The RS Generalized Lambda Distribution Based Calibration Model

Transcription

1 Internatonal Journal of Statstcs and Probablty; Vol. 2, No. 1; 2013 ISSN E-ISSN Publshed by Canadan Center of Scence and Educaton The RS Generalzed Lambda Dstrbuton Based Calbraton Model Steve Su 1, Abeer Hasan 2 & We Nng 2 1 Covance Pty Ltd., Sydney, Australa; School of Mathematcs and Statstcs, Unversty of Western Australa, Crawley, Australa 2 Department of Mathematcs and Statstcs, Bowlng Green State Unversty, Bowlng Green, USA Correspondence: Steve Su, Covance Pty Ltd., Sydney, Australa; School of Mathematcs and Statstcs, Unversty of Western Australa, Crawley 6009, Australa. E-mal: dbarro2@gmal.com Receved: December 5, 2012 Accepted: January 10, 2013 Onlne Publshed: January 22, 2013 do: /sp.v2n1p101 URL: Abstract We propose a flexble lnear calbraton model wth errors from RS (Ramberg & Schmeser, 1974) generalzed lambda dstrbuton (GλD). We demonstrate the dervaton of the maxmum lkelhood estmates of RS GλD parameters and examne the estmaton performance usng a smulaton study for sample szes rangng from 30 to 200. The use of RS GλD calbraton model not only provdes statstcal modeller wth a rcher range of dstrbutonal shapes, but can also provde more precse parameter estmates compared to the standard Normal calbraton model or skewed Normal calbraton model proposed by Fgueredoa, Bolfarnea, Sandovala and Lmab (2010). Keywords: generalzed lambda dstrbuton, lnear calbraton model, skew normal dstrbuton, maxmum lkelhood estmaton 1. Introducton The statstcal calbraton model s a reverse regresson technque, where we use the response varable to predct the correspondng explanatory varable. There are number of applcatons of ths technque n scence. For example, we may use radometrc datng to ascertan the age of a tree and further verfy our result usng tree rngs. Our am, however, s to use radometrc datng to estmate age of new trees, and the problem s whether we should mnmze errors n the observaton or mnmze errors n age determnaton. There are many smlar problems n substance concentraton determnaton n bology and chemstry, physcal quanttes determnaton n physcs and blood pressure/cholsterol level measurement n medcne. The lterature on calbraton problem has a long hstory, and one of the earlest works can be found n Esenhart (1939). The usual calbraton experment s a two stage process nvolvng two random varables X and Y. The frst stage s known as the calbraton tral, where we observe the n values of the response varable y 1,, y n from a gven set of explanatory values x 1,, x n and we can estmate the lnk functon between X and Y. The second stage s known as the calbraton experment, where we observe k 1 value(s) of the response varable Y as y 01,, y 0k whch are mapped from some unknown value x 0 from the explanatory varable X. We can express these two stages by the followng equatons. y = α + βx + ε, = 1,, n; y 0 = α + βx 0 + ε 0, = 1,, k, (1.1) We usually assume that the errors ε 1,,ε n,ε 01,,ε 0k are..d and Normally dstrbuted wth mean 0 and varance σ 2. Also, x 1,, x n are known and α, β, x 0 and σ 2 are unknown parameters whch we need to estmate. As an extenson to Normal dstrbuton, Azzaln (1985) ntroduced the skewed Normal dstrbuton. The skewed Normal dstrbuton s defned as g(x; ξ, ω, λ) = 2 ( x ξ ) ( ( x ξ )) ω φ Φ λ, (1.2) ω ω where φ( ) and Φ( ) are the p.d.f. and c.d.f. of a standard normal dstrbuton respectvely. Specally, when ξ = 0 and ω = 1, we obtan the standard skewed Normal dstrbuton. 101

2 Internatonal Journal of Statstcs and Probablty Vol. 2, No. 1; 2013 Based on (1.2), Fgueredoa et al. (2010) defned a skew-normal calbraton model by assumng that ε and ε 0 are..d. and follow a skewed Normal dstrbuton wth ξ = 0 denoted by SN(0,ω,λ). Ths gves us the followng calbraton model: y x SN(α + βx ; ω; λ), = 1., n, y 0 x 0 SN(α + βx ; ω; λ), = 1,, k. (1.3) In (1.3), the condtonal dstrbuton of y gven x and y 0 gven x 0 are governed by skewed Normal dstrbutons. Ths skewed Normal calbraton model allows the modeller to cope wth some degree of skewness n the error dstrbuton. However, ths s stll lmted as the skewed Normal dstrbuton have lmted range of shapes. The skewed Normal dstrbuton stll cannot handle heavy taled, U shape, unform, trangular or exponental upward/downward patterns. These shapes however, can be captured usng GλD (generalzed lambda dstrbutons), and we propose a further extenson to the calbraton model by usng RS GλD. Our artcle s organzed as follows. In Secton 2, we ntroduce the GλD famly. In Secton 3, we outlne the RS GλD calbraton model and dscuss possble ways to estmate parameters of the model usng maxmum lkelhood estmaton. In Secton 4, we demonstrate the estmaton performance of our proposed model across a range of dfferent sample szes from 30 to 200. As a further test to our proposed model to the lterature, we compare the performance of RS GλD calbraton model aganst Normal and skewed Normal calbraton model wth respect to a real lfe dataset used by Fgueredoa et al. (2010) n Secton 5. A dscusson of our proposed method s gven n Secton Generalzed Lambda Dstrbutons The RS GλD (Ramberg & Schmeser, 1974) s an extenson of Tukey s lambda dstrbuton. It s defned by ts nverse dstrbuton functon: F 1 (u) = λ 1 + uλ 3 (1 u) λ 4 0 u 1 (2.1) From (2.1), λ 1,,λ 3,λ 4 are respectvely the locaton, nverse scale and shape 1 and shape 2 parameters. Karan and Dudewcz (2000) noted that GλD s defned only f 0 for 0 u 1. The condtons λ 3 u λ λ 4 (1 u) λ 4 1 for whch RS GλD s a vald p.d.f. are set out n Karan and Dudewcz (2000) and these are also programmed n GLDEX package n R (Su, 2010, 2007a). Fremer, Kolla, Mudholkar and Ln (1988) descrbe another dstrbuton known as FKML GλD. The FKML GλD can be wrtten as: F 1 (u) = λ 1 + u λ 3 1 λ 3 (1 u)λ 4 1 λ 4 0 u 1 (2.2) Under (2.2), λ 1,,λ 3,λ 4 are respectvely the locaton, nverse scale and shape 1 and shape 2 parameters. The fundamental motvaton for the development of FKML GλD s that the dstrbuton s defned over all λ 3 and λ 4 (Fremer et al., 1988). The only restrcton on FKML GλD s > 0. Ths s more convenent to deal wth computatonally than RS GλD and hence t s sometmes the preferred GλD for some researchers. We restrct our attenton n ths artcle to the more dffcult problem of fttng RS GλD calbraton model to data. Wthout loss of generalty, the method we outlned below can be easly adapted to buld FKML GλD calbraton model. 3. Statstcal Model 3.1 GλD Based Calbraton Model We consder the followng usual calbraton model: y = α + βx + ɛ, = 1,, n, (3.1) y 0 = α + βx 0 + ɛ, = 1,, k. (3.2) We assume that ɛ and ɛ are..d. GλD(0,,λ 3,λ 4 ). In general, we consder x 1,, x n to be known and fxed and α, β,,λ 3 and λ 4 are parameters we need to estmate. Our GλD calbraton model takes the followng form: y x GλD(α + βx,,λ 3,λ 4 ), (3.3) 102

3 Internatonal Journal of Statstcs and Probablty Vol. 2, No. 1; 2013 y 0 x 0 GλD(α + βx 0,,λ 3,λ 4 ). (3.4) Consequently, the lkelhood functon for RS GλD s: where L(θ, y, y 0 ) = n =1 + λ 4 (1 z ) λ 4 1 k =1 y = (α + βx ) + zλ 3 (1 z ) λ 4, y 0 = (α + βx 0 ) + zλ 3 (1 z ) λ 4, and 0 z, z 1, θ = (α, β, x 0,,λ 3,λ 4 ). 3.2 Estmaton of Parameters From (3.5), we obtan the followng log lkelhood functon: where log L(θ, y, y 0 ) = n log ( f 1 (θ, y )) + =1 f 1 (θ, y ) = f 2 (θ, y 0 ) = Takng the dervatve of (3.6), we obtan the followng: log L(θ) θ = + λ 4 (1 z ) λ 4 1, (3.5) k log ( f 2 (θ, y 0 ) ) (3.6) =1 + λ 4 (1 z ) λ 4 1, n =1 + λ 4 (1 z ) λ f 1 θ + k =1 1 f 2 θ, (3.7) where θ = (α, β, x 0,,λ 3,λ 4 ). Theoretcally, the MLE of θ s the soluton of (3.7) when t s set to be equal to 0. The dervatves θ and θ are gven below. = z y z y = λ λ 3 (λ 3 1)z λ λ 4 (λ 4 1)(1 z ) λ λ 4 (1 z ) λ 4 1 ) 2 + λ 4 (1 z ) λ 4 1 zλ (1 z ) λ 4 2 = [λ 3(λ 3 1)z λ 3 2 λ 4 (λ 4 1)(1 z ) λ4 2 ](z λ 3 (1 z ) λ 4 ) + λ 4 (1 z ) λ 4 1 = ( ) [λ 3(λ 3 1)z λ3 2 λ 4 (λ 4 1)(1 z ) λ4 2 ](z λ 3 log z ) λ 3 + λ 4 (1 z ) λ 4 1 [λ 3 (λ 3 1)z λ3 2 λ 4 (λ 4 1)(1 z ) λ4 2 ]((1 z ) λ 3 log(1 z )) = λ 4 + λ 4 (1 z ) λ 4 1 α = ( λ2 2 )[λ 3(λ 3 1)z λ3 2 λ 4 (λ 4 1)(1 z ) λ4 2 ] + λ 4 (1 z ) λ 4 1 β = ( λ2 2 )[λ 3(λ 3 1)z λ3 2 λ 4 (λ 4 1)(1 z ) λ4 2 ] x + λ 4 (1 z ) λ 4 1 = [λ 3(λ 3 1)z λ 4 (λ 4 1)(1 z ) λ 4 2 ](z λ 3 + λ 4 (1 z ) λ 4 1 (1 z ) λ 4 ) 103

4 Internatonal Journal of Statstcs and Probablty Vol. 2, No. 1; 2013 = ( ) [λ 3(λ 3 1)z λ 3 λ 4 (λ 4 1)(1 z ) λ 4 2 ](z λ 3 + λ 4 (1 z ) λ 4 1 log z ) [λ 3 (λ 3 1)z λ 4 (λ 4 1)(1 z ) λ4 2 ]((1 z ) λ 3 log(1 z )) = λ 4 + λ 4 (1 z ) λ 4 1 α = ( λ2 2 )[λ 3(λ 3 1)z λ 4 (λ 4 1)(1 z ) λ4 2 ] + λ 4 (1 z ) λ 4 1 β = ( λ2 2 )[λ 3(λ 3 1)z = ( 2 x )[λ 3(λ 3 1)z 0 λ 4 (λ 4 1)(1 z ) λ 4 2 ] x 0 + λ 4 (1 z ) λ 4 1 λ 4 (λ 4 1)(1 z ) λ 4 2 ] β + λ 4 (1 z ) λ 4 1 It s dffcult to obtan the exact solutons of settng (3.7) to zero usng the above formulatons, owng to the fact that RS GλD s defned by ts nverse quantle functon and there s a hgh degree of complexty nvolved n solvng the above equatons. As an alternatve, we carry out the maxmum lkelhood estmaton by maxmsng (3.6) drectly usng Nelder-Mead optmsaton algorthm as s customary done for maxmum lkelhood estmaton problems nvolvng GλD (see Su, 2010, 2007a, 2007b). Ths s a preferred and more relable method of estmaton as opposed to tryng to satsfy the exact condtons to whch all of the above equatons equal to zero. The GLDEX package n R (Su, 2010, 2007a) facltates the Nelder-Mead optmsaton algorthm for GλD. Our algorthm s as follows: 1) Generate a set of ntal values for α, β, x 0,,λ 3,λ 4. There are a number of strateges that can be used to determne the best set of ntal values. One strategy s to generate ntal values α, β, x 0 usng Normal or skewed Normal calbraton model and then generate some low dscrepancy quas random numbers for,λ 3,λ 4 over a range of values and select the set of ntal values that maxmses (3.6). Alternatvely all ntal values can be randomly generated usng low dscrepancy quas random numbers. 2) Set λ 1 = α + βx 0. 3) Check that GλD(λ 1,,λ 3,λ 4 ) s a vald statstcal dstrbuton, ths can be done usng GLDEX package n R. 4) Check the mnmal support of GλD(λ 1,,λ 3,λ 4 ) s lower or equal to the lowest value of y 0. Smlarly, check that the maxmum support of GλD(λ 1,,λ 3,λ 4 ) s greater or equal to the largest value of y 0. Ths s to ensure that the ftted GλD wll span the entre dataset. If these condtons are not met, choose another set of ntal values and repeat from 2). 5) Conduct Nelder Mead optmsaton by maxmsng (3.6) drectly usng the above ntal values to obtan the requred estmates. 4. Smulatons We conduct smulatons to llustrate the performance of our RS GλD calbraton model for sample sze n = 30, 50, 100 and 200 wth α = 3,β = 1.5, x 0 = 15 or 40,λ 3 = 10,λ 4 = 1, and = 2, 5, 10. We further generate x 1, x 2,, x n from Un f orm(10, 30), and we set k = 1. We use the true parameters as our ntal values to kck start the optmsaton process to obtan our MLE estmate for x 0. We repeat ths process 1000 tmes, whch gve us 1000 ˆx 0m estmates of x 0. The mean ˆx 0, Bas(x 0 ) and MSE(x 0 ) are calculated as follows: ˆx 0 = ˆx 0m 1000 m=1 Bas(x 0 ) = (ˆx 0m x 0 ) 1000 m=1 MSE(x 0 ) = (ˆx 0m x 0 ) m=1

5 Internatonal Journal of Statstcs and Probablty Vol. 2, No. 1; 2013 The results of above smulatons are shown n Tables 1 and 2. As expected, the MSE decreases as we ncrease the sample sze or ncrease the value of nverse scale parameter. In terms of bas, we observe that the performance appear to be farly consstent across sample szes, ths gves confdence n the use of RS GλD calbraton model for smaller samples, even though there are are more parameters that need to be estmated from ths model. There also appears to be a tendency for RS GλD calbraton model to slghtly overestmate as nearly all the bas results are postve. Increasng the shape parameter λ 3 does not always result n ncrease n MSE, ths s because the shape parameter spaces of λ 3 and λ 4 for RS GλD are farly complex. Table 1. Smulatons results wth x 0 = 15, α = 3, β = 1.5, λ 4 = 1 = 2 = 5 = 10 n λ 3 ˆx 0 Bas MSE ˆx 0 Bas MSE ˆx 0 Bas MSE Table 2. Smulatons results wth x 0 = 40, α = 3, β = 1.5, λ 4 = 1 = 2 = 5 = 10 n λ 3 ˆx 0 Bas MSE ˆx 0 Bas MSE ˆx 0 Bas MSE Table 3. Smulatons results wth x 0 = 15,α = 3,β = 1.5, true error dstrbuton GEV(0.1860, , ) s approxmated by RS GλD wth λ 1 = 0, ,λ ,λ n ˆx 0 Bas MSE We further consdered usng RS GλD to approxmate generalzed extreme value dstrbuton (GEV) wth locaton, scale and shape parameters beng , , respectvely. We choose RS GλD wth λ 1 = 0, ,λ ,λ for ths demonstraton (Fgure 1). We then generate smulated data based on GEV and use our approxmated RS GλD to estmate x 0 wth α = 3,β= 1.5 and repeat ths over 1000 smulaton runs. The result of ths smulaton s gven n Table 3. We observe that the RS GλD calbraton model tends to overestmate the true x 0 by a small margn, but the bas appears to decrease as sample sze ncreases. 105

6 Internatonal Journal of Statstcs and Probablty Vol. 2, No. 1; 2013 Generalzed Extreme Value Dstrbuton approxmated by RS GλD GEV(0.1860, , ) GλD(0, , , ) Densty x Fgure 1. Approxmatng GEV usng RS GλD 5. Applcaton We apply the RS GλD calbraton model to a dataset whch measures teenager testcular volume (ml 3 ). Ths dataset s from Chpkevtch, Nshmura, Tu and Galea-Raas (1996) and conssts of 42 observatons. Fgueredoa et al. (2010) consdered two measurement methods from Chpkevtch et al. (1996): dmensonal measurement wth a calper (DM) and measurement by ultrasonography (US) and the data s gven n Table 4. In ther paper, Fgueredoa et al. (2010) consder the x 0 value of 16.4, whch s observed twce by ultrasonography. They subsequently treated ths value as unknown, wth correspondng y 0 values of y 01 = 10.3 and y 02 = Then, they estmate x 0 usng ther skewed Normal calbraton model and compared ths wth the standard Normal calbraton model. We dd the same usng the RS GλD calbraton model and our results are shown n Table 5. Table 4. Measurements obtaned by dmensonal measurement wth a calper (DM) and by ultrasonography (US) from the rght tests for 42 teenagers, n ml 3 DM US DM US DM US DM US DM US DM US Table 5. A comparson of lnear calbraton models RS GλD model SN model Normal model Parameter Estmate Stdev. Estmate Stdev. Estmate Stdev. α β σ x λ λ λ AIC BIC HQ The theoretcal dervaton of the varablty of our estmates under RS GλD s not readly tractable as n the cases of skewed Normal and Normal dstrbutons. As we need to numercally derve our calculatons, small errors n 106

