AN EFFICIENT ESTIMATION METHOD FOR THE PARETO DISTRIBUTION

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1 Joural of Statistics: Advaces i Theory ad Applicatios Volue 3, Nuber, 00, Pages 6-78 AN EFFICIENT ESTIMATION METHOD FOR THE PARETO DISTRIBUTION Departet of Matheatics Brock Uiversity St. Catharies, Otario Caada LS 3A Caada e-ail: huag@brocku.ca Abstract This paper studies estiatio of the Pareto distributio. We use a weighted epirical distributio fuctio (WEDF) for estiatig the populatio distributio fuctio of the Pareto distributio. The exact efficiecy fuctio of this estiator relative to classical epirical distributio fuctio (EDF) is derived. Properties of this efficiecy fuctio are studied. Mote Carlo siulatio results cofir the theoretical coclusios, which idicate that the WEDF is ore efficiet tha the EDF i ay situatios. Applicatio to kerel ad paraetric desity estiatio is also give.. Itroductio The Pareto distributio is a power-tailed distributio. It has ay applicatios i ecooics, actuarial sciece, survival aalysis, queueig etworks, ad other stochastic odels. It is iportat to explore estiatio ethods for this distributio. 00 Matheatics Subject Classificatio: Priary 6G30; Secodary 6G05. Keywords ad phrases: efficiecy, order statistics, weighted epirical distributio fuctio. Research of author is supported by a NSERC Caada grat. Received March 8, Scietific Advaces Publishers

2 6 Several types of Pareto distributio have bee defied (Kleiber ad Kotz [4]). I this paper, we oly discuss the oe-paraeter Pareto type II. Its desity fuctio (d.f.) is defied as α f ( x) = ( ) ( ), x 0, α > 0, (.) α+ + x ad the cuulative distributio fuctio (c.d.f.) is F ( x) =, x 0, α > 0. (.) α ( + x) We wat to estiate the c.d.f. F ( x) i (.) by usig a oparaetric ethod, fro a rado saple X, X,, X, 3. I geeral, the epirical distributio fuctio (EDF) S ( x) is a iiu variace ubiased estiator for F ( x), where ( ) ( ]( ), if x A; S x = I, x Xi, where I A = (.3) 0, if x A. Note that S ( x) uses the equal weight for each saple poit. There have bee studies of puttig uequal weights o the data poits, to obtai weighted epirical distributio fuctios or processes (Shorack ad Weller [7], Koul [5], Barbe ad Bertail []). But, there are soe difficulties to deterie what weights should be used for the data poits. Huag [] studied a geeral weighted distributio fuctio (WEDF), aely, ( ), F ( x) = I(, x] X() i p, i (.4) where the p, s are syetric geeral weights. i p, i w, w, ( ( ) w), i i =,,, 0 =,, < w <, ad X() i X( ) X ( ) are the order statistics of the rado saple. Note that

3 AN EFFICIENT ESTIMATION METHOD 63 0 < p, i <, i =,, ; p, i =. I this paper, we use WEDF F ( x) i (.4) to estiate the Pareto distributio c.d.f. F ( x) i (.). Sectio derives a exact efficiecy fuctio of the WEDF F ( x) i (.4) relative to the EDF S ( x ) i (.3), for estiatig the c.d.f. F ( x) i (.). Sectio 3 studies the properties of this exact efficiecy fuctio. Based o these properties, Sectio 4 gives several priciples for deteriig what weights should be used i the F ( x) for obtaiig a efficiet estiator. Sectio 5 discusses applicatios of the F ( x) to kerel ad paraetric desity estiatio. Mote Carlo siulatio results cofir the theoretical coclusios, which idicate that the weighted estiators are ore efficiet tha the classical estiators i ay situatios.. A Exact Efficiecy Fuctio The ea squares error (MSE) fuctio of the WEDF F ( x) i (.4) for estiatig ay populatio c.d.f. F ( x), x R, relative to the EDF S ( x) i (.3) is give by Theore. i Huag [], aely, ( F MSE ) = w F ( F ) + ( w) [( F ) ( F ) F ] 4 ( w)[ F ( F ) + F ( F )], 0. + F (.) Here, for siplificatio, we write F ( x) F, 0 F. I this sectio, we use (.) to derive the MSE fuctio of the WEDF F ( x) for estiatig the Pareto populatio c.d.f. F ( x ) i (.). The result is the followig. Theore.. The MSE fuctio of the WEDF F ( x) i (.4) for estiatig the populatio Pareto c.d.f. F ( x), x 0, α > 0, i (.) is give by

