A New Method of Testing Log-normal Mean

Transcription

1 World Aled Sceces Joural (: , SSN DOS Publcatos, DO: 589/doswasj5 A New ethod of Testg og-oral ea K Abdollahezhad, F Yaghae ad Babaezhad Deartet of Statstcs, Faculty of Sceces, Golesta versty, Gorga, ra Abstract: ths aer, we deal wth hyothess testg for ukow araeters A ew ethod s roosed based o a cobato of the coutatoal aroach test ad axu lkelhood estato (CAT-, troduced by [7] The CAT- does ot requre ay kowledge of the salg dstrbuto t deeds heavly o uercal coutatoal ad ote-carlo sulato study To llustrate the ethodology, we show that the CAT- ca be better tha the two acceted tests, odfed Cox ethod ad geeralzed -value ethod, wth ote Carlo sulatos to test the ea the logoral dstrbuto We also aly our ethod to a set of real data Key words: axu lkelhood estato hyothess araetrc bootstra ower fucto NTRODCTON The ethods of the coutatoal tools have bee owadys aled the stats tcal ferece, artcular the hyothess test for the ukow oulato araeters The a barrer of the hyothess test has always bee o the dstrbuto of the eloyed test statstcs Recetly there has bee efforts wth satsfactory cosequeces to byass ths barrer The dea s qute sle: use ay sulated values for the test statstcs or the araeter estato of the ukow araeter, ad the secfy the test crtcal values Ths ethod s ow kow as: coutatoal aroach test (CAT CAT s cely outled ad eloyed by [7], where the axu lkelhood estato ( are used for the statstcal ferece o the ukow araeters (CAT- We wll cocetrate o testg hyotheses o the ea of the log- oral dstrbuto, sce for ths class we ca effectvely ake coarsos betwee ractcal coutatoal rocedures develoed testg hyothess ore recsely we exae the ower of the test for the ea of the log-oral dstrbuto for odfed Cox ethod, geeralzed -value ethod ad CAT- We observe that the ower of the test for CAT- s uforly sall Ths artcle s orgazed as follows We rovde the geeral coutatoal fraework to hadle a scalar valued araeter based o the ext Secto Secto 3 we aly the CAT- for hyothess testg for log-oral ea Further, we llustrate our aroach usg a real exale Also the ower/sze of our ethod s coared wth the exstg Corresodg Author: K Abdollahezhad, Deartet of Statstcs, Faculty of Sceces, Golesta versty, Gorga, ra 67 ethods Secto 4, soe cocludg rearks are rovded to close the aer The CAT- et X,X,,X s a rado sale fro desty ƒ(x, θ, θ Θ We reset the CAT- for two caseswhe there s o usace araeter ad whe there s (are usace araeter(s Case : Assue θ s scaler valued ad there s o usace araeter The CAT- rocedure for testg H : θ=θ vs H A:( θ< θ or θ>θ or θ θ at a desred level α, s gve through the followg stes: Ste : Derve θ usg the observed sale x,,x, the of θ Ste : ( Assue that H s true, e, set H : θ=θ Geerate artfcal sale X,X,,X d fro desty ƒ(x, θ a large uber of tes (say, tes For each of the relcato calculate the of θ (retedg that θ were ukow ad deote the θ, θ,, θ et θ ( θ ( θ ( be the ordered values of θ, l l Ste 3: ( For testg H : θ=θ agast H A : θ<θ (f such a alteratve s eagful at level α, defe θ = θ ( α Reject H f θ < Alteratvely, calculate the -value as: θ

