper investigates a method by which a continuous multivariate density function fX where

Q

N

X = (X1, . . . , XN ) is a vector of random variables, defined on a compact space Ω = k=1[ak, bk] with defined finite moments, may be approximated by use of a reference distribution function

Q

N

k=1

fν =

fν where fν is a probability density function defined for each k = 1, . . . , N on the

  • k
  • k

closed interval Ωk = [ak, bk] with corresponding orthonormal polynomials {Pk,n

}

, where

nk∈N0

k

nk denotes the order of the polynomial Pk,n and N0 refers to the set of all natural numbers

k

including 0.
These polynomials are defined in a univariate sense for each kth reference distribution as particular polynomial solutions to the Sturm-Liouville differential equations of the form (S´anchezRuiz and Dehesa, 1998) (where λk is some constant and σk is a quadratic polynomial function):

d

[σk(xk)fν (xk)Pk,n (xk)] = λn fν (xk)Pk,n (xk)

for nk = 0, 1, 2, ...

  • k
  • k
  • k
  • k
  • k

dxk

Which when complete, form an orthonormal basis which by the Weierstrass-Stone approximation Theorem (Sz¨okefalvi-Nagy, 1965) and allows for the polynomial approximation of smooth functions.
It should be noted that the first two results of the following section follow from the work outlined in S´anchez-Ruiz and Dehesa (1998) but are given here for reading convenience. We first consider these polynomials Pk,n associated with one dimensional space and subsequently

k

generalise to higher dimensions.

2.1 Univariate Orthonormal Polynomials

Let fν be the density function of the random variable ν with domain Ω = [a, b]. Definition 1. The Hyper-Geometric Type Differential Equation (S´anchez-Ruiz and Dehesa, 1998). Given fν is such that there exists polynomial functions σ(x) and τ(x) of degree no greater than

6

2 and 1 respectively such that:

d

  • [σ(x)fν(x)] = τ(x)fν(x)
  • (1)

(2)

dx

As well as having boundary conditions such that for i = 0, 1, ... lim xiσ(x)fν(x) = lim xiσ(x)fν(x) = 0

  • x→a
  • x→b

Then we define the following equation as the hyper-geometric-type differential equation: σ(x)y00(x) + τ(x)y0(x) − λy(x) = 0

The Hyper-geometric type differential equation is a Sturm-Liouville problem which takes the self-adjoint form (S´anchez-Ruiz and Dehesa, 1998) :

d

  • [σ(x)y0(x)fν(x)] = λy(x)fν(x)
  • (3)

dx

This differential equation has infinitely many eigenfunction solutions with the eigenvalues λn ∈ R

1

given by λn = nτ0 + 2 n(n − 1)σ00 for n = 0, 1, ... (noting that as σ(x) and τ(x) are polynomials of degree no greater than 2 and 1 respectively then σ00 and τ0 are constants) which have corresponding hypergeometric-type polynomial eigenfunctions yn(x) of degree n which are orthogonal on [a, b] with respect to the weight function fν. We note that the solutions yn(x) are polynomial functions and hence will now be denoted as Pn(x).
Furthermore we can obtain a Rodrigues’ (type) Formula (S´anchez-Ruiz and Dehesa, 1998) for the nth eigenfunction of the Hyper-geometric type differential equation (Definition 1) which we will now denote as Pn(x). These eigenfunctions are polynomials with Rodrigues’ formula of the form (for the nth polynomial):

Bn dn

fν(x) dxn

  • Pn(x) =
  • [(σ(x))nfν(x)]

Where Bn ∈ R is the normalising constant in the Rodrigues’ formula and can be chosen fairly arbitrarily in order to show consistency with classical orthogonal polynomial families. In this paper, we will choose Bn as a constant which ensures Pn(x) are orthonormal.
We will now introduce some results related to the orthonormal polynomials Pn(x). These properties will benefit the computation process proposed for estimating joint density functions. Result 1: The expression of the normalising constant Bn. The normalising constant from the Rodrigues’ formula Bn which gives Pn(x) as an orthonormal