4 64 ( ( )) MSE F x = w ( ) α + x ( + x) α + 4 ( w) ( ) ( ) ( ) α α α + x + x + x + ( w), + α α α α ( ) ( ) ( ) ( ) + x + x + x + x (.) The, we defie a exact efficiecy fuctio of F ( x) relative to the EDF S ( x) for estiatig the populatio Pareto c.d.f. F ( x) i (.) by ( ( )) ( ) ( ) ( Var S α α EFF x ( )) + x + x F x = =, x 0, (.3) MSE ( F ( x)) MSE ( F ( x)) where MSE ( F ( x)) is give i (.). 3. Properties of the Exact Efficiecy Fuctio I this sectio, properties of the exact efficiecy fuctio EFF( F ( x)) i (.3) are explored. We ay write the above EFF( F ( x)) as a fuctio F of x, x 0, EFF( ( x)) EFF ( x). We ca prove that Theore 3.. (a) For 3, 0 < w <, >, if w > ; li EFF ( x) = li EFF ( x) = =, if w = ; + x 0 x ( w, ) <, if w <. (3.) (b) The edia of F ( x) is = a x edia. If 3, 0 < w <, the value of EFF ( x edia ) is give by

5 AN EFFICIENT ESTIMATION METHOD 65 a EFF ( ) = [ w + ( w )] <, if =, if >, if w > ; w = ; w <. (3.) (c) For 3, EFF ( x) = has exactly two real roots: z i the ope a iterval ( 0, ), ad z i the ope iterval ( a, ). (d) For 3, whe w >, Whe w <, > o ( 0, z ) ad ( z, ); EFF ( x ) (3.3) < o ( z, z ). < o ( 0, z ) ad ( z, ); EFF ( x ) (3.4) > o ( z, z ). 4. Choice of Weights 4.. Efficiecy exceeds o the tails (, z ) ad ( z ) 0 The efficiecy EFF(x) i (.3) is a cotiuous fuctio of x. Thus, we ca choose the id-weights w to be greater tha (i.e., both the first ad last weights w, to be less tha ). The based o Theore 3.(a), (c), ad (d), the EFF(x) ust exceed o the tails (, z ) ad, 0 ( z, ) of the values of x, where z ad z are the two real roots of EFF ( x ) =. For exaple, i Huag ad Brill [3], a level crossig weighted epirical distributio fuctio (LCEDF) was proposed, which leads to a optial choice of the weights w ad w. It results i ore weight give, to the id-data. The LCEDF estiator F ( x) gives weights o the id-data. It chooses p, i w = >, i =,,, ( )