2 World Al Sc J, (: , ( θ, l l = (uberof θ ( l 's < θ / = l= ( θ ( l < θ ( For testg H : θ=θ agast H A : θ>θ at level α, defe θ = θ (( α Reject H f θ > Alteratvely, calculate the -value as: = (uberofθ ( l 's > θ / = l = ( θ ( l θ > θ ( For testg H : θ=θ agast H A : θ θ defe the cut-off ots as θ = θ (( α/ ad θ = θ (( α/ Reject H f θ < θ < θ Alteratvely, the -value s couted as: =(,, where ( ( = ( uber of θ ' s < θ ( l / ( ( = (uber of θ ' s > θ ( l / Case : Nusace araeter s reset Assue that ( ( θ= ( θ, θ Θ, where θ ( f avalable, s the usace araeter ad θ ( s the araeter of terest The ethodology of the CAT based o to test H : θ = θ vs H :( θ < θ or θ > θ or A θ θ at a desred level α, s gve through the, followg stes: Ste : Derve ( θ = ( θ, θ ( usg the observed sale x,,x the of θ Ste : ( Assue that H s true, e, set H : θ = θ The fd the of θ ( fro the data aga Call ths as the `restrcted of θ ( ' uder H deoted by ( θ R ( Geerate artfcal sale X,X,,X d fro ( ( desty f x, θ, θ a large uber of tes ( et ( R (say, tes For each of these relcated sales, recalculated the of θ θ θ ( ( = (, (retedg that θ were ukow Reta oly the cooet that s relevat for θ ( et these recalculated values of θ be θ, θ,, θ ( ( ( ( θ θ θ be the ordered values of Ste 3 ( For testg H : θ = θ agast H : θ < θ (f A such a alteratve s eagful at level α, defe θ = θ ( α Reject H f θ < Alteratvely, calculate the -value as: = (uberof θ ( l 's < θ θ / = l= ( θ ( l < θ ( To test H : θ = θ agast H : θ > θ at sgfcat level α, defe A (( α θ = θ Null hyothess s rejected f θ > Alteratvely, θ calculate the -value as follows: = (uberofθ ( l 's > θ / = l = ( θ ( l > θ ( To test H : θ = θ agast H :θ θ defe A the cut-off ots as θ = θ (( α/ ad θ = θ (( α/ Reject H f θ < θ < θ Alteratvely, the -value s couted as follows: =(,, where ( ( = ( uber of θ 's < θ / ( l = (uber θ / ( l ( ' s > ( of θ ETHODS FOR TESTNG EAN N OG-NORA ODE et X deote the rado varable that follows a log-oral dstrbuto wth robablty desty fucto f X(x= ex( (lx µ xσ π σ The ea of X s ex( µ+ σ ad the varace of X s ex( µ+σ (ex( σ We let Y deote the log-trasfored, orally dstrbuted varable Y = l (X, that has ea value E(Y = µ ad varace Var(Y = σ Assue that X, = l,, s a deedet rado 673

3 World Al Sc J, (: , sale fro log-oral oulato, e X ~log-oral dstrbuto wth - degree of freedo (stead of (µ,σ, where µ ad σ are ukow Based o the above deedet sales, our terested test s H:= vs H A: (<or> where =ex( µ+ σ Note that the above test s equvalet to the followg test: H : η = η vs H : η (<or> η (3 A ( ( where η = µ+ σ, η =l( Now, set Y=l(X, =,, The Y ~N (µ,σ et η ( be a araeter of terest ad let the usace araeter η = σ Several ethods for testg exresso (3 have bee roosed over the ast decades The ethods clude a ave ethod based o trasforato; a ethod roosed by Cox; a odfed verso of the Cox ethod []; a ethod otvated by large-sale theory ad a ethod based o geeralzed -v alue [6] Accordg to sulato results the odfed Cox ad geeralzed -value are better tha other ethods Therefore, we cosder these two ethods ad coare the wth our roosed CAT- the followg sub-secto, we brefly, exla these ethods odfed cox ethod: Deote the sale ea ad sale varace of Y wth Y ad S u, resectvely, where u = = Y= Y S = (Y Y A estato for η ( s ( η =Y+ Su ( estato for the varace of η s gve by Var( η =S/+ S /( ( 4 u u ad a Cox has suggested that a test statstc for hyothess test (3 ca be derve as 4 S S Z=( η η / + ( stadard oral dstrbuto aled the Cox ethod, e, we reject H f Z>t α /,, where t α /, s the uer α/-level cut-off ot t - -dstrbuto Geeralzed -value: Geeralzed -value ca be used for ferece about araeters whe there s usace araeter [6] suggested the followg rocedure for coutg a geeralzed -value for the log-oral ea; For a gve data set x,,x set y = l(x, =,, ad calculate y ad s u fro the data For j = to, geerate Z~N(, ad ~X Set ( T=y sz/ + ( s / j u u ( let j = f T> η, otherwse j = Set j j= j =, the {, } s a ote Carlo estate of the geeralzed -value for testg (3 The CAT- for ea: The followg stages gve testg of log-oral ea based o CAT- Stage : Get the s of the araeters as ( ( η =Y+ S/ b, =S b, where Y= Y/,S = (Y Y b = = Stage : ( Assue that H s true, e, η = η Y ~ N(,, where η ( s ukow ( ( ( The η η η The s of the araeter η ( = σ whch are called the `restrcted ' s + + η ( ( R = (y = ( Geerate artfcal sale Y,,Y d fro ( ( ( N( η R, R a large uber of tes (say, tes For each of these relcated sa les, recalculated the of η ( et these recalculated values of η ( be The odfed Cox ethod uses the t-studet 674

4 World Al Sc J, (: , ( ( values as η l =Yl+ S bl(,,,,,, N Fally, the ower s aroxated by ( ( et be the ordered values of ( ( ( ( l, l Stage 3 ( For testg H : η = η agast H : η < η (f A such a alteratve s eagful at level α, defe = Reject H f < Alteratvely, ( α calculate the -value as: =(uberof 's< /= (l ( ( ( l= η (l lower ad uer α % cut-off ots ( 5 Now brg the η fro the above stage ad ( get = ( η η η 6 Reeat the above stage through stage 5 a large uber of tes (say, N tes ad get the 675 < ( For testg H : η = η agast H : η > η at A level α, defe = Reject H f (( α > Alteratvely, calculate the -value as: =(uberof 's> /= (l ( ( ( l= η (l > ( For testg H : η = η agast H :η η A defe the cut-off ots as ad (( α/ = (( α/ ( = Reject H f < < Alteratvely, the -value s couted as: = (,, where =(uberof 's< / (l =(uberof 's> / (l The sze ad ower coutato for test H : η = η vsh : η η, log-oral s doe A through the followg stages For fxed, η ( ad η (, geerate d ( ( ( observatos of sze fro N( η η, η, where ( ( η = µ+ σ, η = σ Get η ( ad η ( 3 Set η = η ( H value ad get the restrcted of η ( ( as η R (l (l (l 4 Now geerate Y =(Y,,Y d fro ( ( ( N ( η, η R η R, l =,, Reta oly the values of η be η, η,, η Order these values of η as η, η,, η Get = ( α ( ( ( η η ad η η (these are the = (( α β CAT N = N = Real data exale: The data Table 3 are subjects of a drug roduct wth a log half-lfe tal vestgatos of these data ad of other slar datasets, dcate that a log-oral odel ay be arorate The -values for odfed Cox ethod, geeralzed -value ethod ad the CAT- ( ( ethod, for testg H : η =3 vs H: η 3 are 38, 9 ad 36, resectvely Therefore, the three ethods do ot reject H Table 3 Te subjects of a drug roducts Data Data Nuercal results for sze ad ower coarso: We coare the erforace of our roosed CAT- wth two ethods--the odfed Cox ethod ad the geeralzed -value ethod-- ters of sze ad ower The Table 3: The actual sze of tests whe the oal level s 5 (µ,σ Test 5 5 (, β C β GP β C (5,9 β C β GP β C (,8 β C β GP β C (,6 β C β GP β C (µ,σ Test 3 5 (, β C β GP β C (5,9 β C β GP β C (,8 β C β GP β C (,6 β C

5 β GP β C owers of our roosed CAT- (C, the geeralzed -value (GP ad the odfed Cox (C are deoted by β C, β GP ad β C resectvely For ths urose we geerated sales wth szes =, 5,, 5,3, 5,, fro a log-oral Table 33: The ower of tests whe the oal level s 5 wth η ( =55 (µ,σ Test 5 5 (, β C β GP β C (5, β C β GP β C (,9 β C β GP β C (,7 β C β GP β C (µ,σ Test 3 5 (, β C β GP β C (5, β C β GP β C (,9 β C β GP β C (,7 β C β GP β C dstrbuto wth araeters µ =, 5,, ad ( σ =( η µ 5, relcato were used We ( ( cosder the test H : η = 5 vs H : η 5 The actual sze of tests are gve Table 3 ad the ower of tests for η ( = 55, 6 are gve Table 33 ad 34, resectvely t ca be observed fro the below tables that the actual sze of test of CAT- ethod s always less tha the oal level, however, ths ca ot be hae the other ethods the geeralzed -value ethod the actual sze are as good as the CAT- ethod, World Al Sc J, (: , 676 but the actual szes of ths ethod are ore tha the oal level soecases Table 34: The ower of tests whe the oal level s 5 wth η ( =6 (µ,σ Test 5 5 (, β C β GP β C (5, β C β GP β C (, β C β GP β C (,8 β C β GP β C µ Test 3 5 (, β C β GP β C (5, β C β GP β C (, β C β GP β C (,8 β C β GP β C The odfed Cox ethod are very lberal oreover, all ethods, the actual sze wll be closer to the oal level, as the sale sze creases Also, accordg to the Table 33 ad 34, we see that the ower of test our ethod s better tha the two acceted tests CONCSON ths study, we aled a ew uercal ethod, CAT-, for testg the ea of log-oral dstrbuto based o the axu lkelhood estato We observed, based o the sulated data ad real data The results shows that the CAT- has suerorty to the odfed Cox ethod ad geeralzed -value ethod ters of the test ower behavor ad the test sgfcat level The CAT- does ot requre the salg dstrbuto ad oulato dstrbuto forulato ACKNOWEDGEENTS

6 Authors exress ther scere arecatos to the referees for valuable coets REFERENCES Ahed, SE, RJ Toks ad A Volod, Test of hoogeety of arallel sales fro logoral oulatos wth uequal varaces Joural of Statstcal Research, 35 (: 5-33 Chag, CH ad N Pal, 8 A Revst to the Behres-Fsher Proble: Coarso of Fve Test ethods Coucatos statstcs-sulato ad Coutato, 37: Chag, CH, N Pal, W ad JJ, Coarg several oulato eas: a araetrc bootstra ethod ad ts coarso wth usual ANOVA F test as well as ANO Coutatoal Statstcs, 5 (: Crow, E ad K Shzu, 988 ogoral dstrbuto, arcel Dekker: New York 5 Gll, PS, 4 Sall sale ferece for the coarso of eas of logoral dstrbuto, Boetrcs, 6: 37-4 World Al Sc J, (: , 6 Krshaoorthy, K ad T athew, 3 fereces o the eas of logoral dstrbutos usg geeralzed -values ad geeralzed cofdece tervals Joural of statstcal lag ad ferece, 5: 3-7 Pal, N, WK ad CH g, 7 A coutatoal aroach to statstcal fereces J Al Probab Statst, (: Tsu, KW ad S Weerahad, 989 Geeralzed -values sgfcace testg of hyothess the resece of usace araeters J A Statst Assoc, 84: Weerahad, S, 993 Geeralzed cofdece tervals J A Statst Assoc, 88: Zhou, XH ad S Gao, 997 Cofdece tervals for the log-oral ea Statstcs edce, 6: Zhou, XH ad W Tu, 999 Coarso of several deedet oulato eas whe ther sales cota logoral ad ossbly zero observatos Boetrcs, 55: