Confidence intervals for the difference and the ratio of Lognormal means with bounded parameters

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1 Songklanakarn J. Sc. Technol. 37 () 3-40 Mar.-Apr Orgnal Artcle Confdence nterval for the dfference and the rato of Lognormal mean wth bounded parameter Sa-aat Nwtpong* Department of Appled Stattc Facult of Appled Scence Kng Mongkut Unvert of Technolog North Bangkok Bang Sue Bangkok 0800 Thaland. Receved: Augut 04; Accepted: 8 Januar 05 Abtract In th paper confdence nterval for the dfference between lognormal mean and the rato of lognormal mean wth retrcted parameter pace are propoed. The modfed confdence nterval perform well both for the coverage probablt and the expected length. We how thee reult va Monte Carlo mulaton. Keword: generalzed confdence nterval mple confdence nterval for lognormal mean mple confdence nterval for rato of lognormal mean Monte Carlo mulaton. Introducton In man practcal applcaton n varou area uch a engneerng cence and ocal cence t known that there ext bound on the value of unknown parameter. For example n engneerng proce control: the lfe tme of machne and the value of ome meaurement for controllng machne are bounded n cence: the weght or heght of ubject or the blood preure of patent are bounded. Alo n ocal cence the retrement age of publc ervant bounded. Stattcal nference for bounded parameter well known; ee e.g. Mandelkern (00) Feldman and Coun (998) Roe and Woodroofe (003) Wang (008) Wang (006) Wang (007) Gampaol and Snger (004) Nwtpong (03a 03b 03c). When nterval etmaton condered n a tuaton where the parameter to be etmated bounded t ha been argued that the clacal Neman procedure for ettng confdence nterval unatfactor. Th due to the fact that the nformaton regardng the retrcton mpl gnored. It therefore of gnfcant nteret to contruct confdence nterval for the parameter * Correpondng author. Emal addre: nw@kmutnb.ac.th that nclude the addtonal nformaton on parameter value beng bounded to enhance the accurac of the nterval etmaton. Recent paper b Mandelkern (00) Wang (006) Wang (007) Wang (008) Gampaol and Snger (004) are concerned manl wth the normal mean but n practce man applcaton hown that data from lognormal dtrbuton are ftted to the problem n envronment bolog health cence and phcal cence ee e.g. Nwtpong (03a 03b 03c) Kumar and Tpath (004) Chen and Zhou (006) Krnamoorth and Mathew (003) Kumar and Gbbon (004) Thpath et al. (009). Therefore n th paper we extend the work of Nwtpong (03a) to propoe new confdence nterval for the dfference between lognormal mean and the rato of lognormal mean the populaton mean are bounded b ung the approach preented b Zou et al. (009) and Donner and Zou (00). The propoed nterval are evaluated n term of coverage probablt and expected length va Monte Carlo mulaton.. Lognormal Dtrbuton and the Parameter of Interet be a random var- X LN and are repectvel the mean and varance of Y = Let X X X... X n able havng lognormal dtrbuton ~

2 3 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () ln(x ) ~ N lognormal dtrbuton. The probablt dent functon of the ln X exp ; 0 for X f X X 0 ; otherwe. The mean and the varance of X are repectvel exp / E X Var X exp exp. The parameter of nteret of th paper are repectvel and the parameter mean are bounded: a b a b where a b are contant. 3. Confdence Interval for Lognormal Mean Zhou and Gao (997) propoed the confdence nterval for lognormal mean whch exp cox where 4 4 S S S S S S cox Y Z / Y Z / [ lcox ucox] n( n ) n( n ) and Z / n j j S ( n ) Y Y ( / )00 th percentle of tandard normal dtrbuton. Krhnamoorth and Mathew (003) alo propoed the confdence nterval for the lognormal mean ung the generalzed confdence nterval propoed b Weerahand (993) and th confdence nterval gven b exp gc where T ( / ) T ( / ) [ l u ] gc gc gc T ( Y ) U ( n ) S / Z n U / n / ( n U n ) a tandard devaton of the oberved Y. and T( / ) the ( / *00)th percentle of T. In th cae the generalzed tet varable T ( Y ) a functon of ( Y ) and requred to atf the followng condton: A. For a fxed the probablt dtrbuton of T ( Y ) free of unknown parameter. A. The oberved value of T ( Y ) at Y mpl. A3. For the gven value of and the dtrbuton of T ( Y ) monotone dtrbuton n Krhnamoorth and Mathew (003) found that the confdence nterval exp gc ha a good coverage probablt compared to the confdence nterval exp. cox Zou et al. (009) propoed a mple confdence nterval for the um of parameter whch and the confdence nterval for (( ˆ ˆ ) Z var( ˆ ) var( ˆ ) ( ˆ ˆ ) 0 / Z var( ˆ ) var( ˆ )) / ˆ ˆ are etmator of n th cae ˆ Y ˆ / S Suppoe L U lu the confdence nterval for Zou et al. (009) found that th confdence nterval ˆ ˆ ˆ ˆ L l l ˆ ˆ ˆ ˆ U u u () For / the confdence nterval for are ( l u ) Y t S / n Y t S / n / / ( n )S ( n )S ( l u ) () / / Putng () nto () we fnall have the requred confdence nterval for the lognormal mean and we denote th confdence nterval a rov. 4. Confdence Interval for the Dfference between Lognormal Mean From confdence nterval we now contruct the confdence nterval for ung the ngle confdence nterval for each of 4. The generalzed confdence nterval for the dfference between lognormal mean Ung the ame dea a n ecton 3 we now propoed the generalzed confdence nterval for the dfference between lognormal mean whch T ( / ) T ( / ) [ l u ] GD gcd gc d gc rov