7

R

polynomial, such that b Pn(x)Pm(x)fν(x)dx = δn,m where δn,m denotes the Kronecker symbol,

a

is given by:

vuuut

  • h
  • i

Q

R

n−1
1

(−1)n

τ0+ 12 (n+i−1)σ00 i=0

Bn =

for n = 1, 2, . . . and B0 = 1

b

n! (σ(x))nfν(x)dx

a

  • ꢀ
  • ꢁ

Proof. S´anchez-Ruiz and Dehesa (1998) gives the orthogonality constant An = Eν Pn2(ν) for a general Bn as

  • ꢂ
  • ꢃ

Z

n−1

b

Y

1

  • An = (−1)nn!Bn2
  • τ0 + (n + i − 1)σ00

(σ(x))nfν(x)dx

2

a

i=0

  • ꢀ
  • ꢁ

Hence as Eν Pn2(ν) > 0 and by the definition of orthogonality An = 0 then it can be ascertained that the expression of Bn defined above ensures An = 1 for all n = 0, 1, . . . .

As the polynomials Pn(x) are defined to be orthonormal in this paper we know Bn is given by the expression of Bn shown in Result 1. Result 2: The completeness of the orthogonal base {Pn}n∈N Consider the L2 space L2(Ω, µν) where µν is a measure in (Ω, F) given by µν(ω) = ω fν dx

0

R
R

for ω ∈ F, then Pn(x) are complete in L2(Ω, µν) space with Pn ∈ L2(Ω, µν) as Ω Pn2dµν
=

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  • The Moment Problem Associated with the Stieltjes–Wigert Polynomials

    The Moment Problem Associated with the Stieltjes–Wigert Polynomials

    J. Math. Anal. Appl. 277 (2003) 218–245 www.elsevier.com/locate/jmaa The moment problem associated with the Stieltjes–Wigert polynomials Jacob Stordal Christiansen Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark Received 17 May 2001 Submitted by T. Fokas Abstract We consider the indeterminate Stieltjes moment problem associated with the Stieltjes–Wigert polynomials. After a presentation of the well-known solutions, we study a transformation T of the set of solutions. All the classical solutions turn out to be fixed under this transformation but this is not the case for the so-called canonical solutions. Based on generating functions for the Stieltjes– Wigert polynomials, expressions for the entire functions A, B, C,andD from the Nevanlinna (n) parametrization are obtained. We describe T (µ) for n ∈ N when µ = µ0 is a particular N-extremal solution and explain in detail what happens when n →∞. 2002 Elsevier Science (USA). All rights reserved. Keywords: Indeterminate moment problems; Stieltjes–Wigert polynomials; Nevanlinna parametrization 1. Introduction T.J. Stieltjes was the first to give examples of indeterminate moment problems. In [18] he pointed out that if f is an odd function satisfying f(u+ 1/2) =±f(u),then ∞ − unu loguf(log u) du = 0 0 E-mail address: [email protected]. 0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII:S0022-247X(02)00534-6 J.S. Christiansen / J. Math. Anal. Appl. 277 (2003) 218–245 219 for all n ∈ Z. In particular, ∞ − unu logu sin(2π log u) du = 0,n∈ Z, 0 so independent of λ we have ∞ ∞ 1 − 1 − + 2 √ unu logu 1 + λ sin(2π logu) du= √ unu logu du= e(n 1) /4.
  • Moment Problems and Semide Nite Optimization

    Moment Problems and Semide Nite Optimization

    Moment Problems and Semidenite Optimization y z Dimitris Bertsimas Ioana Pop escu Jay Sethuraman April Abstract Problems involving moments of random variables arise naturally in many areas of mathematics economics and op erations research Howdowe obtain optimal b ounds on the probability that a random variable b elongs in a set given some of its moments Howdowe price nancial derivatives without assuming any mo del for the underlying price dynamics given only moments of the price of the underlying asset Howdoweob tain stronger relaxations for sto chastic optimization problems exploiting the knowledge that the decision variables are moments of random variables Can we generate near optimal solutions for a discrete optimization problem from a semidenite relaxation by interpreting an optimal solution of the relaxation as a covariance matrix In this pap er we demonstrate that convex and in particular semidenite optimization metho ds lead to interesting and often unexp ected answers to these questions Dimitris Bertsimas Sloan Scho ol of Management and Op erations ResearchCenter Rm E MIT Cambridge MA db ertsimmitedu Research supp orted by NSF grant DMI and the Singap oreMIT alliance y Ioana Pop escu Decision Sciences INSEAD Fontainebleau Cedex FRANCE ioanap op escuin seadfr z Jay Sethuraman Columbia University DepartmentofIEORNewYork NY jayccolumbiaedu Intro duction Problems involving moments of random variables arise naturally in many areas of mathe matics economics and op erations research Let us give some examples that motivate the
  • The Classical Moment Problem

