Generalized Likelihood Functions and Random Measures

Transcription

1 Pure Mathematical Sciece, Vol. 3, 2014, o. 2, HIKARI Ltd, Geeralized Likelihood Fuctio ad Radom Meaure Chrito E. Koutzaki Departmet of Mathematic Uiverity of the Aegea Karlovai GR , Samo, Greece Copyright c 2014 Chrito E. Koutzaki. Thi i a ope acce article ditributed uder the Creative Commo Attributio Licee, which permit uretricted ue, ditributio, ad reproductio i ay medium, provided the origial work i properly cited. Abtract The deity etimator we cotruct i thi paper, called geeralized likelihood, rely o pecific radom meaure. The key idea i to approximate a ukow deity fuctio -give a ample of it obervatio through aother fuctio which i made by aother family of kow deity fuctio. Uder thi poit of view, we coclude that uder ome aumptio the clae of ditributio, which cotitute the maximum domai of attractio ca be viewed to be two itead of three ad we dicu the quetio of differece betwee the clae of equivalece of probability ditributio. Mathematic Claificatio: 62K05; 60GG57; 62K20 Keyword: Radom meaure; Deity etimatio 1 Itroductio May method for the etimatio of a ukow deity fuctio f by mea of fuctio of i.i.d radom variable X 1, X 2,..., X have bee propoed i recet year. Thee iclude the kerel method tudied i [4], the characteritic fuctio approach tudied i [5] ad the propertie for Lévy Radom meaure tudied by [3] have how that i certai cae thee etimate are bet poible. I [4], the problem of etimatio of a probability deity fuctio wa examied ad the problem of determiig the mode of a probability deity

2 88 Chrito E. Koutzaki fuctio, a well. The cotructio of a family of etimate of f(x) wa idicated, ad of the mode, which are coitet ad aymptotically ormal. The problem of etimatig the mode of a probability deity fuctio i omewhat imilar to the problem of maximum likelihood etimatio of a parameter. Alo, a radom meaure i aid to be a ubmeaure of a ecod radom meaure if it probability law i abolutely cotiuou with repect to that of the ecod. I [3], it wa proved that if the ecod meaure i a Lévy radom meaure the the ubmeaure i Lévy if ad oly if the Rado-Nikodym derivative atifie a atural factorizatio coditio. I thi paper, we actually how that the Weibull ad Fréchet ditributio familie do ad do ot belog repectively, to the equivalece cla of deitie defied by the kerel of Gumbel ditributio. For thi we ue a radom meaure i accordace with the defiitio give i [3]. Thee familie are importat, ice they are the Maximum Domai of Attractio, but we alo dicu other ditributio familie, too. 2 Radom Meaure ad Kerel Deitie We coider a probability pace (Ω, F, µ) ad a meaurable pace (E, E). A radom meaure i [3] ξ o (E, E) over the probability pace (Ω, F, µ) i a map ξ : E Ω R +, uch that: 1. the map ω ξ(a, ω) i a radom variable for ay A E 2. the map A ξ(a, ω) i a meaure o E, µ- almot urely i Ω. I the ext, the meaurable pace (Ω, F) ad (E, E) are idetified to (R, B R ), while µ = λ, where λ deote the Lebegue meaure o R. I [5] ad [4] a fuctio K : R R i called kerel if i atifie the followig propertie - i the ext called kerel coditio: (i) up{ K(x), x R} <, K(x) dx <, R (ii) lim xk(x) = 0, K(x)dx = 1, x R 1 (iii) R xi K(x)dx = 0, i = 1, 2,..., m 1, R x m p K(x) dx <, m N, p R ++.

3 Geeralized likelihood fuctio ad radom meaure 89 The Sobolev pace W p (m) (M), where M > 0, p > 0 ad m N i a fuctio pace over the domai R for example which coit of the fuctio i abolutely cotiuou for {f : R R f (k), k = 1, 2,..., m 1, f (m) L p (R), f (m) p M}. A fuctio f : I R, where I i a iterval of R i abolutely cotiuou o I if for ay ε > 0, there i a δ > 0 uch that wheever a fiite family of ub-iterval (x k, y k ), k = 1, 2,..., m of I exit uch that m (y i x i ) < δ, the m (f(y i ) f(x i )) < ε hold. A fuctio f i abolutely cotiuou o R if it i abolutely cotiuou o every iterval of R. A criterio for the abolute cotiuity i the exitece of the firt derivative f λ-almot everywhere ad that f i Lebegue itegrable. The etimator of the deity f W p (m) (M) which relie o the ordered ample t 1, t 2,..., t of it ad alo o the kerel K i equal to ˆf (x) = 1 K( x t i ). > 0 i elected o that 0,. For example = 1, N. If we uppoe that m = 2, p = 1, K(x) 0, x R, the the above coditio become up{k(x), x R} <, lim x K(x) = 0, K(x)dx = 1, x R xk(x)dx = 0. R Accordig to [5, Theorem 4.1], R (f(x) ˆf (x)) 2 dx 0,

