Joural of Moder Alied tatistical Methods Volume 0 Issue Article 9 --0 Estimatio of Parameters of Johso s ystem of Distributios Florece George Florida Iteratioal iversity, fgeorge@fiu.edu K. M. Ramachadra iversity of outh Florida, ram@usf.edu Follow this ad additioal works at: htt://digitalcommos.waye.edu/masm Part of the Alied tatistics Commos, ocial ad ehavioral cieces Commos, ad the tatistical Theory Commos Recommeded Citatio George, Florece ad Ramachadra, K. M. (0 "Estimatio of Parameters of Johso s ystem of Distributios," Joural of Moder Alied tatistical Methods: Vol. 0 : Iss., Article 9. DOI: 0.37/masm/300480 Available at: htt://digitalcommos.waye.edu/masm/vol0/iss/9 This Regular Article is brought to you for free ad oe access by the Oe Access Jourals at DigitalCommos@Wayetate. It has bee acceted for iclusio i Joural of Moder Alied tatistical Methods by a authoried editor of DigitalCommos@Wayetate.
Joural of Moder Alied tatistical Methods Coyright 0 JMAM, Ic. November 0, Vol. 0, No., 494-504 538 947//$95.00 Estimatio of Parameters of Johso s ystem of Distributios Florece George Florida Iteratioal iversity Miami, F K. M. Ramachadra iversity of outh Florida Tama, F Fittig distributios to data has a log history ad may differet rocedures have bee advocated. Although models like ormal, log-ormal ad gamma lead to a wide variety of distributio shaes, they do ot rovide the degree of geerality that is frequetly desirable (Hah & hairo, 967. To formally rereset a set of data by a emirical distributio, Johso (949 derived a system of curves with the flexibility to cover a wide variety of shaes. Methods available to estimate the arameters of the Johso distributio are discussed, ad a ew aroach to estimate the four arameters of the Johso family is roosed. The estimate makes use of both the maximum likelihood rocedure ad least square theory. The ew ME-east quare aroach is comared with other two commoly used methods. A simulatio study shows that the ME-east square aroach rovides better results for, ad families. Key words: Johso distributio, ubouded, bouded, logormal, estimatio. Itroductio Ay data set with fiite momets ca be fitted by a member of the Johso families such as, or. The most commoly used methods to estimate the arameters of the Johso distributio are the ercetile aroach (hairo, 980 ad Quatile method (Wheeler, 980. A ew aroach is roosed for the estimatio of Johso arameters ad is comard to other methods. For additioal reereces, see Draer (95, Hill (976, Hah ad hairo (967, George, et al (009. The Johso Traslatio ystem Give a cotiuous radom variable X whose distributio is ukow ad is to be aroximated, Johso roosed three ormaliig trasformatios havig the geeral Florece George is a Assistat Professor i the Deartmet of of Mathematics ad tatistics, Florida Iteratioal iversity. Email: fgeorge@fiu.edu. K. M. Ramachadra is a Professor i the Deartmet of Mathematics ad tatistics, iversity of outh Florida. Email: ram@usf.edu. form: X ξ Z γ + f (. f (. deotes the trasformatio fuctio, Z is a stadard ormal radom variable, γ ad are shae arameters, is a scale arameter ad ξ is a locatio arameter. Without loss of geerality, it is assumed that > 0 ad > 0. The first trasformatio roosed by Johso defies the logormal system of distributios deoted by : X ξ Z γ + l, X > ξ γ + l( X ξ, X > ξ curves cover the logormal family. The bouded system of distributios is defied by (. X ξ Z γ + l, ξ < X < ξ + ξ + X (.3 494
