A GENERALIZED ERROR DISTRIBUTION

Transcription

1 A GENERALIZED ERROR DISTRIBUTION GRAHAM L. GILLER ABSTRACT. W rviw th proprtis of a univariat probability distribution that is a possibl candidat for th dscription of financial markt pric changs. This distribution is an rror distribution that rprsnts a gnralizd form of th Normal, posssss a natural multivariat form, has a paramtric kurtosis that is unboundd abov and posssss spcial cass that ar idntical to th Normal and th doubl xponntial (Laplac) distributions.. THE UNIVARIATE GENERALIZED ERROR DISTRIBUTION.. Dfinition. Th Gnralizd Error Distribution is a symmtrical unimodal mmbr of th xponntial family. Th domain of th p.d.f. is x [, ] and th distribution is dfind by thr paramtrs: µ (, ), which locats th mod of th distribution; σ (, ), which dfins th disprsion of th distribution; and, (, ), which controls th skwnss. W will us th notation x G(µ, σ, ) to dfin x as a variat drawn from this distribution. (A suitabl rfrnc for this distribution is [].) Th probability distribution function, F (x), is givn by () df (x µ, σ, ) = x µ σ + σγ( + ) dx. This function is rprsntd in Figur, on th following pag. It is clar from this dfinition that th mod of th p.d.f. is µ and that it is unimodal and symmtrical about th mod. Thrfor th mdian and th man ar also qual to µ. If w choos = thn Equation is rcognizd as th p.d.f. for th univariat Normal Distribution, i.. G(µ, σ, ) = N(µ, σ ). If w choos = thn Equation is rcognizd as th p.d.f. for th Doubl Exponntial, or Laplac, distribution, i.. G(µ, σ, ) = L(µ, 4σ ). In th limit th p.d.f. tnds to th uniform distribution U(µ σ, µ + σ)... Th Cntral Momnts. Th cntral momnts ar dfind by Equation. () µ r = E(x µ) r = + σγ( + ) x= (x µ) r x µ σ Th odd momnts clarly all vanish by symmtry. For th vn momnts, Equation may b writtn as Equation 3, in which w rcogniz that th intgral is a rprsntation of th gamma function. (3) µ r = r σ r Γ() t (r+) t dt = r r Γ{(r + )} σ Γ() Dat: August 6, 5. Gillr Invstmnts Rsarch Not: 3/. Somtims just calld Th Error Distribution k dx.

2 GRAHAM L. GILLER df(x,,).4 = /4 = / = =. - x FIGURE. Th Gnralizd Error Probability Dnsity Function Thrfor, th distribution has th paramtrs: (4) (5) (6) (7) man = µ; varianc = σ Γ() ; skw, β = ; and, kurtosis, β = Γ(5)Γ() Γ. (3) For < th distribution is platykurtotic and for > it is lptokurtotic. Th xcss kurtosis, γ, is tnds to 6/5 as and is unboundd for >. Th lptokurtotic rgion is illustratd in Figur, on th nxt pag. W may us Stirling s formula for Γ(z) to obtain th following approximation for th kurtosis: (8) γ () 3 ( ) A Standardizd Gnralizd Error Distribution. It is oftn convnint to work with th p.d.f. which is standardizd. By this it is mant that that population man is zro and th population varianc is unity. W s from Equation 5 that th varianc of th G.E.D. p.d.f., as dfind in Equation, is a vry strong function of. Howvr, it is trivial to rscal th varianc to transform Equation into an quivalnt p.d.f. with constant varianc σ. Lt us introduc th scaling paramtr ξ and mak th For this rason, som authors writ c/ for, paramtrising th Normal distribution as c =.

3 A GENERALIZED ERROR DISTRIBUTION 3 γ () FIGURE. Excss Kurtosis Masur γ for > substitution σ σξ in Equation. Th normalizd p.d.f. now has th form (9) df (x µ, σ, ; ξ) = This p.d.f. has th varianc () σ ξ Γ(). ξ x µ σ + σξ Γ( + ) dx. If w choos ξ to liminat all dpndnc of th varianc on, thn w may dfin a homoskdastic p.d.f. as () df H (x µ, σ, ) = { } { Γ() ( x µ Γ() σγ( + ) σ ) } A standardizd p.d.f. is thrfor trivially givn by df S (x ) = df H (x,, ). With this formulation th xtrmly rapid incras in kurtosis, as is incrasd from th Normal rfrnc valu of, is clarly dmonstratd in Figur 3, on th following pag. dx.. A MULTIVARIATE GENERALIZATION.. Construction of a Multivariat Distribution. Th p.d.f. of Equation is of th form suitabl for th construction of a multivariat p.d.f. using th rcip of rfrnc []. This

