Median Stable Distributions and Schröder s Equation

Transcription

1 Median Stable Distibutions and Schöde s Equation Gib Bassett August 7, 218 Abstact: Median stable distibutions ae an extension of taditional (mean) stable distibutions. The extension is pat of a eseach pogam begun many yeas ago with Roge Koenke. The pogam seeks genealizations of well-known statistics to new domains.(if least squaes is the linea model vesion of the sample mean, then whee ae the linea model vesions of othe statistics, like the quantiles). In the cuent implementation, the taditional definition of stability (in tems of sums of iid andom vaiables) is ecast as a condition on the sampling distibution of an estimato. Fo the taditional (mean) stable distibution, the sample mean s (escaled) sampling distibution is identical to the distibution of the iid data. Median stable distibutions ae defined similaly by eplacing the sample mean with the sample median. Since the sampling distibution of the median is a functional its stable distibution is the solution to a functional equation. It tuns out that this defining functional equation is an instance of a famous equation due to Schöde fom 187. The fame of the equation is due to the way it incopoates iteation of functions, a key featue of what many yeas late would become dynamic systems analysis. The cuent pape eviews median stable distibutions in light of the connection to Schöde s functional equation. 1

2 1 Intoduction: Median Stable Distibutions Median stable distibutions ae an extension of the idea of (mean) stable distibutions [1]. The taditional definition of stability (in tems of sums of iid andom vaiables) is ecast as a condition on the sampling distibution of an estimato. 1 This makes the taditional analysis a special case of the moe geneal poblem of the conditions unde which an estimato has the same distibution as the data. This extension continues a eseach pogam begun many yeas ago with Roge Koenke whee well known statistics ae genealized to new domains; [2], [7]. Let ˆX = ˆX(F) denote the sample mean based on iid data, X i (F) = X(F) whose common cdf is F. A mean stable distibution is a cdf H so that (afte centeing at a µ and scaling by a λ) the distibution of the estimato and the data ae the same: λ( ˆX(H) µ) = d X(H) µ (1) Familia examples of mean stable distibutions ae the nomal (with µ as the expected value and λ= n), and the Cauchy (with µ as the median and λ = 1). Replacing the sample mean with the sample median in (1) defines a median stable distibution. To fix ideas conside the simplest case of the median with n = 3 i.i.d obsevations. Let ˆX = ˆX(F) denote the sample median and its sampling cdf by M(x) = M(x : F) = P[ ˆX < x], which is given by, M(x) = G(F(x)) (2) whee the G-function is 2, G(w) = 3w 2 2w 3, w [,1]. (3) The G-function is depicted in Figue 1 along with schematics illustating how G maps 1 Identifying a stable distibution in tems of an estimato is not standad. The usual pesentation, motivated by the sample mean, is in tems of sums of i.i.d andom vaiables; fo example, [6]. Reintepeting the standad definition in tems of the sample mean has the advantage of genealization to new contexts. 2 ˆX < x if: (i) two-out-of-thee, o (ii) thee-out-of-thee of the X i ae less than x. Two-out-of-thee has pobability, F(x) 2 (1 F(x)), and can occu in thee-choose-two equals 3 ways. Thee-out-of-thee can occu in 1 way, which has pobability, F(x) 3. So, M(x) = F(x) 3 +3F(x) 2 (1 F(x)) = 3F(x) 2 2F(x) 3 = G(F(x)). 2

3 F to M. (Note that G(w) is the distibution of the sample median when the data is unifomly distibuted on [,1]). 1 G(x) x=y F.5 F F F Figue 1: Mapping fom F to M, via G It is eadily veified that the median of G(F(x)) and F(x) ae the same. The sample median is median-unbiased simila to the sample mean being mean-unbiased. Without loss of geneality the data is heeafte centeed so that eithe its unique median is, o is in the inteval of medians, [µ, µ + ]. A median stable distibution theefoe is a cdf H with a λ-scaling facto satisfying (1), o, H(x) = M[x : H(λ 1 x)], o on substituting fo M, H(x) = G(H(λ 1 x)). (4) 3