6 66 p, = p, = w, = < ( ). Therefore, the EFF exceeds o the tails ( 0, z ) ad ( z, ) of the x values, where z ad z are the two real roots of EFF ( x ) =. 4.. Efficiecy exceeds o the iddle ( z z ), Suppose, we choose the id-weights w to be saller tha both the first ad last weights (i.e., w, are greater tha ). The based o Theore 3.(a), (c), ad (d), the EFF ust be less tha o the tails ( 0, z ) ad ( z, ) of the x values, where z ad z are the two real roots of EFF ( x ) =. For exaple, suppose we choose o the id-data. The p, i w = <, i =,,, ( + ) p, = p, = w, = > ( ) +. Therefore, the EFF exceeds o the iddle ( z, z ) of the x values, where z ad z are the two real roots of EFF ( x ) =. For these two exaples, weights w = > ad w = ( ) < have bee used, respectively. I Appedix A, Tables,, ( + ) 3, 4, ad Figures,, 3, 4 show the exact efficiecy fuctios EFF(x) of F ( x) relative to the ordiary epirical distributio fuctio S ( x ), for x =,, 4, 6, 8, 0, ad = 3, 5, 0, 0, 30, 40, 50, 60, 70, 80, 90, 00, α = ad 3 cases. Also, the roots z ad z of EFF(x) = are give. The results of these two exaples are cosistet with Theore 3..

7 4.3. Geeral suggestio AN EFFICIENT ESTIMATION METHOD 67 I this paper, our geeral suggestio is as follows. If we are ore iterested i the tails of the distributio, the we use w >, to gai efficiecy o the tails of the SWEDF. If we are iterested o the iddle of the distributio, the we use w <, to gai efficiecy o the iddle of the SWEDF. Of course, it is iterestig to study other choices of the weights. 5. Applicatios to Desity Estiatio ad Siulatios 5.. Kerel desity estiators We apply the F ( x) to a soothig techique. It is well kow that the classical kerel desity estiator for true desity fuctio (d.f.) f is x Xi fk, ( x) = K, x R, (5.) h h where K () is a syetric desity fuctio, ad h > 0 is a badwidth. We defie a kerel desity estiator based o the WEDF F ( x) as follows. Defiitio 5.. The weighted kerel desity estiator (WKDE) is defied by x X () i f * ( x) =,,,,, p ik x R (5.) K h h where p, i is defied i (.4), X() i, i,, = are the order statistics. 5.. Paraetric desity estiators I this sectio, we obtai a weighted paraetric desity estiator f w ( x ) based o the F ( x). By the d.f. i (.), the populatio ea is µ =, α >. (5.3) α

8 68 The, we have the followig oet, axiu likelihood, ad weighted estiators for the shape paraeter α, X α X =, X = Xi, X is the saple ea; (5.4) α MLE log( ), = = Xi + (5.5) i X w α w =, X w (5.6) where Xw = p i, Xi, is the weighted saple ea, ad p, i is give i (.4). Based o the above three estiators for α, we have the followig desity estiators. Defiitio 5.. Three desity estiators for the populatio d.f. f(x) are defied by α f ( ) ( ) ( X X x = ), x > 0,, (5.7) α + + x X α MLE fmle ( x) = ( )(, x > 0,, MLE + x α + ) α w fw ( x) = ( ) (, x > 0,, + x α w + ) (5.8) (5.9) where α, α MLE, ad α X w are defied i (5.4), (5.5), ad (5.6), respectively Siulatios We ra siulatios o estiatig two Pareto distributios with paraeters α = ad 3. We used the estiators F ( x), S ( x), f, ( x), K f ( x), f ( ), f ( x ) ad f ( x ) defied i (.4), (.3), (5.), (5.), K, X x MLE, w

9 AN EFFICIENT ESTIMATION METHOD 69 (5.7), ad (5.8), (5.9), respectively. We geerated =,000,000 saples for each case usig saple sizes of 0, 0, 30, ad 50. The siulatio ea square errors are defied by SMSE ( F ( x) ) = F ()( i x) F ( x) ( ), ( ), SMSE ( S ( x)) = S()( i x) F ( x) ( ( ) ) ( () ( ) ( ) ) * SMSE fk, x = fki *, x f x, i = SMSE ( f ( )) ( () ( ) ( ) ) K, x = fki, x f x, i = ( ), SMSE ( f X ( x) ) = f () ( x ) f ( x X i ) SMSE ( f ( )) ( ()( ) ( )) MLE x = fmle i x f x, i = ( ) ( SMSE f ( )) w x = fwi ()( x) f( x), i = where F ()( x ), S ()( x ), f ()( x ), f ()( x ), f i i i i X () i ( x), f MLE () i ( x ), ad f wi () ( x ) are estiated values fro the i-th siulatio, i =,,,. f ( x) ad F ( x) are give i (.) ad (.). by The siulatio efficiecy of F ( x) relative to the EDF S ( x ) is give SMSE ( S ) SEFF ( F ( x) ) =. (5.0) SMSE F ( x) ( )

10 70 The siulatio efficiecy of f ( x) relative to f ( x) is give by SMSE ( f, ( x)) SEFF ( f K K, ( x)) = SMSE ( f ( x)). (5.) K, The siulatio results are give i Table 5 ad Figure 5, Appedix B for the choice of weight w = ( ) oly. We calculate the siulatio efficiecies SEFF ( F ( x)) ad SEFF ( f ( x)) by usig (5.3) ad (5.4) at selected values x =,, 3, 4, 5, 6, 8, 0, for α = ad 3 cases. We use a stadard oral kerel ad optial badwidth (Silvera [6], p. 45) h 4 / 5 / 5 = The uerical siulatio results i Tables 5 ad Figures 5 show that: (a) The siulatio efficiecies of the estiators WEDF F ( x) ad the WKDE f ( x) obtaied fro,000,000 rus by usig (5.3) ad (5.4), for K, each of 44 cases are very close to the exact efficiecies give i Tables ad, respectively, which are directly coputable by usig (.3). (b) The estiators WEDF F ( x) ad the WKDE fk, ( x ), whe usig weight w = ( ) ( > ) are ore efficiet relative to the classical estiators S ( x) ad f ( x) i 9 out of 44 (8.6%) cases K, overall. For log tail values of x, the efficiecy of F ( x) relative to ( x ) always exceeds, which is cosistet with Theore 3.. The siulatio efficiecy of f w ( x) relative to f X ( x) ad f ( x ) are give by ( ( ) ( )) SMSE( f ( x)) SMSE( X fw X x f ( x)) MLE SEFF =, (5.) w S

11 AN EFFICIENT ESTIMATION METHOD 7 ( ( ) ( )) SMSE( f MLE ( x)) SEFF fw MLE x = (. (5.3) SMSE f ( x)) The siulatio results are give i Table 6, Appedix B for the choice of weight w = ( ) oly. We calculate the siulatio efficiecies SEFF ( f ( ) ( x )) ad ( ( )( x)) w X f w MLE w SEFF by usig (5.) ad (5.3) at selected values x =,, 3, 4, 5, 7, 9,, for α = ad 3 cases. The weighted estiator f w ( x), whe usig weight w = ( ) ( > ) are ore efficiet relative to the classical estiators f X ( x) ad f MLE ( x ) i 05 out of 44 (7.9%) cases overall. For log tail values of x, the efficiecy of f w ( x) relative to f X ( x) ad f MLE ( x ) always exceeds, which is cosistet with Theore 3.. The siulatios were ru by usig C++ ad MAPLE coputer progras. Ackowledgeet J. Tag cotributed excellet coputatioal siulatio work. Refereces [] P. Barbe ad P. Bertail, The Weighted Bootstrap, Spriger-Verlag, New York, 995. [] M. L. Huag, The efficiecies of a weighted distributio fuctio estiator, Proc. Aer. Statist. Assoc., Noparaetric Statistics Sectio (003), [3] M. L. Huag ad P. H. Brill, A distributio estiatio ethod based o level crossigs, Joural of Statistical Plaig ad Iferece 4() (004), [4] C. K. Kleiber ad S. Kotz, Statistical Size Distributio i Ecooics ad Actuarial Scieces, Joh Wiley & Sos, New York, 003. [5] H. L. Koul, Weighted Epirical ad Liear Models, Istitute of Matheatical Statistics, Lecture Notes-Moograph Series, Vol., Hayward, Califoria, 99. [6] B. W. Silvera, Desity Estiatio for Statistics ad Data Aalysis, Chapa ad Hall, New York, 986. [7] G. R. Shorack ad J. A. Weller, Epirical Processes with Applicatios to Statistics, Joh Wiley & Sos, New York, 986.

12 7 Appedix A: The Exact Efficiecies Table. The exact efficiecies of F ( x) relative to ( x ), w = ( ), α =, edia = 0.44 S Saple size Two roots of EFF = x z z Note. Efficiecies i bold are greater tha. Figure. The exact efficiecies of F ( x) relative to ( x ), w =, α =. ( ) S

13 AN EFFICIENT ESTIMATION METHOD 73 Table. The exact efficiecies of F ( x) relative to ( x ), w = ( ), α = 3, edia = S Saple size Two roots of EFF = X z z ` Note. Efficiecies i bold are greater tha. Figure. The exact efficiecies of F ( x) relative to ( x ), w =, α = 3. ( ) S

14 74 Table 3. The exact efficiecies of F ( x) relative to ( x ), w =, α =, edia = 0.44 ( + ) S Saple size Two roots of EFF = x z z Note. Efficiecies i bold are greater tha. Figure 3. The exact efficiecies of F ( x) relative to ( x ), w =, α =. ( + ) S

15 AN EFFICIENT ESTIMATION METHOD 75 Table 4. The exact efficiecies of F ( x) relative to ( x ), w =, α = 3, edia = ( + ) S Saple size Two roots of EFF = X z z Note. Efficiecies i bold are greater tha. Figure 4. The exact efficiecies of F ( x) relative to ( x ), w =, α = 3. ( + ) S

16 76 Appedix B: Siulatio Results Table 5. The siulatio efficiecies of F ( x) relative to S ( x ) saple size = 0, 0, 30, 50; geerated =,000,000 ties, w = ( ), SMSE( S ( )) SMSE ( ( ) ) ( ( ) ( ), ( )) * x fk x SEFF F * x = ; SEFF fk, x = ( ) SMSE F* x SMSE f* ( x ) ( ) (A) Pareto Distributio, α = ( K, ) x P{ X x} = f K, ( x) = = = = ( x) F = = = Note. There are 9 4 = 7 cases. Overall, the weighted ethod has 54 out of 7 (75%, i Bold) cases with efficiecies >. (B) Pareto Distributio, α = 3 x P{ X x} = f K, ( x) = = = = F ( x) = = = Note. There are 9 4 = 7 cases. Overall, the weighted ethod has 65 out of 7 (90.3%, i Bold) cases with efficiecies >.

17 AN EFFICIENT ESTIMATION METHOD 77 (A) Pareto Distributio, α =. (B) Pareto Distributio, α = 3. Figure 5. The siulatio efficiecies of F ( x) relative to ( x ), w = saple size = 0, 30, 50; geerated =,000,000 ties. ( ) S

18 78 Table 6. The siulatio efficiecies of f w relative to f X ad f MLE saple size = 0, 0, 30, 50; geerated =,000,000 ties, w = ( ) SMSE( f ( ( ) ( )) ( x)) SEFF ( X fw X x = ; SMSE f ( x)) ( ( ) ( )) SMSE fmle ( x) SEFF fw MLE x =. SMSE f ( x) w ( ) ( w ) (A) Pareto Desity, α = x P{ X x} = f w ( X ) ( x) = = = = f w ( MLE )( x) = = = Note. There are 9 4 = 7 cases. Overall, the weighted ethod has 4 out of 7 (58.3%, i Bold) cases with efficiecies >. (B) Pareto Desity, α = 3 x P{ X x} = f w ( X ) ( x) = = = = f w ( MLE )( x) = = = Note. There are 9 4 = 7 cases. Overall, the weighted ethod has 63 out of 7 (87.5%, i Bold) cases with efficiecies >. g