4 90 Chrito E. Koutzaki while ad for m = 2, p = 1 the above itegral i le tha c > 0, if f W (2) 1. c 2 3, where We may ue other deity fuctio which atify the previou coditio of the kerel i order to approximate aother ukow deity fuctio, by takig obervatio of it. We coider a radom meaure uch that: ξ : B R Y R +, 1. the map y ξ(a, y) i a radom variable for ay A B R 2. the map A ξ(a, y) i a meaure o E, λ- almot urely i Y, 3. λ i the Lebegue meaure retricted o the ope et Y R. Hece by thi way for a igle-parametric family of deitie atifyig the kerel coditio, whoe parametric pace i the et Y, we may defie the radom meaure ξ K whoe the meaure -part i ξ K (y)(x) = K y (x), x R, y Y. If we take a igle-parametric family of deity kerel whoe parameter take the value y or ele if we take ay meaure-value of the radom meaure ξ K o Y, the the covergece reult [5, Theorem 4.1], take the form: Theorem 2.1 If f(x) i a ukow deity i W (2) 1 ad t 1, t 2,..., t i a ordered ample from it, the R (f(x) ˆf (x)) 2 dx 0, ad if m = 2, p = 1 the above itegral i le tha c 1, where c 2 1 > 0 ad 3 ˆf (x) = 1 ). > 0 i elected o that 0, if y Y ad every K y atifie the kerel coditio. Proof: The proof arie from [5, Theorem 4.1]. Defiitio 2.2 ˆf (x) = 1 ) i called geeralized likelihood fuctio for f(x).

5 Geeralized likelihood fuctio ad radom meaure 91 3 Clae of Equivalece for Deitie The cloure of the cla K Y = {K y, y Y, Y R}, i F K = {f(x) W 1,+ (2) (f(x) ˆf Ky,) 2 0, + }, for ome ordered ample t 1, t 2,...t,... from f(x) ad for ome equece ( ) N, uch that > 0 ad 0, while +. We may defie a biary relatio betwee the deitie i W (2) 1 accordig to the familie of kerel, which atify the kerel coditio ad approximate them. Thi relatio i defied a follow: f g f, g F K. Theorem 3.1 The relatio i a equivalece relatio, amely reflexive, ymmetric ad traitive. Proof: (i) If f F K, the f, f F K, which implie f f (Reflexivity). (ii) If f, g F K, amely f g, we otice that g, f F K, hece g f (Symmetry). (iii) If f, g F K, amely f g, while g, z F K, amely g z, we deduce that f, z F K, amely f z. Theorem 3.2 The relatio divide the et of the deitie for which f W (2) 1 ito dijoit clae of equivalece. Proof: The proof of the Theorem relie o the fact that if a ukow deity f W (2) 1 belog to a pair of differet clae F K, F L, for the ame ε > 0 we obtai: η ) f(x)) 2 dx < ε2 8, 1(ε), L y ( x τ i ) f(x)) 2 dx < ε2 η 8, 2(ε), where L i aother kerel, τ i, i = 1, 2,..., are ample poit from the ukow deity, too (η ) N i a equece of real umber uch that η > 0 ad η 0, while η +. We coider 0 (ε) = max{ 1 (ε), 2 (ε)}. For thi 0 (ε) ad by the Mikowki Iequality, we obtai ) 1 η L y ( x τ i )) 2 < ε2 η 4, = 0(ε).

6 92 Chrito E. Koutzaki The lat iequality implie that 1 ) = 1 η L y ( x τ i η ), = 0 (ε), x R, a cotradictio. Theorem 3.3 The family of ormalized Gumbel ditributio ca be ued a kerel for the approximatio of a Weibull ditributio accordig to Theorem 1. Proof: Thi arie from the fact that a ormalized Gumbel deity f ( R xf(x)dx = 0) atifie the kerel coditio ad the abtract Weibull deity h W (2) 1. The fact that the ormalized Gumbel deitie K(x) = 1 x m exp( z exp( z)), z =, b > 0, m R, x R, beig actually the oe b b with R xk(x)dx = 0, R x2 K(x)dx = 1, atify the ret kerel coditio up{k(x), x R} <, lim x x K(x) = 0, R K(x)dx = 1, i verified a follow.: The third coditio i verified by the fact that K(x) i a deity fuctio. The firt property i verified by the fuctioal form of K(x) ad pecifically from the iequality K(x) exp( exp( z)). The ecod oe i verified by the iequality x K(x) e z 1 b e z exp( exp( z)) = 1exp( exp( z)). b Corollary 3.4 There i at leat oe o-heavy tailed ditributio deity approximated by heavy-tailed ditributio i the ee of Theorem 1. Proof: Normalized Gumbel ditributio belog to the cla of heavytailed ditributio. O the other had, the Weibull ditributio deity f(x) = k( x a a )k 1 exp ( x a )k, x 0, for k = 1 i equal to a expoetial ditributio deity, which i the deity of a o-heavy tailed ditributio. About clae of heavy-tailed ditributio, ee for example i [2]. Theorem 3.5 The family of Fréchet ditributio caot be approximated by ormalized Gumbel ditributio accordig to Theorem 1. Thi hold becaue if k(x) i a Fréchet deity, thi doe ot belog to the Sobolev pace W (2) 1. Proof: The Fréchet ditributio deity i: f(x) = a( x m x > m, > 0. The firt derivatio of f(x) i: f (x) = ( a( m+x exp ( m x ) a 1 ) a ) = a(am+a ax+)exp ( m x (m x) 2 ) a ( x m ) a m x ) f a(am + a ax + )exp( a ( x m ) a (x) =. (m x) 2 The ecod derivatio of f(x) i: ) a 1 x m ( ) exp a,

7 Geeralized likelihood fuctio ad radom meaure 93 f (x) = a(a2 (m + x) 2 + a(2m + 3 2x) )exp ( ) a ( x m ) a. 2 (m x) 3 m For x > m ad > 0 : a(a2 (m + x) 2 + a(2m + 3 2x) )exp ( ) a ( x m ) a dx =. 2 (m x) 3 The Fréchet ditributio deity i ot abolutely cotiuou o R. Fréchet deity doe ot belog to the Sobolev pace W (2) Defiitio 3.6 If two ditributio familie K, L, are called differet if there i at leat oe member(deity) f L, uch that f / F K hold. Corollary 3.7 (Normalized) Gumbel family i differet from Fréchet family. Proof: See the Proof of the previou Theorem. We may provide ome more Example of differece of ditributio familie. Theorem 3.8 Expoetial family i differet from the Gauia family of ditributio. m x Proof: Let u coider the family of caoical kerel where σ > 0. The kerel coditio K σ (x) = 1 2πσ exp{ 1 2σ 2 x2 }, up{k σ (x), x R} <, lim x K σ (x) = 0, x R K σ(x)dx = 1, R xk σ(x)dx = 0 hold for the caoical kerel, hece we may examie if the expoetial ditributio f(x) = u exp{ ux}h 0 (x), where H 0 (x) i the Heaviide tep fuctio, relative to 0 ad u > 0. We otice that f W (2) 1. We coider the ordered ample poit t 1, t 2,..., t from the expoetial ditributio, where t 1 = mi{t i i = 1, 2,..., } > 0. We are goig to how that the followig o-covergece reult i true: 1 σ 2π exp{ (x t i) 2 } f(x)) 2 dx +, h 2σ 2 h 2 = 1, m x So

8 94 Chrito E. Koutzaki The above itegral i greater or equal tha the followig um of itegral: 1 ( ( 0 2πσ 2 exp{ x2 σ }dx = 1 σ π 2 2πσ 2 2 = σ π =. Hece, the family of expoetial ditributio i differet tha the oe of the caoical oe. We otice that both the caoical ad the expoetial ditributio are ot heavy -tailed. Theorem 3.9 The family of Pareto -type ditributio i differet from the Gauia family of ditributio. Let u coider the family of caoical kerel where σ > 0. The kerel coditio K σ (x) = 1 2πσ exp{ 1 2σ 2 x2 }, up{k σ (x), x R} <, lim x K σ (x) = 0, x R K σ(x)dx = 1, R xk σ(x)dx = 0 hold for the caoical kerel, hece we may examie if the Pareto-type ditributio g(x) = 2 H x 3 1 (x), where H 1 (x) i the Heaviide tep fuctio, relative to 1. We otice that g W (2) 1. We coider the ordered ample poit t 1, t 2,..., t from the expoetial ditributio, where t 1 = mi{t i i = 1, 2,..., } > 0. We are goig to how that the followig o-covergece reult i true: 1 σ 2π exp{ (x t i) 2 } g(x)) 2 dx +, h 2σ 2 h 2 = 1, +. The above itegral i greater or equal tha the followig um of itegral: 4 ( ( 1 σ 2πx 3 exp{ t ix 2 } dx σ 2 2 σ 2πexp{ 1 } = +. σ 2 We alo otice that thi Pareto -type ditributio i a heavy-tailed oe, while a Gauia ditributio i ot a heavy -tailed ditributio.

9 Geeralized likelihood fuctio ad radom meaure 95 Referece [1] Aliprati C.D., Border K.C. (1999). Ifiite Dimeioal Aalyi, A Hitchhiker Guide, (ecod editio). Spriger. [2] Cai, J., Tag, Q. (2004). O max-um equivalece ad covolutio cloure of heavy-tailed ditributio ad their applicatio. Joural of Applied Probability, [3] Karr A. (1978). Lévy Radom Meaure, Aal of Probability 6, [4] Parze E. (1962). O Etimatio of a Probability Deity ad Mode. Aal of Mathematical Statitic 33, [5] Wahba G. (1975). Optimal Covergece Propertie of Variable Kot, Kerel, ad Orthogoal Serie Method for Deity Etimatio. Aal of Statitic 3, Received: March 18, 2014