A Comparison of Hypothesis Testing Methods for the Mean of a Log-Normal Distribution

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1 World Applied Scieces Joural (6): ISS IDOSI Publicatios A Copariso of Hypothesis Testig ethods for the ea of a og-oral Distributio 3 F. egahdari K. Abdollahezhad ad A.A. Jafari Islaic Azad iversity eyriz Brach Ira Departet of Statistics Golesta iversity Gorga Ira 3 Departet of Statistics Yazd iversity Yazd Ira Abstract: This paper deals with testig the ea of a log-oral populatio. We apply a ewly developed Coputatioal Approach Test (CAT) which is essetially a paraetric bootstrap ethod. A advatage of the CAT is that it does ot require the explicit kowledge of the saplig distributio of the test statistic. The CAT is the copared with three accepted tests- Cox ethod odified Cox ethod ad geeralized p-value ethod with ote Carlo siulatios. Our detailed studies idicate soe iterestig results icludig the facts that the size ad power of CAT is better tha other ethods. sig real data we have illustrated our ethod. Key words:coputatioal Approach test og-oral distributio Geeralized p-value Testig hypothesis ITRODCTIO For exaple icoe data ca ofte be cosidered to be log-oral. Oe way of aalyzig such data is to logthe Accessibility coputatioal resources has trasfor the origial variable X ad to base the iferece cotributed to the fudaetal researches to be o the trasfored variable Y I(X). This eas the carried out i ay areas of atheatical scieces. distributio fro which our data eerges ca be Coplex theoretical results ca ow be better approxiated with a log-oral distributio. I this paper realized through uerical coputatios ad/or we have discussed the hypothesis tests of the arithetic ote Carlo siulatios well before they ca be ea value of X i a log-oral distributio. It is true that verified aalytically. Recetly [] developed a siple the edia is ofte used to describe the average of coputatioal techique called Coputatioal skewed distributios like icoe distributios. However Approach Test (CAT) for hypothesis testig there are situatios whe the arithetic ea is a probles. The CAT looks siilar to paraetric paraeter of iterest. bootstrap but has soe great differeces-especially et X deote the rado variable that follows a logthe way it exploits the uisace paraeter(s). Aog oral distributio with probability desity fuctio the oteworthy aspects of this ethod it is observed that: (i) the CAT is a paraetric ethod which does ot require fx ( x) exp{ (l x ) } the kowledge of the exact saplig distributios of the x paraeter estiators; ad as a result (ii) the CAT ca be used eve for a very coplicated paraetric odel which E( X ) exp( + ). We let Y deote the logtrasfored ofte relies heavily o the asyptotic results oly. [] applied the CAT for Behres-Fisher proble ad orally distributed variable Y I(X) that capared it with four test ethods ad [3] showed that has ea value µ ad variace. for the oe-way AOVA the CAT ca provide alost as This article is orgaized as follows. I Sectio we uch power as the classical F-test uder the propose a CAT for testig hypothesis for log-oral hooscedastic oral odel. ea. I Sectio 3 we illustrate our approach usig a real I applied statistics classes we soeties coe exaple. The size of our ethod is copared with other across data that eed to be trasfored prior to aalysis. existig ethods i Sectio 4. Correspodig Author: A.A. Jafari Departet of Statistics Yazd iversity Yazd Ira E-ail: aajafari4@yahoo.co. 845

2 World Appl. Sci. J. (6): ethods for Testig ea: Assue that X i i... is a idepedet rado saple fro log-oral populatio i.e. X i ~ log-oral (µ ) where µ ad are ukow.based o the above idepedet saples the our iterested test is exp( + ) equivalet to the followig test: where. ote that the above test is A H : vs. H : (< or >) Cox ethod ad odified Cox ethod: Deote the saple ea ad saple variace of Y with Y ad S u respectively where i u i i i Y Y S ( Y Y). A estiatio for is ad a 4 u Su H : vs. H : (< or >) A where + l( ). ow set Y i I(X i) i... The Y i ~ (µ ). et be a paraeter of () iterest ad let the uisace paraeter There are several ethods for testig. The ethods iclude a ave ethod based o trasforatio; a ethod proposed by Cox; a odified versio of the Cox ethod; a ethod otivated by large-saple theory; ad a ethod based o geeralized p-value. Accordig to siulatio results the Cox ethod odified Cox ethod ad geeralized p-value are better tha other ethods. Therefore we cosider these three ethods ad copare the with our proposed CAT. estiatio for the variace of Y + S u S +. Cox has suggested that a test statistic for ( ) hypothesis test ca be derive as[4 5] 4 Z. S S + ( ) is give by A approxiate distributio for this statistics is stadard oral distributio. Therefore we ca reject H if Z > z a/ where z a/ is the upper -level cut-off poit stadard oral distributio. Also we ca use t-studet with - degrees of freedo. Therefore we reject H if Z > t a/- where t a/- is the upper -level cut-off poit t--distributio. Geeralized P-value: Geeralized p-value ca be used for iferece about paraeters whe there is uisace paraeter. [6] suggested the followig procedure for coputig a geeralized p-value for the log-oral ea; For a give dataset x i...x set yi I (x i) i... ad calculate y ad s u fro the data. For j to Geerate Z ~ () ad ~ ( ). Set u s ( ) u s Tj y Z +. et I j if > otherwise I j Set p j T j I j the () ( ) {p -p } is a ote Carlo i estiate of the geeralized p-value for testig. The Cat for ea: et X X...X is a rado saple fro desity f(x ) where the fuctioal for of f is assued to be kow. The paraeter is partitioed as () () ( ) where if available is the uisace paraeter ad is the paraeter of iterest. The ethodology of the CAT for testig H : at a desired level a is give through the followig steps (for ore detail see []. Step : Obtai Step : the E of Assue that H is true i.e. set H :. The () fid the E of fro the data agai. Call this as () the `restricted E of ' uder H deoted by () R. 846