7 Internatonal Journal of Statstcs and Probablty Vol. 2, No. 1; 2013 numercal procedures could accumulate nto large errors even f we could evaluate the exact theoretcal soluton. As a workaround, we adopt the followng procedure. Once we obtaned the parameters of our model, α, β, x 0,, λ 3, λ 4, we conduct smulatons to estmate the varablty of our estmate. We use our estmated parameters from the RS GλD calbraton model and x (excludng x = 16.4) from the orgnal data to randomly generate y 0 and y accordng to (3.1) and (3.2). We then maxmse the lkelhood n (3.6) usng Nelder Mead Smplex algorthm wth ntal values beng our orgnal estmated parameters. We repeat the process 1000 tmes and calculate the sample standard devatons of our estmated parameters. Table 5 lsts the estmated parameters and ther standard devatons from RS GλD, skewed Normal and Normal calbraton models. We compute the Akake, Bayesan and Hannan-Qunn nformaton crteron (AIC, BIC, and HQ) to allow model selecton between three models. All three crteron favors the RS GλD calbraton model. In addton, the RS GλD model s much more effcent compared to the other models, wth the smallest varablty n ts parameter estmates. 6. Concludng Remarks We propose a new calbraton model wth RS GλD errors, whch s an extremely flexble model that can cope wth a wde range of dfferent error dstrbutons. Our method also lends to the development of FKML GλD calbraton model, whch may have better propertes wth regard to numercal convergence. Our smulatons studes suggest our proposed model perform well for small sample szes across a range of nverse scale and shape parameters of RS GλD. We further demonstrate that the RS GλD calbraton model can outperform skewed Normal or Normal calbraton model, wth lower AIC, BIC and HQ nformaton crteron and lower varablty n our parameter estmates n the context of a real lfe data. These smulaton results are promsng and future statstcal models should am to develop statstcal technque that are talored to data, rather than requrng emprcal data to satsfy a partcular statstcal model. One possble extenson of our model s the development of a mxture RS GλD calbraton model, whch would extend the flexblty of our model even further but also present a very challengng problem for data wth small samples. References Azzaln, A. (1985). A class of dstrbutons whch ncludes the normal one. Scandnavan Journal of Statstcs, 12, Chpkevtch, E., Nshmura, R., Tu, D., & Galea-Raas, M. (1996). Clncal measurements of testcular volume n adolescents: Comparson of the relablty of 5 methods. The Journal of Urology, 156, Esenhart, C. (1939). The nterpretaton of certan regresson methods and ther use n bologcal and ndustral research. Annals of Mathematcal Statstcs, 10, Fgueredoa, C., Bolfarnea, H., Sandovala, M., & Lmab, C. (2010). On the skew-normal calbraton model. Journal of Appled Statstcs, 37(3), Fremer, M., Kolla, G., Mudholkar, G. S., & Ln, C. T. (1988). A study of the generalsed tukey lambda famly. Communcatons n Statstcs-Theory and Methods, 17, Karan, Z. A., & Dudewcz, E. J. (2000). Fttng statstcal dstrbutons: The generalzed lambda dstrbuton and generalsed bootstrap methods. New York: Chapman and Hall. Ramberg, J. S., & Schmeser, B. W. (1974). An approxmate method for generatng asymmetrc random varables. Communcatons of the Assocaton for Computng Machnery, 17, Su, S. (2007a). Fttng sngle and mxture of generalsed lambda dstrbutons to data va dscretzed and maxmum lkelhood methods: GLDEX n R. Journal of Statstcal Software, 21(9). Su, S. (2007b). Numercal maxmum log lkelhood estmaton for generalzed lambda dstrbutons. Computatonal Statstcs and Data Analyss, 51(8), Su, S. (2010). Handbook of dstrbuton fttng methods wth R. In E. Karan, & Z. Dudewcz (Eds.), Fttng GLD to data Usng the GLDEX n R (Chap. 15). CRC Press. 107