
Journal of Mathematical Chemistry, Vol. 42, No. 3, October 2007 (© 2006) DOI: 10.1007/s10910-006-9134-5 Expressing a probability density function in terms of another PDF: A generalized Gram-Charlier expansion Mario´ N. Berberan-Santos Centro de Qu´ımica-F´ısica Molecular, Instituto Superior Tecnico,´ 1049-001 Lisboa, Portugal E-mail: [email protected] Received 21 March 2006; revised 11 April 2006 An explicit formula relating the probability density function with its cumulants is derived and discussed. A generalization of the Gram-Charlier expansion is presented, allowing to express one PDF in terms of another. The coefficients of this general expan- sion are explicitly obtained. KEY WORDS: probability density function, cumulant, Gram-Charlier expansion, Hermite polynomials AMS subject classification: 60E10 characteristic functions; other transforms, 62E17 approximations to distributions (non-asymptotic), 62E20 asymptotic distribution theory 1. Introduction The moment-generating function of a random variable is by definition [1–3] the integral ∞ M(t) = f(x) etxdx, (1) −∞ where f(x) is the probability density function (PDF) of the random variable. It is well known that if all moments are finite (this will be assumed through- out the work), the moment-generating function admits a Maclaurin series expan- sion [1–3], ∞ tn M(t) = m , (2) n n! n=0 where the raw moments are ∞ n mn = x f(x) dx (n = 0, 1,...,). (3) −∞ 585 0259-9791/07/1000-0585/0 © 2006 Springer Science+Business Media, Inc. 586 M.N. Berberan-Santos / Generalized gram-charlier expansion On the other hand, the cumulant-generating function [3,4] K(t) = ln M(t), (4) admits a similar expansion ∞ tn K(t) = κ , (5) n n! n=1 where the κn are the cumulants, defined by (n) κn = K (0), (n = 1, 2,...,). (6) The recurrence relation for the moments, obtained from ∞ ∞ tn tn M(t) = m = exp κ (7) n n! n n! n=0 n=1 by taking nth order derivatives at t = 0, is n n m + = m − κ + . (8) n 1 p n p p 1 p=0 The raw moments are therefore explicitly related to the cumulants by m1 = κ1, = κ2 + κ , m2 1 2 = κ3 + κ κ + κ , m3 1 3 1 2 3 = κ4 + κ2κ + κ2 + κ κ + κ , m4 1 6 1 2 3 2 4 1 3 4 (9) = κ5 + κ3κ + κ κ2 + κ2κ + κ κ + κ κ + κ , m5 1 10 1 2 15 1 2 10 1 3 10 2 3 5 1 4 5 = κ6 + κ4κ + κ2κ2 + κ3 + κ3κ + κ κ κ + κ2 m6 1 15 1 2 45 1 2 15 2 20 1 3 60 1 2 3 10 3 + κ2κ + κ κ + κ κ + κ , 15 1 4 15 2 4 6 1 5 6 Equation (8) can be solved for κn+1, n−1 n κ + = m + − m − κ + . (10) n 1 n 1 p n p p 1 p=0 An explicit formula giving the cumulants in terms of the moments is also known [3]. The first four cumulants are κ1 = m1 = µ, κ = − 2 = σ 2, 2 m2 m1 κ = 3 − + = γ σ 3, (11) 3 2m1 3m1m2 m3 1 κ =− 4 + 2 − 2 − + = γ σ 4, 4 6m1 12m1m2 3m2 4m1m3 m4 2 M.N. Berberan-Santos / Generalized gram-charlier expansion 587 where γ1 is the skewness and γ2 is the kurtosis. Of all cumulants, only the second is necessarily non-negative. With the exception of the delta and normal cases, all PDFs have an infinite number of non-zero cumulants [3,5]. A third canonical function of the PDF is the characteristic function [1–3], defined by ∞ Φ(t) = f (x) eitxdx (12) −∞ that is simply the Fourier transform of the PDF. The PDF can therefore be writ- ten as the inverse Fourier transform of its characteristic function, ∞ − 1 − f(x) = F 1 [Φ(t)] = Φ(t) e itxdt. (13) 2π −∞ The characteristic function admits the Maclaurin expansion, ∞ (it)n Φ(t) = m . (14) n n! n=0 Taking into account that ∞ ( ) 1 − δ n (x) = (−it)ne itxdx, (15) 2π −∞ where δ(n)(x) is the nth derivative of the delta function, termwise Fourier inver- sion of equation (14) yields the following formal PDF expansion, obtained in a different way by Gillespie [6] ∞ m ( ) f(x) = (−1)n n δ n (x). (16) n! n=0 From the characteristic function a cumulant generating function can also be defined, C(t) = ln Φ(t), (17) whose series expansion gives ∞ (it)n C(t) = κ . (18) n n! n=1 Under relatively general conditions [3,7], the moments (cumulants) of a dis- tribution define the respective PDF, as follows from the above equations. It is therefore of interest to know how to build the PDF from its moments (cumu- lants). An obvious practical application is to obtain an approximate form of the 588 M.N. Berberan-Santos / Generalized gram-charlier expansion PDF from a finite set of moments (cumulants). This is an important problem that has received much attention for a long time (see [3] and references therein). In this work, an explicit formula relating a PDF with its cumulants is obtained, and its interest and limitations discussed. A generalization of the Gram-Charlier expansion that allows to express a PDF in terms of another PDF is obtained in two different ways. The procedure for the calculation of the coeffi- cients of the general expansion is also given. 