4 The solutions to this functional equation define the n = 3 vesion of median stable distibutions. If the data is distibuted like H then so too is the median. Median stable distibutions fo n > 3, n = ae defined similaly: substitute G fo G in (4) whee G is the distibution of the sample median given iid unifomly distibuted data; w G (w) = C g(t) dt. (5) whee g is the deivative of G 3, and the constant C makes G(1) = 1; C = 6 (2+1)!!! ; see, e.g., [5]. The sampling cdf of the median is G (F(x)). Schöde. I ecently discoveed that the defining functional equation fo median stability is an instance of an old and famous functional equation due to Schöde in 187, [13]. As in ou case, the given is G, and the Schöde poblem is to solve the functional equation (4) fo the unknown H. The fame of the equation is due its cental ole in what would many yeas late become complex dynamic systems. It aises because of the neat way that the equation featues iteation of functions. To illustate, suppose H has an invese (in ou case, quantile) function H 1 so that G in (4) expessed in tems of H and H 1 is: G(w) = H(λH 1 (w)). (6) (Notice that this shows what G looks like fo a given H; see Schöde s Opposite Appoach below). Composing G(w) twice gives, G (2) (w) = G(G(w)) = H(λH 1 (H(λH 1 (w)) = H(λ 2 H 1 (w)) and continuing, G (k) (w) = H(λ k H 1 (w)). (7) This k-composition woks when k is a positive intege. But it also woks fo any eal k thus defining composition fo non intege k. Fo example, G ( 1 2 ) (w) = H(λ 1 2 H 1 (w)) (because G ( 2 1 ) (G ( 1 2 ) (w)) = G(w)). Altenatively: G ( 2) (w) = H(λ 2 H 1 (w)), and G ( 1) (w) = H(λ 1 H 1 (w)). 4

5 Newton s method fo appoximating the oots of a polynomial was the poblem Schöde was woking on that led to his functional equation. A polynomial detemines an iteating Newton G -function whose fixed point is a oot of the polynomial. Unde the ight conditions on the polynomial and a stating w the iteation G (k) (w ) conveges to a oot of the polynomial. Solving Schöde s equation(4) is not staightfowad. In fact, despite intoducing the functional equation Schöde neve succeeded in finding methods that guaanteed solutions... He moe o less admitted defeat, settling instead fo what one might call the Opposite Appoach, wheein one begins with [H] and λ, and then defining [G] [via (6)], theeby obtaining eady made examples of pe-solved Schöde equations ([4], p.214). It was not until 1884 that Koenigs [9] showed how to get H fo a given G. Thee is typically no closed fom solution and H is expessed as the limit of a functional composition opeation involving the known G function. 3 Discussion of Schöde s Equation and median stable distibutions is continued in the next section. The equation is used to deive popeties of H aound zeo and in the tails. Section 4 discusses connections between median stable distibutions and the emedian, an estimato defined ecusively as the median of medians; see [12]. Many well known featues of mean stable distibutions have diect extensions to the emedian. This is because the defining ecusive featue of the emedian is a popety of the sample mean (the mean of means is a mean), and many featues of mean stable distibutions deive fom this ecusive popety. The final section depats fom the median poblem with its G and takes up Schöde s Opposite Appoach in which G-functions ae deived fom known H distibutions. In paticula, the G functions coesponding the the Paeto and Laplace distibution ae deived.it thus eveals how the distibutions ae encapsulated in the associated G 3 Many esults ae summaized in [15] and [1]. Fo an histoical oveview of the ole of the functional equation in analyzing complex systems see, [8], and its ole in the histoy of composition opeatos see, [14]. The canonical poblem in the liteatue has a fixed point of (athe than ou fixed point of 1/2), and the known function coesponds to what is ou G 1 ( w); see [15], p