    The Classical Moment Problem

    The classical moment problem Sasha Sodin∗ March 5, 2019 These lecture notes were prepared for a 10-hour introductory mini-course at LTCC. A cap was imposed at 50 pages; even thus, I did not have time to cover the sections indicated by an asterisk. If you find mistakes or misprints, please let me know. Thank you very much! The classical moment problem studies the map S taking a (positive Borel) measure µ on R to its moment sequence (sk)k≥0, Z k sk[µ] = λ dµ(λ) . The map is defined on the set of measures with finite moments: −∞ −k µ(R \ [−R, R]) = O(R ) , i.e. ∀k µ(R \ [−R, R]) = O(R ) . The two basic questions are 1. existence: characterise the image of S, i.e. for which sequences (sk)k≥0 of real numbers can one find µ such that sk[µ] = sk for k = 0, 1, 2, ··· ? 2. uniqueness, or determinacy: which sequences in the image have a unique pre- image, i.e. which measures are characterised by their moments? In the case of non-uniqueness, one may wish to describe the set of all solutions. The classical moment problem originated in the 1880-s, and reached a definitive state by the end of the 1930-s. One of the original sources of motivation came from probability theory, where it is important to have verifiable sufficient conditions for determinacy. Determinacy is also closely related to several problems in classical analysis, particularly, to the study of the map taking a (germ of a) smooth function f to the sequence of (k) its Taylor coefficients (f (0))k≥0.
  • The Truncated Moment Problem

    The Truncated Moment Problem

    The Truncated Moment Problem Philipp J. di Dio To my parents They gave me the freedom to worry only about mathematics. The Truncated Moment Problem Von der Fakult¨atf¨urMathematik und Informatik der Universit¨atLeipzig angenommene DISSERTATION zur Erlangung des akademischen Grades DOCTORRERUMNATURALIUM (Dr. rer. nat.) im Fachgebiet Mathematik vorgelegt von Dipl.-Math. B.Sc. Philipp J. di Dio geboren am 25.02.1986 in Lutherstadt Wittenberg. Einreichungsdatum: 10. Januar 2018. Die Annahme der Dissertation wurde empfohlen von: 1. Prof. Dr. Konrad Schm¨udgen,Universit¨atLeipzig 2. Prof. Dr. Claus Scheiderer, Universit¨atKonstanz Die Verleihung des akademischen Grades erfolgt mit Bestehen der Verteidigung am 28. Mai 2018 mit dem Gesamtpr¨adikat magna cum laude. Contents 1 Introduction 1 2 Preliminaries 5 2.1 Basic Definitions for the Moment Problem and Carath´eodory Numbers . .5 2.2 Three Theorems of Hans Richter . .9 2.3 Apolar Scalar Product and Real Waring Rank . 10 3 Carath´eodory Numbers CA and CA;± 11 3.1 The Moment Cone SA and its Connection to CA ........................ 11 3.2 Continuous Functions . 15 3.3 Differentiable Functions . 18 3.4 Upper Bound Estimates on CA;± and the Real Waring Rank . 27 3.5 One-Dimensional Monomial Case with Gaps . 28 3.6 Multidimensional Monomial Case . 38 4 Set of Atoms, Determinacy, and Core Variety 47 4.1 Set W(s) of Atoms . 47 4.2 Determinacy of Moment Sequences . 54 4.3 Core Variety . 57 5 Weyl-Circles 59 5.1 Preliminaries . 59 5.2 Main Tool . 61 5.3 Hamburger Moment Problem . 62 5.4 Stieltjes Moment Problem on [a; 1).............................