3 World Appl. Sci. J. (6): Geerate artificial saple X X...X i.i.d. fro desity f( x ) a large uber of ties (say l () R ties). For each of these replicated saples () recalculated the E of ( ) (pretedig that were ukow). Retai oly the copoet that is relevat for et these recalculated E values of be. et... ()... ( ) Step 3: For testig H agaist : H A : < (if such a alterative is eaigful) at level a defie ( ). Reject H if <. Alteratively calculate the p-value as: (uber of ( )'s < ) l p I. < ) : ( l ( l ) For testig H agaist : > at (( ) ) H A level a defie. Reject H if >. Alteratively calculate the p-value as: (uber of ( )'s > ) l p I. > ) I.. (( /) ) (uber of ( l) 's < ) p ad be the ordered values of ( l ( l ) () For testig H : agaist H : defie the cut-off poits as ad Reject H if (( /) ) < <. Alteratively the p-value is coputed as: p i(p p ) where (uber of ( l) 's > ) p. The followig steps give the ipleetatio of our proposed CAT for hypothesis testig of log-oral proble. Step : Get the Es of the paraeters as where () Y + S b S b Step : ( l ) i b i i i Y Y S ( Y Y). Assue that is true i.e. The H Yi ~ ( () () () ) where is ukow. The Es of the paraeter () () which are called the restricted E' is () R + + yi i ( ). Geerate artificial saple Y...Y i.i.d. fro ( () () R R ) a large uber of ties (say ties). For each of these replicated saples recalculated the E of. et these recalculated E values of. be... ( l Yl + S ). bl et ()... be the ordered values of ( ) l. Step 3: The sae as Step 3 above with the exceptio that 's are used istead of ad is used i place of. ( l ) The size ad power coputatio for log-oral is doe through the followig stages. () For fixed ad geerate iid observatios of ( ) size fro () () where () +. Get () ad. Set ( value) ad get the restricted E () R H () of as. 847

4 World Appl. Sci. J. (6): Table : Carbo ooxide levels at a oil refiery i Califoria. Date 9//99 /4/99 /3/99 /3/99 //99 CO level Date 8/6/99 9//99 9//99 3/3/993 CO level Table : The actual size of tests whe the oial level is µ Test Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT µ Test Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT ow geerate Y (Y...Y ) i.i.d. fro ( () () R R ) I... Retai oly the E values of be.... Order these E values of as.... Get ( ) () ( ) (( ) ) get I I( ). ad (these are the lower ad upper a % cut-off poits). ow brig the fro the above Step- ad Repeat the above Step- through Step-5 a large uber of ties (say ties) ad get the I values as I I...I. Fially the power is approxiated by Table 3: the power of tests whe the oial level is with µ Test Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT µ Test Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT CAT I i i Real Exaple: The data i Table are ie easureets of carbo ooxide levels i the air. The easureets were ade close to a Califoria oil refiery i We will use these data to tests the ea carbo ooxide level. Iitial ivestigatios of these data ad of other siilar datasets idicate that a log-oral odel ay be appropriate. The data are posted at lib.stat.cu.edu/das/. The p-values for Cox ethod odified Cox ethod geeralized p-value ethod ad CAT for testig H : 3 vs H : 3 are ad.878 respectively. Therefore The four ethods do ot reject H. Siulatio Study: A siulatio study is perfored to copare size ad power test of four ethods; i) the Cox ethod ii) the odified Cox ethod iii) the geeralized p-value ethod ad iv) the CAT. 848

5 World Appl. Sci. J. (6): Table 4: The power of tests whe the oial level is with µ Test Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT µ Test Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT Cox odified Cox geeralized CAT For this propose we geerated saples with sizes fro a log-oral distributio with paraeters µ.5 ad ( - µ) 5 replicatio were used. We cosider the test H : 3 vs. H A : 3. The actual size of tests are give i Table ad the power of tests for 3.54 are give i Tables 3 ad 4 respectively. It ca be observed fro the below tables that the actual size of test of CAT ethod is always less tha the oial level however this ca ot be happe i the other ethods. I the geeralized p-value ethod the actual size is as good as the CAT ethod but its actual size is ofte ore tha the oial level. The other two ethods are very liberal. oreover i all ethods the actual size will be close to the oial level as the saple size icreases (Table ). REFERECES. Pal. W.K. i ad C.H. ig 7. A coputatioal approach to statistical ifereces J. Appl. Probab. Statist. : Chag C.H. ad. Pal 8. A Revisit to the Behres-Fisher Proble: Copariso of Five Test ethods Couicatios i statistics-siulatio ad Coputatio 37: Chag C.H.. Pal W. i ad J.J. i. Coparig several populatio eas: a paraetric bootstrap ethod ad its copariso with usual AOVA F test as well as AO Coputatioal Statistics 5: Zhou X.H. ad S. Gao 997. Cofidece itervals for the log-oral ea Statistics i ed. 6: Zhou X.H. S. Gao ad S.. Hui 997. ethods for coparig the eas of two idepedet log-oral saples Bioetrics 53: Krishaoorthy K. ad T. athew 3. Ifereces o the eas of logoral distributios usig geeralized p-values ad geeralized cofidece itervals J. Statistical Plaig ad Iferece 5: