Estimating the Parameters of the Three-Parameter Lognormal Distribution

Transcription

1 Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School Estimatig the Parameters of the Three-Parameter Logormal Distributio Rodrigo J. Aristizabal Florida Iteratioal Uiversity, DOI: 0.548/etd.FI04308 Follow this ad additioal works at: Recommeded Citatio Aristizabal, Rodrigo J., "Estimatig the Parameters of the Three-Parameter Logormal Distributio" (0). FIU Electroic Theses ad Dissertatios This work is brought to you for free ad ope access by the Uiversity Graduate School at FIU Digital Commos. It has bee accepted for iclusio i FIU Electroic Theses ad Dissertatios by a authorized admiistrator of FIU Digital Commos. For more iformatio, please cotact dcc@fiu.edu.

2 FLORIDA INTERNATIONAL UNIVERSITY Miami, Florida ESTIMATING THE PARAMETERS OF THE THREE-PARAMETER LOGNORMAL DISTRIBUTION A thesis proposal submitted i partial fulfillmet of the requiremets for the degree of MASTER OF SCIENCE i STATISTICS by Rodrigo J. Aristizabal 0

3 To: Dea Keeth Furto College of Art ad Scieces This thesis, writte by Rodrigo J. Aristizabal, ad etitled Estimatig the Parameters of the Three-Parameter Logormal Distributio, havig bee approved i respect to style ad itellectual cotet, is referred to you for judgmet. We have read this thesis ad recommed that it be approved. Florece George Gauri L. Ghai Zhemi Che, Major Professor Date of defese: March 30, 0 The Thesis of Rodrigo J. Aristizabal is approved. Dea Keeth Furto College of Arts ad Scieces Dea Lakshmi N. Reddi Uiversity Graduate School Florida Iteratioal Uiversity, 0 ii

4 DEDICATION I dedicate this thesis to my wife, Lilyaa, ad my two childre, Rodrigo ad Katheri; they have ispired me to serve others. iii

5 ACKNOWLEDGMENTS I thak God, who gave me the stregth ad settled everythig to accomplish this thesis. To him, be the glory forever. I would like to thak the members of my committee, Dr. Zhemi Che, Dr. Florece George, ad Dr. Gauri Ghai for their support ad patiece. I am especially thakful to my Major Professor, Dr. Zhemi Che, for his guidace ad ecouragemet. I would like to express my gratitude to all other professors i the Departmet of Mathematics ad Statistics who have taught me about this woderful field of sciece ad ispired me to teach it. I wish to thak to all my frieds at FIU for their ecouragemet ad ucoditioal support, especially to my fried Zeyi Wag. Fially, I am very grateful to my woderful wife who has bee my compaio durig coutless log ights i this jourey. iv

6 ABSTRACT OF THE THESIS ESTIMATING THE PARAMETERS OF THE THREE-PARAMETER LOGNORMAL DISTRIBUTION by Rodrigo J. Aristizabal Florida Iteratioal Uiversity, 0 Miami, Florida Professor Zhemi Che, Major Professor The three-parameter logormal distributio is widely used i may areas of sciece. Some modificatios have bee proposed to improve the maximum likelihood estimator. I some cases, however, the modified maximum likelihood estimates do ot exist or the procedure ecouters multiple estimates. The purpose of this research is focused o estimatig the threshold or locatio parameter γ, because whe γˆ is kow, the the other two estimated parameters are obtaied from the first two MLE equatios. I this research, a method for costructig cofidece itervals, cofidece limits, ad poit estimator for the threshold parameter γ is proposed. Mote-Carlo simulatio, bisectio method, ad SAS/IML were used to accomplish this objective. The bias of the poit estimator ad mea square error (MSE) criteria were used throughout extesive simulatio to evaluate the performace of the proposed method. The result shows that the proposed method ca provide quite accurate estimates. v

7 TABLE OF CONTENTS CHAPTER PAGE. INTRODUCTION AND BACKGROUD.... Itroductio.... Backgroud Research Hypothesis Method ad Statistical Aalysis 6.5 Two-Parameter Logormal Distributio Review..7.6 Three-Parameter Logormal Distributio Review INTERVAL AND POINT ESTIMATION FOR γ.3. Pivotal Quatity..3. Critical Values of for Ay Sample Size..5.3 Cofidece Iterval for the Parameter γ.8.4 X(, γ ) As a Icreasig Fuctio of γ.3.5 Upper Cofidece Limit for the Parameter γ Poit Estimator of the Parameter γ SIMULATION RESULTS AND DISCUSSION CONCLUSION. 44 LIST OF REFERENCES..46 vi

8 LIST OF TABLES TABLE PAGE Table. Critical Values for the Pivotal Quatity..6 Table.3. Sample from a Logormal Distributio ( μ = 4, σ =, ad γ = 00) Table.3. Values of for a fix sample whe γ icrease from 0 to X ()..3 Table.5. Percetage of Samples from a Logormal Distributio for which the Cofidece Limits caot be calculated Table.6. Radom Sample from a Logormal Distributio ( μ = l(50), σ = 0.40, adγ = 00 )...36 Table.6. Summary of Estimates of 4 differet methods Table 3. Effect of o the Performace of MLE ad Proposed Methods.. 40 Table 3. Effect of the Shape Parameter σ o the Performace of the MLE ad Proposed Methods. 4 Table 3.3 Effect of the Scale Parameter μ o the Performace of the Two Methods...4 Table3.4 Effect of the Threshold Parameterγ o the Performace of the Two Methods. 4 Table 3.5 Performace of the Modified Method vii

9 . INTRODUCTION AND BACKGROUD. Itroductio The three-parameter logormal distributio is a skewed distributio that is useful for modelig cotiuous positive radom variables with support set [, for some 0. Limpert, Stahel ad Abbt (00) illustrated how the log-ormal distributios are widespread through the sciece. I this article they preseted a table which summarizes some fields where the logormal distributio is applied: i Geology (Cocetratio of elemets Co, Cu, Cr, 6 Ra, Au, ad U), i medicie (latecy periods of ifectious diseases, survival times after cacer diagosis), i evirometal sciece (raifall, air pollutio, the distributio of particles, chemicals ad orgaism i the eviromet), food techology (size of uit), ecology (species abudace), liguistics (legth of spoke words i phoe coversatios, legth of seteces), social scieces (age of marriage, farm size i Eglad ad Wales, Icomes), i operatio research (time distributio i queuig applicatios), i fiace (a log term discout factor), ad it has bee used i other diverse areas of the scieces. They cocluded: We feel that doig so would lead to a geeral preferece for the log-ormal or multiplicative ormal, distributio over the Gaussia distributio whe describig origial data. Although the logormal distributio is used for modelig positively skewed data, depedig o the values of its parameters, the logormal distributio ca have various shapes icludig a bell-curve similar to the ormal distributio. The maximum likelihood method is the most popular estimatio techique for may distributios because it selects the values of the distributio s parameters that give the observed data the greatest

10 probability, ad it has bee used for estimatig the parameters of the twoparameter logormal distributio. The maximum likelihood estimator of the locatio or threshold parameter for the three-parameter logormal distributio is the smallest order statistic, which is always greater tha. The iclusio of the third parameter, the locatio parameter, may create some trouble i the estimatio of the three-parameter logormal distributio. Some modificatio has bee proposed to improve the maximum likelihood estimator, but the iterative procedure that they suggest, for some samples the estimates did ot exist. This research proposes a ew approach for costructig exact cofidece itervals of the locatio or threshold parameter.. Backgroud The probability desity fuctio of the three-parameter logormal distributio is [ l( x γ ) μ] f ( x; μ, σ, γ ) = exp, (..) ( x γ ) σ π σ where 0 γ < x, < μ <, σ > 0. μ, σ ad γ are the parameters of the distributio. The distributio becomes the two-parameter logormal distributio whe γ = 0. The three-parameter logormal distributio has bee studied extesively by Yua (933), Cohe (95), Hill (963), Harter ad Moore (966), Muro ad Wixley (970), Giesbrecht ad Kempthome (976), Kae (978, 98), Cohe ad Whitte (980, 98), Wigo (984), ad Cohe ad White ad Dig (985), ad by others. I 933 Yua obtaied momet estimators of the three parameters of the logormal distributio

11 by equatig the first three sample momets to correspodig populatio momets ad used iterative procedure. I 963 Hill demostrated that the global maximum likelihood led to the iadmissible estimates =, =, =, regardless of the sample. The solutio of the likelihood equatio led to local-maximum-likelihood estimates, but i some cases the local maximums do ot exist. Harter ad Moore (966) proposed a iteractive procedure for obtaiig maximum-likelihood estimates of the parameters based o complete, sigly cesored, ad doubly cesored samples. Muro ad Wixley (970) examied the estimator based o order statistics of small samples from a threeparameter logormal distributio. Wigo (984) suggested a improve algorithm for computig local-maximum likelihood estimators (LMLE), which ivolves oly a sigle umerical maximizatio of a suitably trasformed coditioal log-likelihood fuctio (LLF). Cohe, Whitte ad Dig ( 985) preseted modificatio of the momet estimators (MME) ad modificatio of the maximum likelihood estimators (MMLE) i which the third momet ad the third local maximum likelihood estimatig equatios were respectively replaced by fuctios of the first order statistic..3 Research Hypothesis I this research, all parameters ( μ, σ, ad γ ) are assumed to be ukow. Let,,,, be the observatios of a sample from the three-parameter logormal distributio ad,,,, the correspodig order statistics. Let X = (,,,, ) a vector that represets a sample from the three-parameter logormal distributio. 3

12 Our attetio is focused o estimatig parameter, because whe is kow, the it follows from the first two maximum likelihood estimatio equatios that, (.3.) ad. (.3.) I order to fid a cofidet iterval for the parameter, a pivotal quatity,,, is defied as [ ] 3 l( xi γ ) l( xi γ ) + X, γ ) =,.3.3 ( 3 l( x + i γ ) [ ] l( x + i γ ) where 3 represets the iteger part of 3., ca be rewritte as [ l( x ] () ) 3 i γ μ l( x() i γ ) μ + σ σ X(, γ ) = l( ) [ 3 x γ μ ]. (.3.4) l( x γ ) μ + σ + σ 4

13 Sice l( x γ ) μ = z σ ~ N(0,), the pivotal quatity ca be writte as + [ ] 3 z [ ] X, γ ) = (.3.5) ( 3 z z + z + The pivotal quatity,, as a mathematical fuctio, is strict mootoic fuctio of, ad as a radom variable, the probability distributio of, does ot deped o ay parameters of the populatio. The, has the same probability distributio for all values of, μ, ad σ. Mote Carlo simulatio ca be used to fid quatiles of,. For fixed value of ad, we ca fid umbers ad satisfyig,,.3.6 where represets the empirical quitile of the radom variable,. The defiitio of the th quitile of, is..3.7 Sice, is a icreasig fuctio of, the the cofidece iterval of ca be costructed from the Equatios (.3.8) ad (.3.9), which were obtaied from the equatio (.3.6). 5

14 ,.3.8,..3.9 Equatios (.3.8) ad (.3.9) ca be solved to fid ad. Both ad must be smaller tha. Therefore, the hypothesis for this research is that a table for empirical quatiles of the radom variable, ad the equatios (.3.8) ad (.3.9) ca be used to fid cofidece iterval for the parameter. Cofidece iterval of γ will be squeezed to obtai poit estimate of γ..4 Method ad Statistical Aalysis Mote Carlo simulatios will be used to establish the quatiles of the radom variable, for a give sample size ad α. A large umber of pseudo-radom samples were geerated from a ormal distributio with parameters μ = 0 ad σ =.The pseudo radom sample of size draw from the ormal distributio N(0,) will be sorted i ascedig order, ad the value of, is calculated for each sample usig equatio (.3.5). This procedure is repeated 0,000,000 times for each sample size. The all, values will be sorted i ascedig order to fid the percetiles of the pivotal quatity for each. I order to use the percetiles or critical values ad to fid cofidece iterval for, it ca be show that, is a strictly icreasig fuctio of. 6

15 A radom sample Xwill be draw from a populatio that follows a logormal distributio with kow parameters,, ad σ usig the followig formula i SAS/IML x = exp( μ + σ * raor( repeat(0, )) + γ. The ( α)% cofidece iterval for γ will be calculated usig the critical values ad for a give ad usig the equatios (.3.8) ad (.3.9). The equatios ca be solved for usig the bisectio method. Statistical simulatio will be used to evaluate the performace of the estimatio method proposed, ad the proposed method will be compared with the existig methods i the literature..5 Two-Parameter Logormal Distributio Review The probability desity fuctio (pdf) of the two-parameter logormal distributio is: ;,, 0,, If X is a radom variable that has a log-ormal probability distributio, the Y = lx is ormally distributed with scale parameters μ ad shape parameter σ, which are the mea ad stadard deviatio of the radom variable Y = lx. Casella ad Berger preseted i Statistical Iferece (00) page 8 a iterestig property of the logormal distributio that all momets exist ad are fiite. The r th momet of X about the origi is /.5. 7

16 However, the logormal distributio does ot have a momet geeratig fuctio. The expected value of a radom variable X that follows a two-parameter logormal distributio ca be calculated usig (.5.). Usig (.5.) agai to calculate the secod momet of X, the variace of X is give by. If the expected value ad the variace of X are kow, we ca obtai equivalet expressio for the parameters. The cumulative distributio fuctio of the log-ormal distributio (.5.) with two parameters is ; μ,σ ) ;Φ. Sice the media of X is such a poit where ; μ,σ ) = 0.5, the it is ot difficult to prove that. The mode of X is the poit of global maximum of the 8

17 probability fuctio (.5.) which satisfy the equatio. The the mode of X is. The relatioship amog the mea, the media ad the mode is The geometric mea of the two-parameter log-ormal distributio is. = Geometric-Mea(X). Figure.5. Logormal Desity Plots for μ = 0, γ = 0, ad some values of σ. 9

18 A importat property of the log-ormal distributio is that, for very small values of the shape parameter 3, the shape of the logormal distributio is early close to a ormal distributio. Figure.5. shows this property. The method of maximum likelihood is the most commoly used techique of parameter estimatio for the two-parameter logormal distributio, because it fids the parameter estimators that make the data more likely tha ay other value of the parameter that would make them. The likelihood fuctio of the two-parameter logormal distributio for a give sample is derived by takig the product of the probability desities of the idividual observatios X i.,,,, l, The values of ad σ that maximize, also maximize,. To fid the maximum likelihood estimators of, the gradiet of with respect to is calculated, ad equate to

19 0 l The, the maximum likelihood estimators for are To verify that these estimators maximize,, the secod derivative matrix of is calculated

20 The Hessia or secod derivative matrix is give by 0 0. The determiat of the Hessia matrix is 0. O the other had for all o-zero vector because,,, are all positive. Sice H >0 ad the Hessia is egative-defiite, the log-likelihood fuctio ad the likelihood fuctio L have a local maximum whe take the values i equatio (.5.3) ad (.5.4) respectively.

21 .6 Three-Parameter Logormal Distributio Review The probability desity fuctio (pdf) of the three-parameter logormal distributio is: [ l( x γ ) μ] f ( x; μ, σ, γ ) = exp, (.6.) ( x γ ) σ π σ where x > γ 0, < μ <, σ > 0, ad γ is the threshold parameter or locatio parameter that defies the poit where the support set of the distributio begis; μ is the scale parameter that stretch or shrik the distributio ad σ is the shape parameter that affects the shape of the distributio. If X is a radom variable that has a three-parameter log-ormal probability distributio, the Y = l( X γ ) has a ormal distributio with mea μ ad variaceσ. The two-parameter logormal distributio is a special case of the three-parameter logormal distributio wheγ = 0. I 933, Yua derived the mea, variace, third stadard momet (skewess), ad forth stadard momet (kurtosis) of the logormal distributio as a fuctio of μ, σ ad γ. σ Mea (X) = E(X) = γ + exp μ + (.6.) Var(X) = exp( + σ )( exp( σ ) ) μ. (.6.3) α = exp( σ ) * (exp( σ ) ) (.6.4) 3 + α = exp(4σ ) + exp(3σ ) + 3exp(σ ) 3. (.6.5) 4 3

22 If the expected value ad the variace of X are kow, we ca solve these two equatios to obtai equivalet expressio for the parameters μ adσ. μ l ( E( X )) l + Var( X ) ( E( X )) = σ l + Var( X ) ( E( X )) = The cumulative distributio fuctio of the three-parameter log-ormal distributio is F X ( x; μ, σ, γ ) l = Φ ( x γ ) μ σ. The media of X is a value that satisfies the equatio ( x; μ, σ, γ ) = 0. 5 F X, the l( x γ ) μ = 0 σ Media ) + μ ( X = γ e. The mode of X is the poit of global maximum of the probability fuctio (.6.) which ' satisfies the equatio f ( x; μ, σ, γ ) = 0. ( ) ( μσ Mode X = γ + e ). The relatioship amog the mea, the media ad the mode is Mode ( X ) < Media( X ) < E( X ). 4

23 Estimatig the parameters of the three-parameter logormal distributio is more complicated tha estimatig the parameters of the two-parameter logormal distributio. The likelihood fuctio of the three-parameter logormal distributio for a give sample { x x,... } =Xis derived by takig the product of the probability desities of the, x idividual observatios Xi. [ f ( x i μ, σ, )] L( μ, σ, λ X ) = γ ( l( x γ ) μ) i = μ σ γ ( γ ) i,, X ) =. x i.exp. I mi{ x,..., } > γ σ π = σ i x i L ( (.6.6) XXL,, the values of μ,σ, ad γ that maximize If ( μ σ, γ ) = l( L( μ, σ, γ ) XL ( μ, σ, γ ) also maximize L ( μ, σ, γ )X. Cohe (95) obtaied local maximum likelihood estimators (LMLE) equatig the partial derivatives of the log-likelihood to zero. Differetiatig the log-likelihood fuctio ad equatig to zero, we obtai the local maximum likelihood estimatig equatios. Xl L( μ, σ, γ ) = lσ l π l( x γ ) i ( l( x γ ) μ) i σ l L μ μσ = x i i) = [ l( γ ) μ] 0 5

24 l( xi ˆ) γ ˆ μ = (.6.7) l L ii ) = + [ l( x i γ ) μ] σ σ σ = 0 3 ˆ σ (.6.8) = x i ˆ ( ) l( ˆ) γ μ l L l( xi γ ) μ iii) = + = 0 γ x γ σ γ i xi (.6.9) It ca be see that the maximum likelihood estimator (MLE) of γ is mi{ x,..., x } = x Hill (963) demostrated the existece of paths alog which the (). likelihood fuctio of ay ordered sample, x (),...,. x ( ), teds to as (γ, μ, σ ) approach ( x (),, ). This global maximum thereby leads to the iadmissible estimates γ = (), μ = α, X ad σ = +α, regardless of the sample. See Cohe ad Whitte (980). However, Cohe foud a local maximum likelihood estimate for γ by replacig (.6.7) ad (.6.8) i the last equatio (.6.9) to obtai a equatio iγ. γ l i l( xi γ ) ( l( xi γ )) + l( xi γ ) xi xi ( x γ ) γ = 0 (.6.0) Equatio (.6.0) ca be solved iteratively to obtai the local maximum likelihood estimator of γ. Cohe ad Whitte (980) preseted a modified maximum 6

25 likelihood estimator by replacig the third equatio (.6.0) for aother equatio of γ a little less complicated; however, whe multiple admissible roots occur, Cohe ad Whitte suggest choosig as LMLE for γ, the root which results i the closest agreemet betwee the sample mea x ad E ( X ) = ˆ μ = ˆ γ + exp( ˆ μ + ˆ σ x ). I this research, all the parameters μ, σ ad γ, are assumed to be ukow. Our attetio will be focused o estimatig parameter γ because whe γ is kow, equatios (.6.7) ad (.6.3) ca be used to estimate μ adσ. The objective of this research is to replace the third equatio of the maximum likelihood with aother equatio which has oly oe root for γ. Cofidece itervals ad oe-side cofidece limits ca also be obtaied. To fid a good estimator ad to costruct a cofidece iterval of γ is a difficult problem. The log-ormal pdf s shape varies from almost symmetric, as a uptured T, to highly skewed, as a L. Figure.6 shows the diversity of shape of the three-parameter log-ormal distributio. The shape parameter σ affects the skewess ad peakedess of the pdf. Equatio (.6.4) ad (.6.5) cofirm that the skewees ( α 3 ) ad peakedess ( α 4 ) oly deped o σ. O the other had, whe the scale parameter μ is icreased to 8, the dispersio of the radom variable also icreases, but the shape of the pdf does ot vary. The last two graphs show the same pdf, but with a differet scale i the vertical axis. The pdf is very close to the horizotal axis, ad the value of this pdf fuctio is 0.00 whe x=mode. The larger the scale parameter μ, the more spread out the distributio. The threshold parameter λ basically shifts the pdf. 7

26 Figure.6 Differet shapes of the Three-Parameters Log-Normal Distributio. μ σ γ Mode media mea Skewess Kurtosis μ σ γ Mode media mea Skewess Kurtosis

27 μ σ γ Mode media mea Skewess Kurtosis μ σ γ Mode media mea Skewess Kurtosis

28 μ σ γ Mode media mea Skewess Kurtosis μ σ γ Mode media mea Skewess Kurtosis

29 μ σ γ Mode media mea Skewess Kurtosis μ σ γ Mode media mea Skewess Kurtosis E+08.83E E

30 μ σ γ Mode media mea Skewess Kurtosis E E E μ σ γ Mode media mea Skewess Kurtosis E E E

31 . INTERVAL AND POINT ESTIMATION FOR γ Let x, x, x3,..., x be the observatios of a sample from a three-parameter logormal distributio ad x (), x( ), x( 3),..., x( ) the correspodig order statistics. Let ( x x, x x ) X =,..., be a vector that represets a sample poit of a logormal, 3 () ( ) ( ) ( ) distributed populatio.. Pivotal Quatity As metioed i Chapter, a pivotal quatity ca be defied i order to estimate parameterγ. [ ] 3 l( x() i γ ) l( x() i γ ) + i = X(, γ ) =, (..) l( x γ ) l( x γ ) i = + i = + [ ] 3 where represets the iteger part of 3. A radom variable Q ( X, θ) = Q( X, X,..., X, θ) is said to be a pivotal quatity (or pivot) if the distributio Xof (, θxq ) is idepedet of all parameters. That is, if ~ F ( θx), the (, θxq ) has the same distributio for all values of θ. See G. Casella ad R. Berger, Statistical Iferece (00). The expressio of (,Xγ ) i (..) ca be rewritte as 3

32 [ l( x ] () ) 3 i γ μ l( x() i γ ) μ + σ σ X(, γ ) = [ ]. 3 l( x γ ) μ l( x γ ) μ + σ + σ (..) l( xi γ ) μ Sice l( x i γ ) ~ N( μ, σ ) the = Zi σ ~ N (0,) ad l( x γ ) μ = Z σ (). i The pivotal quatity ca also be express as a fuctio of the order statistics, [ ] 3 Z Z + X(, γ ) = Z Z + + (..3) The pdf of the order statistics Z is f Z! ( i )!( i)! i () [ ] [ ] i i z = ( ) f Z ( z) FZ ( z) FZ ( z ) (..3) where f Z (z) ad F Z (z) are the pdf ad cdf respectively of the stadard ormal distributio which are idepedet of all parameters. The probability distributio of the pivot does ot deped o μ, σ orγ. 4

33 Aother importat property of the pivotal quatity X(, γ ) is that, for a fix sample, the pivotal quatity is a strictly icreasig fuctio ofγ. This property of Xthe pivotal quatity allows us to costruct a cofidet iterval forγ. Theoretical proof of this property ca be obtaied, but we will discuss this property i the Sectio.4 usig a fix sample.. Critical Values of for Ay Sample Size It has bee show above that satisfy the defiitio of pivotal quatity. Mote Carlo simulatio method is used to fid the critical values of the pivotal quatity for differet sample sizes from =5 to =50. To accomplish this objective, 0,000,000 pseudosamples of size were geerated each time from a log-ormal distributed populatio. Sice the critical values of ad its probability distributio does ot deped o ay parameters, the followig parameter combiatio μ = 0, σ =, ad γ = 0 was used i order to simplify the computatio. Each pseudo-sample was sorted i ascedig order, ad X(, γ ) was calculated for each sample usig Equatio (..). The, 0,000,000 pivotal quatities were obtaied ad sorted i ascedig order. The followig percetiles of were calculated, 0.05, 0.05, 0.0, 0.5, 0.30, 0.50, 0.70, 0.75, 0.90, 0.95, 0.975,

34 The critical values of were calculated as ( ) if ( ) is a iteger mα mα α = ( [ mα ] ) + ( [ mε ] + ) otherwise ad ( m(α ) ) = ([ m ]) ([ m ] ) α ( α ) + ( ε ) + if m( α ) is a iteger otherwise This procedure was repeated for each size of samples. Table (..) shows the critical values of for sample sizes from = 5 to = 50. Table. Critical Values for the Pivotal Quatity

35 Table. Critical Values for the Pivotal Quatity

36 Table. Critical Values for the Pivotal Quatity X(, γ ) As a Icreasig Fuctio of γ The method of fidig cofidet iterval for γ usig a pivotal quatity is possible oly if the pivotal quatity X(, γ ) is a strictly mootoic fuctio of γ. It ca be show that the pivotal quatity fuctio 8

37 [ ] 3 l( x() i γ ) l( x() i γ ) + i = X(, γ ) = l( x γ ) l( x γ ) i = + i = + [ ] 3 is a strictly icreasig fuctio of γ whe we fix the sample X. Pivotal quatity icrease ifiitely whe γ approaches from the left to the first order statistic X (). I fact, the umerator of X(, γ ) cotais a term that has the form [l( X γ the () )] γ X() ( X γ ) lim, =. O the other had, the pivotal quatity X(, γ ) reaches its miimum value whe ( ) γ approaches from the right to its miimum value 0 0 < γ < X (). [ ] 3 l( x() i γ ) l( x() i γ ) + mi = lim ( X, γ ) = lim γ γ l( x γ ) l( x γ ) + + [ ] 3 [ ] 3 l x() i l x() i i = + i = mi = (.3.) l x l x i = + i = + 9

38 Sice X(, γ ) is idepedet of all parameters, we ca select the fix sample Xfrom a logormal distributed populatio with ay combiatio of parameters μ,σ ad γ. A sample of 30 observatios was geerated from a logormal distributio with parameters μ = 4, σ =, adγ = 00, ad its data are listed below. Table.3. Sample from a Logormal Distributio ( μ = 4, σ =, ad γ = 00) The first order statistic of the sample is X () = ad the 30 th order statistic is X ( 30 ) = The miimum value for X(, γ ) is 0 l x 0 l x 0 l x 0 mi = = l x We ca graph the pivotal quatity X(, γ ) versus γ usig the data of this example. The results are preseted i Table.3. ad i Figure.3. 30

39 Table.3. Values of for a fix sample whe γ icrease from 0 to X (). γ γ γ γ Gamma Figure.3 Graphic of as a strictly icreasig fuctio of γ. 3

40 We ca fid through Table.3. ad Figure.3 that X(, γ ) is a strictly icreasig fuctio of γ. Additioally we ca verify the tedecy of X(, γ ) whe γ approaches to its boudaries 0 ad X () = lim ( X, γ ) = mi = ad + γ 0 lim γ X ( ) ( X, γ ) =. Whe γ is very close to the first order statistic X (), the graph of the pivotal quatity X(, γ ) is almost a vertical lie. Sice pivotal quatity X(, γ ) is a strictly icreasig fuctio of γ, we ca costruct the cofidet iterval for γ..4 Cofidece Iterval for the Parameter γ For a give sample size ad cofidece level values α ad α. α we ca fid correspodig critical ( α < ( XP, γ ) < α ) = α. We are ( α)00% cofidet that the true value of X(, γ ) is at least α but ot greater tha α. Sice X(, γ ) is a strictly icreasig fuctio of γ, we ca obtai a ( α)00% cofidet iterval for γ,which has the form XXL (, α ) < γ < U (, α ). 3

41 Here L (, α X) = γ L ad U (, α ) = γ U Xare the lower ad upper cofidet limits for γ. I fact, γ L ad γ U are the solutios of the equatios i = i = l( x + l( x [ ] 3 γ ) l( x() i γ ) i = 3 [ ] l( x + i = + [ ] 3 γ ) γ ) = α (.4.) ad i = i = l( x + l( x [ ] 3 γ ) l( x() i γ ) i = 3 [ ] l( x + i = + [ ] 3 γ ) γ ) = α (.4.) Equatios (.4.) ad (.4.) ca be solved to fid γ L ad γ U. Both, the lower ad upper cofidet limits of γ must be smaller tha, sice the first order statistic of the radom variable X is greater tha the threshold parameterγ. Now, we ca calculate the cofidece iterval of γ for the radom sample preseted i Sectio.3 usig the Equatios.4. ad.4.. To solve these equatios, the bisectio method ad SAS/IML was used. We ca solve the Equatios.4. ad.4. because mi is smaller tha the critical values of. 33

42 = mi = 0.375< ad mi 0.375<. 735 = =. The 95% cofidet iterval for γ, obtaied usig SAS/IML ad the bisectio method, is ( , ). If mi (.3.) is greater tha α or α, the Equatios (.4.) or (.4.) caot be solved, ad two-side cofidece iterval for γ caot be costructed i those cases..5 Upper Cofidece Limit for the Parameter γ It was metioed above that if mi is greater tha α or α, the Equatios (.4.) or (.4.) have o solutio i the iterval 0, X () ) [. Mote Carlo simulatio ad SAS/IML were used to estimate the percetage of samples for which we caot calculate the upper ad lower cofidece limits. For several sample sizes,,000,000 samples were geerated from a logormal distributio with parameters ( μ =, σ = 4, adγ = 0 ). It was foud that the lower cofidece limit caot be calculated for more tha % of the samples whe is smaller tha 30; i cotrast, the upper cofidece limit ca be calculated for more tha 99% ( 0.99%) of the sample whe is greater tha 7. The results were tabulated i Table.5.. Sice the two-side cofidece caot always be costructed, a oe-side upper cofidece limit of γ is recommeded. I fact, the proposed method is good for fidig upper cofidece limits of γ, ot for lower cofidece limits. 34

43 Table.5. Percetage of Samples from a Logormal Distributio for which the Cofidece Limits caot be calculated. N Replicatios μ σ γ %No Low %No Up % 0.00% % 0.00% % 0.00% % 0.00% % 0.00% % 0.03% % 0.0% % 0.39% % 0.99% %.5% %.56%.6 Poit Estimator of the Parameter γ The poit estimator of the threshold or locatio parameter γ ca be obtaied by squeezig the cofidece iterval described i the Sectio.4. A sigificace level α =00% produces 0% cofidece iterval which is a poit estimator of the media of γ because γ = γ γ α α = The pivotal quatity X(, γ ) ca be calculated as a fuctio of γ usig (..) for each sample. The critical value correspodig to the sample size ca be foud i Table.3.. The, the threshold parameter γ ca be estimated by solvig the Equatio 35

44 l( x + l( x [ ] 3 l( x l( x + + [ ] 3 ˆ) γ ˆ) γ ˆ) γ ˆ) γ = 0.50 (.6.) The proposed method ca be illustrated usig the example cosidered i a previous study by Cohe ad Whitte (980). The data set cosists of 0 observatios from a populatio with μ = l(50), σ = 0.40, adγ = 00 the Table.6... The sample data are listed i Table.6. Radom Sample from a Logormal Distributio ( μ = l(50), σ = 0.40, γ = 00 ). ad Summary statistics of this data set are: x =5. 30, s =9. 954, media=50.68, x () = 7., x ( 0 ) = 0. 45, = 0. mi 78. The critical values of the pivotal quatity whe =0 are 0. 55,. 00, ad = 0.50 = = 36