GEORGE & RAMACHANDRAN curves cover bouded distributios. The distributios ca be bouded o the lower ed, the uer ed or both eds. This family covers Gamma distributios, eta distributios ad may others. The ubouded system of distributios is defied by / X ξ X ξ Z γ + l + +, < X < X ξ γ + sih (.4 The curves are ubouded ad cover the t ad ormal distributios, amog others. sig the fact that, after the trasformatio i (., Z follows stadard ormal distributio, the robability desity fuctio (df of each of the family i the Johso system ca be derived. If X follows the Johso distributio ad X ξ Y the, for family, the df is ( ex [.l( ] y γ + y, π y ξ < X < +. similarly, for the family, the df is, ( y y ex γ +.l( π [ y/( y] y ξ < X < + ξ +. The df for the family is y ( π y + + + + < X < +. I geeral the df of X is give by, ex γ.l( y y, ' ( x g ( ex π for all ad x H ' g y, ( for the family y for the family [ y( y] for the family y + g( y l( y for the family l( y/ ( y for the family l[ y+ y + ] for the family. The suort H of the distributio is: H [ ξ, + for the family [ ξξ, + ] for the family (, + for the family. γ +. g( (.5 (.6 Parameter Estimatio of the Johso ystem: Percetile Matchig Percetile matchig ivolves estimatig k required arameters by matchig k selected quatiles of the stadard ormal distributio with corresodig quatile estimates of the target oulatio. For give ercetages { α : k}, the corresodig quatiles { } ad { x } α α are give resectively by 495
ETIMATION OF PARAMETER OF JOHNON YTEM OF DITRITION ad α x α Φ F ( α ( α Φ (. is the stadard ormal distributio fuctio ad F is the target distributio fuctio. Oce the fuctioal form f (. amog systems give by equatios.-.4 has bee idetified, the method of ercetile matchig attemts to solve the k equatios α xˆ α ξ γ + f (, k xˆ α is a estimator of the quatile x α based o samle data. lifker ad hairo (980 itroduced a selectio rule, which is a fuctio of four ercetiles for selectig oe of the three families, to give estimates of the Johso arameters. The fit arameters for the trasformatio are calculated by solvig the trasformatio equatio for the chose distributio tye at the four selected ercetiles. Choose ay fixed value ( 0 < < of a stadard ormal variate; the four oits ± ad ± 3 determie three itervals of equal legth. Determie the ercetile P ζ corresodig to ζ 3,,, 3 resectively. For examle, if 0.5 the P 0.69500 69.5. et 0.5 x3, x, x, x 3 be the ercetiles of data values corresodig to the four selected ercetiles of the ormal distributio. The tye of Johso distributio chose is based o the value of the discrimiat d calculated as follows. m d x x, m x3 x, x x 3. distributio is chose. A discrimiat equal to or betwee the two values results i a logormal fit. The fit arameters for the trasformatio are calculated by solvig the trasformatio equatio for the chose distributio tye at the four selected ercetiles. The arameter estimates for the Johso distributio are: ad ˆ, m cosh + γ ˆ m m ˆ sih, / / m ˆ /, m m + + + m ˆ x x + ξ +. m + The arameter estimates for the are distributio ˆ / ; cosh + + m ( > 0, If the calculated discrimiat d is greater tha.00, the a ubouded distributio is chose; if the value is less tha 0.999, the a bouded 496