4 4 GRAHAM L. GILLER f S (x ).5.5 = = = / = /4.5 - x FIGURE 3. Univariat Standardizd Gnralizd Error Distribution procdur is applid to th standardizd univariat p.d.f., f(x ), dfind for our distribution as () f(x ) = df { } S(x ) = Γ() x }. dx Γ( + ) Γ() Rplacing x in Equation by th Mahanalobis distanc Σ (x, µ), givs (3) df (x µ, Σ, ) = A dn x Γ( + ) { } { } xp Γ() Γ() (x µ)t Σ (x µ). Th constant A is introducd to maintain th normalization of th nw function. It is givn by (4) A = πn Σ Γ( + )Γ( n ) = π n Γ(n) Σ Γ()Γ( n ) { Γ() } { Γ() } n Γ() } g k g n dg. Substituting this rsult into Equation 3 givs (5) df (x µ, Σ, ) = dn x Γ( + n ) { } n { } xp πn Σ Γ( + n) Γ() Γ() (x µ)t Σ (x µ).

5 A GENERALIZED ERROR DISTRIBUTION 5 V/Σ 4 3 n = 5 n = n = FIGURE 4. Varianc Scal Factor for Constructd Multivariat Distributions.. Momnts of th Constructd Distribution. Using rsults of rfrnc [], w s that th p.d.f. of Equation 3 is unimodal with mod µ. This is also qual to th man of th distribution. Th covarianc matrix, V, is qual to th matrix Σ multiplid by th scal factor (6) n Γ() } g g n+ dg Γ() } g g n dg = Γ{(n + )}Γ( + ). Γ( + n) 3 Not that in th limit this bcoms n+. Th strong dpndnc of this factor on, for svral valus of n, is shown in Figur 4, abov. Th skw of th distribution is zro by construction (β,n = ) and th multivariat kurtosis paramtr is (7) β,n = n Γ() } g g n+3 dg [ Γ() } g g n+ dg Γ{(n + 4)}Γ(n) = n Γ. {(n + )} Γ() } g ] g n dg Th lptokurtotic rgion is illustratd in Figur 5, on th following pag.

6 6 GRAHAM L. GILLER γ,n () 3 n = n = n = FIGURE 5. Excss Kurtosis Masur γ,n for >.3. Th Multivariat Kolmogorov Tst Statistic. Lt {G i }N i= rprsnt an ordrd st of sampl valus of Σ (x, µ). From rfrnc [], w know that G (8) Pr(g < G ) = F (G g= ) = f(g ) g n dg g= f(g ) g n dg. Substituting our xprssion for f( ), Equation, givs [ { } ] γ n, (9) F (G Γ() G ) =, Γ(n) whr γ( ) is th lowr incomplt gamma function[3]. W may us th Kolmogorov statistic () d N = max S i F (G i ), i whr {S i } N i= ar th ordr statistics associatd with th sampl, to tst th null hypothsis that a givn datast is rprsntd by Equation MAXIMUM LIKELIHOOD REGRESSION Givn a st of N i.i.d. random vctors, {X i } N i=, ach drawn from th Gnralizd Error Distribution, th joint probability, or liklihood, of a particular ralization, {x i } N i=,

7 A GENERALIZED ERROR DISTRIBUTION 7 is givn by () L(µ, Σ, ) = i= N df (x i µ, Σ, ). Th commonly usd liklihood function, L = ln L, is thrfor givn by N { } L(µ, Σ, ) = Γ() (x i µ) T Σ N (x i µ) + ln Σ () + Nn i= i= πγ() Γ( + n) ln + N ln Γ( + n ). W may also writ this xprssion in trms of th covarianc matrix, V, as blow. N [ ] Γ{(n + ))} L(µ, V, ) = (x i µ) T V N (x i µ) + ln V Γ( + n) (3) + Nn πγ( + n) Γ( + n) ln + N ln Γ{(n + )} Γ( + n ). REFERENCES [] Evans, M., Hastings, N., & Pacock, B., Statistical Distributions, 3rd. Edn., pp , John Wily & Sons, Inc.,. [] Gillr, G.L., On th Construction, Intgration and Idntification of Multivariat Probability Dnsity Functions, p. 8, Gillr Invstmnts Rsarch Not 33/, [3] Gradshtyn, I.S. & Ryzhik, I.M., Tabl of Intgrals, Sris, and Products, 6th. Edn., p. 89, Acadmic Prss,.