45 Usig SAS/IML ad the bisectio method, Equatios.4.,.4. ad.6. ca be solved. I this case, the lower cofidece limit caot be foud because Equatio.4. has o solutio i the iterval [0, 7.). The upper cofidece limit of γ is 5.47 ad the poit estimate of γ is γ = Table.6. summarizes the poit estimates obtaied for 4 differet methods metioed by Cohe ad Witte i their article. The estimate, obtaied by the proposed method, is amog the three best estimates. Table.6. Summary of Estimates of 4 differet methods Method Estimator γˆ Bias of γˆ Maximum Likelihood Estimator Local Maximum Likelihood Estimator Modified Maximum Likelihood Estimator I Modified Maximum Likelihood Estimator I Modified Maximum Likelihood Estimator I Modified Maximum Likelihood Estimator II Modified Maximum Likelihood Estimator II Modified Maximum Likelihood Estimator II Momet Estimator Modified Momet Estimator I Modified Momet Estimator I Modified Momet Estimator I Modified Momet Estimator II Modified Momet Estimator II Modified Momet Estimator II Pivotal Quatity Method (Proposed) Populatio 00 37

46 3. SIMULATION RESULTS AND DISCUSSION The performace of the proposed method to estimate the threshold parameter γ is evaluated usig the Mea Square Error (MSE) criteria. Mote Carlo simulatio was used to geerate r samples from a three-parameter logormal distributio ad γ was estimated for each sample usig the procedure described above. If γ, γ,..., γ k are poit estimates of γ obtaied from the k samples, the estimates of the mea, the variace, ad the MSE of the radom variable γ are k ˆ γ i ˆ μ γ = k, (3.) ˆ ˆ γ = σ k ( ˆ γ ˆ i μ γ ) k, (3.) Bias γ = μ γ ( 3.3) γ γ ad MSE = k I = ( ˆ γ γ ) i k = ˆ σ + ( Bias γ ) ˆ γ γ (3.4) where γ is the true value of the threshold parameter ad k is the umber of samples with size take from the populatio. The bias of the poit estimator γ is the differece betwee the expected value of γ ad the parameter γ. 38

47 Statistical simulatio was coducted to ivestigate the effect of differet sample sizes o the performace of the method proposed i this research, ad it was compared with the performace of the MLE method. It was metioed, i Sectio.6, that the MLE estimator of γ is the first order statistic x (). Mote Carlo simulatio ad SAS/IML were used to geerate,000,000 samples for each sample size cosidered from a populatio with parameters μ =, σ =, adγ = The pdf of this populatio is preseted i Figure Figure 3. Pdf of the Populatio with Parameters μ =, σ =, adγ = 0. mu sig gam mode media mea Skewess Kurtosis E The mea, the variace, the MSE, ad the bias forγˆ were calculated for several sample sizes usig both methods. The proposed method has smaller bias, eve for small samples. Whe is 00 or bigger, the MSE of the proposed method is better tha MLE method. Table 3. shows the results. 39

48 Table 3. Effect of o the Performace of MLE Method ad Proposed Method N K μ σ γ PARAMETERS MLE PROPOSED BIAS BIAS ˆ μ ˆ γ ( ˆ ˆ γ ) μ ˆ γ ( ˆ γ ) MSE (γˆ ) MSE (γˆ ) Whe the shape parameter σ icreases, the skewess ad the kurtosis of the distributio also icrease. The skewess measures the lack of symmetry of the pdf ad the kurtosis measures the peakedess ad heaviess of the tail of the pdf. The proposed method performs better tha MLE method whe the data shows low skew ad o heavy tail; this occurs whe the pdf looks more like a ormal distributio. However, the MLE method performs better whe the data shows high skewess ad heavy tail or the pdf looks like a L-shape. Table 3. shows the effect of σ (shape parameter) o the performace of the two methods. Whe scale parameter μ icreases, the dispersio of three-parameter logormal radom variable icreases. The proposed method performs better tha MLE whe the data is more dispersed. Whe μ = 8, the MSE of MLE method is,439,494 ad the 40

49 estimate of γ is 3,0, which is 3 times bigger tha the real parameter. Table 3.3 shows the results. Table 3. Effect of the Shape Parameter σ o the Performace of the MLE ad Proposed Method. N k μ σ γ PARAMETERS MLE PROPOSED BIAS BIAS μ ˆ γ ( ˆ ˆ γ ) μ ˆ γ ( ˆ γ ) MSE (γˆ ) MSE (γˆ ) Table 3.3 Effect of the Scale Parameter μ o the Performace of the Two Methods. k μ σ γ PARAMETERS MLE PROPOSED BIAS BIAS ˆ μ ˆ γ ( ˆ ˆ γ ) μ ˆ γ ( ˆ γ ) MSE (γˆ ) MSE (γˆ ) Variatios i the threshold parameter γ do ot sigificatly affect the performace of both methods. Table 3.4 shows the results. The MSE of the proposed method could be improved if we solve Equatio (.6.) for γˆ startig the bisectio method at a value closer to the data. The samples of Table 3.4 were selected from a populatio with 4

50 parameters μ = ad σ =, ad 99% of the values of this populatio are i the iterval ( e + γ, e + γ ). Whe γ = 000, 99% of the values of this populatio are i the iterval (000., 035.7). It does ot make sese to start the bisectio method from 0. Whe,000,000 samples with =30 are geerated, we expect about 50,000 observatios less tha 000. ad some of these samples could have smaller skewess tha the populatio. For this kid of sample, the bisectio method obtais estimates very far from,000. Oly oe of these estimates icreases the mea square error sigificatly. Table3.4 Effect of the threshold parameter γ o the performace of the two methods. k μ σ γ PARAMETERS MLE PROPOSED BIAS BIAS ˆ μ ˆ γ ( ˆ ˆ γ ) μ ˆ γ ( ˆ γ ) MSE (γˆ ) MSE (γˆ ) Table 3.5 Performace of the modified method. K μ σ γ ˆ μ ˆ γ BIAS ( ˆ γ ) MSE (γˆ ) % NO γˆ % NO γˆ U Start Bisectio x() 3( ) rage x() ( ) rage x() ( ) rage 4

51 The proposed method could be improved if we start the bisectio method at a poit equivalet to x () (three, two or oe time the data rage). Simulatios were performed with this modificatio ad the MSE was reduced sigificatly. If we start bisectio method for each sample at x() ( ), the MSE is 0.94 ad it is possible to estimate γ for almost all radom samples as you ca see i the colum %NO γˆ i Table

52 4. CONCLUSION The three-parameter logormal distributio is widely used i may areas of sciece. Some statisticias have maifested their preferece for this distributio whe describig origial data. Fidig a good estimator ad costructig a cofidece iterval of this distributio is a difficult problem. Some modificatios have bee proposed to improve the maximum likelihood estimator. I some cases, however, the modified maximum likelihood estimates do ot exist or the procedure ecouters multiple estimates. The three-parameter logormal pdf s shape varies from almost symmetric, as a uptured T, to highly skewed, as a L. The purpose of this research is focused o estimatig the threshold parameterγ, because whe is kow, the the estimatio of the other two parameters ca be obtaied from the first two maximum likelihood estimatio equatios. Cosequetly, better estimate of γ leads to a better estimate of the other two parameters. I this research a method for costructig cofidece itervals, cofidece limits, ad poit estimators for the threshold or locatio parameter γ is proposed. I order to fid a cofidet iterval for the parameter, a pivotal quatity, X(, γ ), was defied. Two importat properties of X(, γ ) were show. The first oe is that the probability distributio of X(, γ ) is idepedet of all parameters, which allows us to costruct a table for critical values of the pivotal quatity. The secod oe is that the pivotal quatity fuctio (γ ) is a strictly icreasig fuctio of γ, which allows us to costruct cofidet itervals for γ. Mote-Carlo simulatio was used to fid the critical values α ad α. Equatios (, γ ) =X ad α (, γ ) =X α were solved usig 44

53 the bisectio method i order to fid the cofidece iterval of γ. The proposed method is good for fidig upper cofidece limits ofγ, ot for lower cofidece limits. The poit estimator of the threshold or locatio parameter γ ca be obtaied by squeezig the cofidece iterval ad usig the critical value The bias of the poit estimator ad mea square error (MSE) criteria were used throughout extesive simulatio to evaluate the performace of the proposed method. Compared with other methods, the pivotal quatity method performs quite well obtaiig poit estimators of γ with lower bias tha most of them, eve i cases where other methods cofrot difficulties such as small samples ad populatio with low skewess. I fact, the proposed method produces a lower bias of γˆ tha the maximum likelihood estimator (MLE) method does i all simulatios summarized i Table 3. through 3.5, where a wide combiatio of the three parameters was cosidered. Usig the MSE criteria, the proposed method shows better performace tha the MLE method whe the populatio skewess is low ad the spread of the distributio is high. The MSE of the proposed method ca be improved if we do ot cosider the few atypical samples, which are much less tha % of the samples, for which the bisectio method produces a big bias. Moreover, the pivotal quatity could be slightly modified i order to get better MSE. The method itroduced i this research provides a more accurate estimate of the locatio parameter i most of the origial data, especially whe the populatio skewess is low ad the spread of the distributio is high. 45

54 LIST OF REFERENCES Casella, G., ad R. L. Berger, Statistical Iferece, (00). Cohe, A Clifford, Estimatig Parameters of Logarithmic Normal Distributios by Maximum likelihood, Joural of the America Statistical Associatio, 46, (95), 06-. Cohe, A.C., B.J. Whitte, Estimatio i the Three-Parameter Logormal Distributio, Joural of the America Statistical Society, 75 (980) Cohe, A.C., B.J. Whitte ad Y. Dig, Modified Momet Estimatio for the Three Parameter Logormal Distributio, Joural of Quality Techology, 7 (985) Harter, H.L. ad A.L. Moore, Local-Maximum-Likelihood Estimatio of the Parameter of Three-Parameter Logormal Populatio From Complete ad Cesored Samples, Joural of the America Statistical Society, 6 (966) Hill, M.B., The Three-Parameter Log-Normal Distributio ad Bayesia Aalysis of a Poit-Source Epidemic, Joural of the America Statistical Associatio, 58, (963) Limpert, E., W. Stahel ad M. Abbt, Log-Normal Distributio Acroos the Scieces: Keys ad Clues, BioSciece, 5 (5), (00) Muro A.H. ad R.A.J. Wixley, Estimatio o Order Statistics of Small Samples From a Three-Parameter Logormal Distributio, Joural of the America Statistical Society, 65 (970) -5. Wigo, D.R., Fittig Three-Parameter Log-Normal Models by Numerical Global Optimizatio a Improved Algorithm, Computatio Statistical Data Aalysis, (984) 3-5. Yua, Pae-Tsi, O the Logarithmic Frequecy Distributio ad Semi-Logarithmic Correlatio Surface, Aals of Mathematical Statistics, 4, (933),