2. Calculation of a PDF from its cumulants Using equations (13), (17), and (18), the PDF can be written as ∞ ∞ ( )n −1 C(t) 1 it f(x) = F [e ]= exp κn exp (−ixt) dt, (19) 2π −∞ n! n=1 assuming that the series has a sufficiently large convergence radius. Taking into account that the PDF is a real function, equation (19) can be simplified to give 1 ∞ t2 t4 t3 f(x) = exp −κ2 + κ4 −··· cos xt − κ1t + κ3 −··· dt.(20) 2π −∞ 2! 4! 3! Finally, because the integrand is of even parity, equation (20) becomes 1 ∞ t2 t4 t3 f(x) = exp −κ2 + κ4 −··· cos xt − κ1t + κ3 −··· dt. (21) π 0 2! 4! 3! Equations (20 and 21) explicitly relate a PDF with its cumulants and allow – at least formally – its calculation from the cumulants, provided the series involved are convergent in a sufficiently large integration range. Equation (20) was previously obtained for one-sided PDFs [8] by means of analytical Laplace transform real inversion [9]. 3. Finite number of nonzero cumulants: dirac delta and normal PDFs If all cumulants but the first are zero, equation (20) yields the delta function, as expected, 1 ∞ f(x) = cos [(x − κ1)t] dt = δ (x − κ1) . (22) 2π −∞ M.N. Berberan-Santos / Generalized gram-charlier expansion 589 If it is assumed that all cumulants but the first two are zero, the result is now the normal (or Gaussian) PDF, ϕ(x), 1 ∞ 1 1 1 x − µ 2 ϕ(x)= exp − (σt)2 cos [(x − µ)t] dt = √ exp − . 2π −∞ 2 2πσ2 2 σ (23) No further PDFs correspond to a finite set of cumulants. With the excep- tion of the delta and normal distributions, all PDFs have an infinite number of nonzero cumulants, as proved by Marcinkiewicz [3,5]. 4. Derivatives of the normal function and an integral representation for Hermite polynomials An interesting outcome of the integral representation of the normal func- tion, equation (23), is a compact form for its derivatives: ∞ ( ) 1 1 nπ ϕ n (x) = tn exp − (σt)2 cos (x − µ)t + dt. (24) 2π −∞ 2 2 As a simple application of this result, take the Hermite polynomials, defined by the Rodrigues formula, n 2 d − 2 H (x) = (−1)nex e x . (25) n dxn Using equation (24), a known integral representation for these polynomials follows immediately, (− )n ∞ 2 π 1 x2 n t n Hn(x) = √ e t exp − cos xt + dt 2 π −∞ 4 2 n+1 ∞ 2 2 nπ = √ ex un exp −u2 cos 2xu − du. (26) π 0 2 Note that equation (24) also results from the identity for Fourier transforms that enables to express the nth order derivative of a function in terms of its Fou- rier transform, n ∞ ( ) (−1) − f n (x) = (it)n F [ f (x)] e itxdt, (27) 2π −∞ or, given that f(x) is real, n ∞ ( ) (−1) − f n (x) = tn Re in F [ f (x)] e itx dt. (28) 2π −∞ 590 M.N. Berberan-Santos / Generalized gram-charlier expansion 5. Gram-Charlier expansion The usual form of the Gram-Charlier expansion (the so-called Type A series) is an expansion of a PDF about a normal distribution with common µ and σ (i.e., common cumulants κ1 and κ2) [3,10,11]. This expansion that finds applications in many areas, including finance [12], analytical chemistry [13,14], spectroscopy [15], fluid mechanics [16], and astrophysics and cosmology [17– 19], can be straightforwardly obtained from equation (20). Indeed, taking into account equation (23), equation (20) can be rewritten as ∞ 3 4 1 1 2 t t f(x) = exp − (σt) cos (xt−µt) cos κ3 −··· exp κ4 −··· dt + 2π −∞ 2 3! 4! ∞ 3 1 1 2 t − exp − (σt) sin (xt−µt) sin κ3 −··· 2π −∞ 2 3! t4 exp κ −··· dt. (29) 4 4! 3 4 3 Expansion of cos κ t −··· exp κ t −··· and of sin κ t −··· exp 3 3! 4 4! 3 3! κ t4 −··· = 4 4! about t 0gives 3 5 4 6 κ cos κ t −κ t +··· exp κ t − κ t +··· = 1+ 4 t4 − 1 10κ2 + κ t6 +··· 3 3! 5 5! 4 4! 6 6! 4! 6! 3 6 3 5 4 6 κ κ (30) κ t − κ t +··· κ t − κ t +··· = 3 3 − 5 5 +··· sin 3 3! 5 5! exp 4 4! 6 6! 3! t 5! t and therefore, using equation (24), equation (29) becomes the Gram-Charlier series, κ κ κ f (x) = ϕ(x) − 3 ϕ(3)(x) + 4 ϕ(4)(x) − 5 ϕ(5)(x) ! ! ! 3 4 5 1 ( ) + 10κ2 + κ ϕ 6 (x) +··· (31) 6! 3 6 The terms of this series, including the respective numerical coefficients, are fre- quently written as functions of the Hermite polynomials (cf.
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