GEORGE & RAMACHANDRAN ˆ γ sih 4 m + + m m ˆ, + + 4 m ˆ, m ad ˆ x + x m ξ +. m The arameter estimates for the Johso distributio are: ad ˆ, m l ˆ m m ˆ l, / γ m + ˆ x + x ξ. m / Parameter Estimatio of the Johso ystem: Quatile Estimators Wheeler (980 roosed a method to estimate the arameters γ ad i the Johso family usig five quatiles. et ( /, is the samle sie. Deote the quatile of the stadard ormal distributio / corresodig to the cumulative robability by. For examle, if 00, the 0.995, so that.5758. Choose five quatiles x, x k, x 0, x m, x from data corresodig to stadard ormal quatiles,, 0,,. The geeral form of the Johso system ca be writte as γ + l f ( y f ( y y for, f ( y y + (+ y / ; for, f ( y y/( y ; ad for y ( /. Wheeler uses the fact that ay quatity of the form xi x x x r f ( ω f i s f ( ωr f ( ω ( ω ( γ / ω e, does ot deed o ξ or. The arameter estimates for the curves are: ad ad For a ˆ b t t /lb / u + [( u ], t u x x ; x x m k ˆ γ l( a tb t b x x0 ad t. x x curves the arameter estimates are: 0 s 497
ETIMATION OF PARAMETER OF JOHNON YTEM OF DITRITION ad For ˆ / lb, b t t / b + [( b ], ( x x ( x x m 0 b, ( x xm( x0 x t t b a tb curves, ˆ γ l( a, t x x0 ad t. x x ˆ x l t x 0. x0 x To differetiate the three tyes of Johso curves, the ratio t t b u ( x ( x m x0( x x ( x m m 0 0 xk x is used. It is less tha for s, equal to for ad greater tha for. Parameter Estimatio of the Johso ystem: Proosed ME-east quare Aroach A ew algorithm to estimate arameters of Johso s distributio is ow roosed; this algorithm is amed the ME-east quare Aroach, because both maximum likelihood ad least square aroaches were emloyed to estimate the four arameters. Although the maximum likelihood equatios for γ ad were derived by torer (987, there are o closed form solutios for ξ ad. The idea of combiig both a maximum likelihood aroach ad least square theory makes the derivatio of all four arameters more tractable aalytically. The robability desity fuctios of the members of the Johso family are kow. First cosider the ad family of the Johso system. sig the geeral form of Johso desities (see equatio.5, the likelihood fuctio is: x ξ ( γ+ g ( ' i / ( π i x ( g( e, ad the log-likelihood is, log log log / log( π x ξ x ξ + g ( ( γ + g( ' i i ettig the artial derivatives with resect to to ero, [ g( ] which ca be writte as, [ g( ] + γ γ g( 0 g( 0 (3. ettig the artial derivatives with resect to γ to ero, ξ γ + g( x 0 which yields, g( ˆ γ. g sig (3.3 i (3.: (3. 498
GEORGE & RAMACHANDRAN ˆ x ξ x ξ [ g( ] [ g( ] var( g (3.3 g is the mea ad var (g is the variace of the values of g defied i (.6. The artial derivatives of the loglikelihood with resect to ξ ad are ot simle. torer (987 resets a legthy strategy for obtaiig the solutios of these arameters. I the maximum likelihood estimatio method, Kamiah, et al. (999 alied the Newto- Rahso iteratio to maximie the log likelihood of the Johso distributio. They observed that, for some samles, the log likelihood fuctio does ot have a local maximum with resect to arameters ξ ad. This o-regularity of the likelihood fuctio caused occasioal ocovergece of the Newto-Rahso iteratio that was used to maximie the log-likelihood (Hoskig, 985 The least squares method is alied herei to estimate arameters ξ ad. From γ (., x ξ + f ( is obtaied. For fixed values of γ ad, this equatio may be cosidered as a liear equatio with arameters ξ ad. The sum of squares of errors is, x f γ ( ξ, [ ξ+ ( ]. To determie the value of ξ ad that miimies ( ξ,, the artial derivatives of ( ξ, with resect to ξ ad are calculated ad these artial derivatives are equated to ero. The followig two equatios, called ormal equatios, are the obtaied: γ x ξ + f ( (3.4 xf γ ( ξ f γ ( + [ f γ ( ] Note that is a stadard ormal variate. The quatiles of x ad the corresodig quatiles of ca be cosidered aired observatios. If there are 00 or more x values, the ercetiles through 99 would be cosidered. If the umber of data oits of x is k k is less tha 00, k quatiles of x ad the corresodig k quatiles of would be cosidered as aired observatios. olvig the ormal equatios results i ad ˆ xf [ f ( ( ] f [ ( x f ] (3.5 ˆ γ ξ x mea[ f ( ] (3.6 x is the mea of x -quatiles ad is the mea of -quatiles used i the above equatios. tartig with some iitial values of ξ ad, these iitial values may be take as the estimates obtaied by ay oe of the revious methods. The estimates of γ ad are the calculated usig equatios (3. ad (3.3. After the estimates of γ ad are obtaied, equatios (3.5 ad (3.6 ca be used to revise the ξ ad estimates. Now these stes may be reeated, each time usig the most recet estimates; the Residual um of quares(r ca be tracked ad, after a few stes, the estimate with miimum R value selected. For the family, cosider the trasformatio i equatio (., so that there are oly 3 arameters icluded. The robability desity fuctio ca be give by, ( x e π ( The likelihood fuctio is, [ γ + l( x ξ ] 499
ETIMATION OF PARAMETER OF JOHNON YTEM OF DITRITION ( x (π / e ( [ γ + l( ξ ] x ettig the artial derivative of log-likelihood with resect to to ero we get, [l( ] which ca be writte as, γ ξ γ ξ [l( ] 0 [l( x ] + [l( x ] 0. (3.8 ettig the artial derivative of log-likelihood with resect to γ to ero, γ + [l( ] 0 which gives, ˆ γ [l( ] (3.9 g. sig (3.9 i (3.8 ad solvig for, results i ˆ [ l( ] [l( ] var( g (3.0 g l(. To estimate ξ, as before, use the method of least squares i the equatio x ξ + f (. The sum of squares of errors is, ( ξ ( + f ( To fid the value of ξ that miimies (ξ, obtai d ( f ( dξ ettig this derivative equal to ero, results i: ˆ ξ x mea[ f ( ] Here the same situatio arises, the estimate of ξ deeds o γ ad ad vice versa; as i the case of the ad distributios. Thus, start with some iitial value of ξ to estimate γ ad, the use these estimated values to estimate ξ. Reeat this rocedure, keeig track of R, ad choose the oe with least R. Results Data of sie,000 were simulated from the, ad distributios to comare differet methods of estimatio. Twety samles of sie,000 were geerated from each of the three secified models. The mea ad the Mea quare Error (ME of the estimated values of the,, ad families are show i Tables, ad 3. It ca be observed that the average of the estimates are close to the true values of the arameters ad, i geeral, the ME of the estimates are smaller i the roosed method tha the other methods. Coclusio A ew aroach that makes use of both the maximum likelihood rocedure ad least square theory was roosed to estimate the four arameters of the Johso family of distributios. The ew ME-east quare aroach is comared with two other commoly used methods. The simulatio study shows that the ME-east square aroach gives better results for the, ad families. The fidigs of this study should be useful for alied ractitioers. 500
GEORGE & RAMACHANDRAN l. No. Table : Mea ad (Mea quare Error-ME of Parameter Estimates for the Johso Family Parameter True Value Percetile Method Quatile Method ME-east quare Aroach γ 0.998(0.67.063(0.409 0.997(0.06.00(0.059.04(0.083 0.997(0.06 ξ 0 0.047(0.085 9.98(0.3 9.93(0.08 0 0.049(5.9 0.40(4.37 0.57(4.99 γ 0.5 0.503(0.009 0.503(0.0493 0.494(0.007 0.5 0.505(0.003 0.59(0.03 0.507(0.00 ξ 0 9.(4.038 9.97(0.077 0.004(0.004 0 0.005(0.85 0.094(.64 9.868(.056 γ.03(0.065.0(0.05.06(0.07 3 0.5 0.507(0.0039 0.5006(0.003 0.509(0.00 ξ 0 9.698(.488 0.00(0.00 0.00(0.00 0 0.355(4.63 0.085(0.69 9.86(0.70 γ 0.5 0.558(0.87 0.539(0.36 0.56(0.65 4.03(0.9.04(0.08.055(0.5 ξ 0 9.8(.097 9.94(0.55 9.9(0.5 0 0.3(5.4 0.30(8. 9.83(0.50 50
ETIMATION OF PARAMETER OF JOHNON YTEM OF DITRITION Table : Mea ad (Mea quare Error-ME of Parameter Estimates for the Johso Family l. No. Parameter True Value Percetile Method Quatile Method ME-east quare Aroach γ 0 0.04(0.3 0.05(0.05 0.05(0.05.4(3.3.08(0.34.05(0.9 ξ 0 0.4(8.9 0.(.5 0.(.4 0.3(99.9 0.5(.6 0.3(0. γ 0.5 0.8(.9 0.5(0. 0.5(0.09.47(3.3.08(0.45.06(0.37 ξ 0.5(64.6 0.06(.79 0.04(.59 0.07(56.5 0.35(.6 0.5(. γ 0-0.003(0.003 0.005(0.00 0.003(0.00 3.033(0.006 0.99(0.003 0.99(0.00 ξ 0 0.03(.43 0.05(0.5 0.06(0.5 0 0.45(.43 9.8(0.7 9.75(0.73 γ 0.5 0.54(0.009 0.488(0.006 0.487(0.007 4.008(0.006 0.999(0.006 0.996(0.006 ξ 0 0.43(.03 9.95(0.9 9.94(.05 0 0.06(0.96 0.06(.3 0.0(.43 50
GEORGE & RAMACHANDRAN l. No. Table 3: Mea ad (Mea quare Error-ME of Parameter Estimates for the Johso Family Parameter γ ( γ, True Value.303 (,0 Percetile Method Quatile Method ME-east quare Aroach -.353(0.05 -.9(0.07.303(0.04.0(0.006 0.97(0.008.0(0.008 ξ 0-0.98(0.4 0.53(0.057 0.53(0.057 γ ( γ, -.3 (0,0 -.4(0.04 -.6(0.0 -.(0.07 0.98(0.003 0.98(0.00 0.98(0.007 ξ 0 0.8(0.4 0.(0.36 0.33(0.8 γ ( γ, -5.9 (,0-6.53(.9-5.6(8.3-5.47(.36 3 3 3.8(.8.66(3.66.87(.4 ξ 0-0.503(5.8 0.7(8.3 0.504(7.7 γ ( γ, -3.45 (,0-3.78(3.6-3.45(0.99-3.45(.63 4.06(0.35.88(0.35.97(0. ξ 0-0.3(4. 0.43(4.4 0.9(.67 Ackowledgmet The authors are grateful to Dr.. M. Golam Kibria for his valuable ad costructive commets which imroved the resetatio of the study. Refereces Draer, J. (95. Proerties of distributios resultig from certai simle trasformatios of the ormal distributio. iometrika, 39, 90-30. 503
ETIMATION OF PARAMETER OF JOHNON YTEM OF DITRITION George, F., Ramachadra, K. M., & ihua,. (009. Gee selectio with Johso distributio. Joural of tatistical Research, 43, 7-5. Hah, J. G., & hairo.. (967. tatistical models i egieerig. Joh Wiley & os, New York. Hill, I. D., Hill, R., & Holder, R.. (976. Fittig Johso curvesby momets. Alied tatistics, 5 80-89. Hoskig, J. R. M., Wallis J. R., & Wood E. F. (985. Estimatio of the geeralied extreme-value distributio by the method of robability-weighted momets. Techometrics, 7 5-6. Johso, N.. (949. ystems of frequecy curves geerated by methods of traslatio. iometrika, 36, 49-76. Kamiah, A. K., Ahmad, M. I, & Jaffiri,. (999. Noliear regresio aroach to estimatig Johso arameters for diameter data. Caadia Joural of Forestry Resources, 9, 30-34. lifker, J., & hairo,. (980. The Johso system: electio ad arameter estimatio. Techometrics,, 39-47. torer, R. H. (987. Adative estimatio by maximum likelihood: Fittig of Johso distributios. ublished Ph.D. thesis, chool of Idustrial ad ystems Egieerig, Georgia Istitue of Techology. Wheeler, R. (980. Quatile Estimators of Johso curve Parameters. iometrika, 67, 75-78. 504