6 function. 2 Schöde s Equation An (H,λ) solution to (4) inheits featues of the G function and will be: inceasing, symmetic about, with H() = 1 2, convex (concave) fo < w < 1/2 (1/2 < w < 1). Futhe, an (H,λ) detemines a scale family of distibutions. If (H(x),λ) is a solution then so is, H(σx) = G(H(λ 1 σx)), σ >. In addition, an (H(x),λ ) fo a given λ detemines solutions fo all λ > 1. If (H(x),λ ) solves (4) then so does (H(x α ),λ α 1 ), α >. (Let H(x λ ) denote the solution with λ. Conside L(x) = H(x α λ ) so that, L(x) = G(H(λ 1 x α )) = G(H((λ 1/α x) α )) = G(L(λ 1/α x)), which says, L(x) with λ 1/α solves the functional equation). Since a solution fo a λ > 1 detemines solutions fo all λ we focus on the case, λ = g( 1 2 ) = s, the slope of G(w) at the fixed point, w = 1 2. This paticula case of the functional equation with λ = s is then: H(x) = G(H(s 1 x)). (8) With λ = s the density of H at x = is 1. (The deivative of (4) is: h(x) = g(h(λ 1 x)h(λ 1 x)λ 1, so h() = g( 1 2 )h()λ 1. If < h() <, then λ = g( 1 2 ) = s. Schöde Solutions. Repeated substitution of the LHS of (8) into the RHS, gives, H(x) = G (k) (H(s k x)) = G (k) ( h()s k x) + o(s k x)). Koenigs [9] showed that the solution to (8) (with h() = 1) is, H(x) = lim k G (k) ( s k x) (9) Fo an H(x) (with h() = 1) the associated G(w) with g( 1 2 ) = s is given by,4 G(w) = H(sH 1 (w)) (1) 4 Analogous to the standad nomal distibution, h() = 1 defines the standad cdf solution to the functional equation. 6

7 3 Popeties of Median Stable Distibutions Seveal featues of H follow fom (9) and (1). The fist concens the density h(x) aound zeo, which like the nomal is popotional to e x2. Theoem 1. h (x) = exp( α()x 2 h () 2 + o(x)) whee, α() = 4 s 2 1. Poof. The deivative of (8) (suppessing the -subscipt) is h(sx) = s 1 g(h(x))h(x), whee g(w) = s[1 4(w 1/2) 2 ], so: log(h(sx)) log(h(x)) = log(1 4(H(x) 1/2) 2 ). Dividing by x 2 and taking limits, x gives, fo the RHS: 4h() 2, while the LHS is: So, whee α() = 4 s 2 1. s lim 2 log(h(x)) log(h(sx)) = (s 2 log(h(x)) 1) lim. x x 2 s 2 x 2 x x 2 log(h(x)) lim = α()h() 2, x x 2 The nomal (with density equal to one at the oigin) has density, e πx2. The density of the median stable distibution (with density equal to one at the oigin) is e α()x2. With n = 3 ( = 1), the coefficient on x 2 is α(1) = 3.2 close to the nomal value of π. As, α() deceases to π. The next esult descibes the tail of H. The tail of H(x) depends on G(w) nea w = (o, nea w = 1). To motivate, conside the simplest case of = 1, n = 2 +1 = 3 whee G(w) = H(sH 1 ) = 3w 2 2w 3 o H(sH 1 ) 3w 2 o changing to x = H 1 and taking limits, = w and this means lim x H(sx)(3H(x)2 ) 1 = 1 lim x x α log3h(x) = 1 whee s α = 2. With s = 3 2, α 1.71, which compaes to the Nomal tail ate of α = 2. 7

8 Fo the geneal case with > 1 the leading tem of G (w) is (2+1)!!( 1)! w+1, and s = 4 (2+1)!!! so Theoem 2. whee, c = (2+1)!!( 1)!, and sα = + 1. lim x x α logch ( x) = 1 Poof. The poof is simila to the above = 1 case and is omitted. A final popety fo H follows fom the the well known nomal limiting distibution of the median: as n, the ( n = scaled) median given iid unifom data is nomal with mean zeo and standad deviation 1 2. In ou notation, = lim G ( x) = N( :, ). In tems of ou functional equation this means, (Veify noting lim G (N(s 1 x :, 2 1)) = N(x :, 1 2 ) lim n s 1 n n = π/2 and N((π/2) 1/2 x :,σ M ) = N(x :, 2 1 )). Hence as n = the median stable distibutions appoach the nomal. 4 Remedian Stable Distibutions Mean stable distibutions do not depend on the numbe of obsevations. The nomal, Cauchy, o any othe mean stable distibution is stable fo any n. The numbe of obsevations only detemines the appopiate λ n scaling facto; fo the nomal, λ n = n; fo the Cauchy, λ n = 1. Mean stable distibutions also have a domain-of-attaction featue concening the limiting distibution of the sample mean. Since mean stability holds fo all n, the limiting distibution of the sample mean with stable data will be the same stable distibution. But even with nonstable data the limiting distibution will be a stable distibution. The best known example is the nomal. With nomal data, 8

9 the limiting distibution of the sample mean is nomal. But with non-nomal data (and finite vaiance) the limiting distibution will be nomal. (If the vaiance is not finite, the limit will be a diffeent stable distibution). Mean stable distibutions not only make the sample mean epoduce the distibution of the data, but they ae also the limiting distibutions of the sample mean when data is not stable. Mean stable distibutions ae the same fo all n, and they ae the limit distibutions of the sample mean. These featues of mean stability ae a consequence of the ecusiveness of the sample mean; the mean of a collection of means is a mean. This ecusive featue does not wok fo the sample median-the median of a collection of medians is not a median. As a esult, and in contast to mean stability, median stable distibutions ae n dependent, and the domain of attaction popety does not hold fo the usual median. While the popeties do not hold fo the median, they do hold fo a diffeent estimato, the emedian. The emedian and mean shae the ecusive popety; the emedian is defined as the median of medians; see [12]. Its base- vesion-(with (2 + 1) k obsevations) is defined ecusively as the median of data, each of which is itself a median, ( ˆX 1 ˆX (k 1), ˆX 2 ˆX (k 1),..., ˆX (2+1) ˆX (k 1) ). Because of this ecusion, the sampling distibution is the k th iteate of the median G function; the sampling distibution of the base emedian with n = (2 + 1) k obsevations is, G (k) (F(x)), [12]. Conside the scaled-by s k -distibution of the emedian; G (k) (F(s k x)). If the data has median stable distibution, H, then the distibution of the emedian is the same as the data, G (k) (H (s k x)) = H (k). In othe wods, the base emedian has a emedian stable distibution that is the same as the median stable distibution H. Since the emedian, like the mean, is ecusive, this emedian stable distibution H is emedian stable fo all n = (2 + 1) k, k = 1,2,... Futhe, while the domain of attaction featue fails fo the median, it woks fo the emedian. Let the data have distibution, F(x), whee, f (x) > in a neighbohood of zeo. The distibution of the emedian is, G (k) (F(s k x)), o G (k) ( f (ωs k x)x), fo ω < 1, which as k is H ( f ()w), the median stable distibution with scale, 1/ f (). (Moe geneally, let F(x) = c(x)xα + o(x), whee c(x) > in a neighbohood 9

10 of. Then, as in the above, the limiting distibution of the emedian is, H (c()x α )). Thus, like mean stability, the limiting distibutions of the sample emedian ae the emedian stable distibutions. 5 Paeto and Laplace G functions This section pesents two examples of pesolved functional equations. It is eminiscent of Schöde s opposite appoach appoach in which, being unable to solve fo H in tems of G, he stats with an H and deives its G. The fist example is the Paeto distibution fo which the esulting G is piecewise linea. The second example has H as the Laplace distibution and deives its associated G. It is inteesting to see how the popeties of the Laplace and Paeto ae encapsulted in thei associated G functions. Paeto. Conside the polyhedal G function, G( w) = sw w < w < 1 2 = sw o + b(w w o ) w o w 1 2 whee s > 1, b = 1 2sw o 1 2w o, and G( 1 2 w) is given by symmety. The associated H(x) is a polyhedal linea in-between the kinks cdf whose value at the kink, x i = s i w o, is: H(s i w o ) = G (i) ( w o) = sw o + (s 1)w o b b i 1 b (Veify using (9), and H k (x) = G (k) ( s k x), whee H k (s i w o ) = G (i) ( w o) = b b sw o + (s 1)w i o 1 b ). Evaluating the density/slope of H(x) gives h(x) = ( b s )i fo, s i w o < x < s i+1 w o, a linea intepolation of the Paeto distibution; that is, h(x) = Ax β, β = 1 logb logs. Laplace. Let H(x) be the standad (h() = 1) Laplace distibution and H 1 (w) the associated invese/quantile function: H(x) = e 2x x > (11) H 1 (w) = 1 2 ln[2(1 w)] w > 1 2 1

11 The distibution fo x < is given by symmety, (H(x) = 1 H( x)), as is the invese fo w < 1 2 ; H 1 (w) + H 1 (1 w) = 1. Fo the associated G-function: G(w) = [2(1 w)λ ] 1 2 < w < 1 (12) = 2 λ 1 w λ < w < 2 1. The G functions fo the Paeto and Laplace distibutions ae compaed with each othe as well as the median G ( = 1,n = 3) in Figue 5. The fat tails of the Paeto ae eflected in the slowe appoach of its G function to and 1. The values nea and 1 fo the median G ae seen to fall between the the Laplace and Paeto thus indicating the tails of the median stable distibution fall between the Laplace and Paeto Median Laplace Paeto Figue 2: G functions Refeences [1] Bassett, G. W. (216). Median Stable Distibutions, in Robust Rank-based and Nonpaametic Methods, by Regina Liu and Joseph W. McKean. Spinge, (216), 11

12 Chapte 14: [2] Bassett, G.W., Koenke, R.: The Asymptotic Theoy of Least Absolute Eo Regession. J. Ame. Statist. Assoc. 73, (1978). [3] Caleson, Lennat; Gamelin, Theodoe W. (1993). Complex Dynamics. Textbook seies: Univesitext: Tacts in Mathematics. New Yok: Spinge-Velag. ISBN [4] Cutight, Thomas, and Cosmas Zachos (29). Evolution pofiles and functional equations. J. Phys. A: Math. Theo. 42.ANL-HEP-PR-9-8 (29). [5] David H. A. (1981). Ode Statistics. Wiley, New Yok (1981). [6] Felle, W. (1971). An Intoduction to Pobability Theoy and Its Applications. Wiley, New Yok (1971). [7] Koenke, R., Bassett, G.W. (1978) Regession Quantiles, Econometica. 46:33-5. [8] Gamelin, T. W. (1996). A Histoy of Complex Dynamics, fom Schde to Fatou and Julia. By Daniel S. Alexande. Wiesbaden (Vieweg). Histoia Mathematica 23.1 (1996): [9] Koenigs, G. (1884). Recheches su les intgales de cetaines quations fonctionelles. Annales Scientifiques de l cole Nomale Supieue 1 (3, Supplment): 341. [1] Kuczma, Maek (1968). Functional equations in a single vaiable. Monogafie Matematyczne. Waszawa: PWN Polish Scientific Publishes. ASIN: B6BTAC2 [11] Kuczma, Maek (1964). Note on Schde s functional equation. Jounal of the Austalian Mathematical Society 4.2 (1964): [12] Rousseeuw, P.J., Bassett, G.W. (199). The Remedian: A Robust Aveaging Method fo Lage Data Sets. J. Ame. Statist. Assoc. 85, (199). 12

13 [13] Schde, Enst (187). Uebe iteite Functionen. Math. Ann. 3 (2): doi:1.17/bf [14] Shapio, Joel H. (1998). Composition opeatos and Schde s functional equation. Contempoay Mathematics 213 (1998): [15] Szekees, G. (1958). Regula iteation of eal and complex functions. Acta Mathematica 1 (34): doi:1.17/bf [1] 13