3 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () T Y Y ( ) exp( ) exp( ) = U Z / n U n S n n ( ) / ~. U / ( n ) 4. The mple confdence nterval for the dfference between lognormal mean Ung the ame dea a n ecton 3 we now propoed the mple confdence nterval for the dfference between lognormal mean baed on Zou et al. (009) whch L U r r where ( L U ) r r L ˆ ˆ ˆ exp( L ) exp( U ) ˆ r exp( ) exp( ) U ˆ ˆ U ˆ ˆ L r S S S ( n ) S L Y t / n n / n S S ( n ) S S U Y t / n n / n 5. Confdence Interval for the Rato of Lognormal Mean 5. The generalzed confdence nterval for the rato of lognormal mean From ecton 3 t ea to ee that the generalzed confdence nterval for the rato of lognormal mean where T ( / ) T ( / ) [ l u ] GS p p gc p gc p T Y Y ( ) p Z U / n n U n S n ( ) / ~. exp( ) exp( ) U / ( n ) (3) 5. The mple confdence nterval for the rato of lognormal mean Donner and Zou (00) propoed the confdence nterval for / b ettng and ettng 0 the confdence nterval for 0 L U where from L ˆ ˆ ˆ exp( L ) exp( U ) ˆ exp( ) exp( ) U ˆ ˆ U ˆ ˆ L. B ettng for L U L U 0 0 we have the confdence nterval rov p z z where exp( L ) exp( U ) exp( L ) exp( U ) ˆ ˆ ˆ ˆ ˆ ˆ L z exp( U )(ˆ exp( U )) z exp( L ) exp( U ) exp( U ) exp( L ) exp( L ) ˆ exp( L ) ˆ ˆ ˆ ˆ ˆ ˆ U (4) L U from (3) and the confdence nterval for the rato of lognormal mean baed on Donner and Zou (00) L U. z z 5.3 The mple confdence nterval for the rato of lognormal mean ung the eparate confdence nterval of Zhou and Gua (997) From ecton (3) we have the confdence nterval for the ngle lognormal mean whch 4 4 S S S S S S Y Z Y Z cox / / n ( n ) n ( n ) [L U ] c c In th cae we et L U (L U ) n c c (4). Hence we have the confdence nterval for the rato of lognormal mean baed on Zhou and Gua (997) whch where L U cox cox L U cox cox L U. z L U n (4) but replacng (L U ) z c c 6. The Confdence Interval for the Dfference between Lognormal Mean wth Retrcted Mean Conder for each bounded populaton mean we have

4 34 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () a b a b b a E(Y ) b E(Y ) E(Y ) a. b a Hence the varance of each data et bounded a well a the dfference between mean bounded; a b b a and b a b a b exp a a exp exp b From (5) we have e f Hence the dfference between mean bounded. Ung Wang (008) the generalzed confdence nterval between lognormal mean where max e l mn f u GDR gc d gc d [ l u ] T ( / ) T ( / ). gc d gcd Alo the mple confdence nterval between lognormal mean wth retrcted mean baed on Zou et al. (009) max e L mn f U r r L ˆ ˆ ˆ exp( L ) exp( U ) ˆ r exp( ) exp( ) U ˆ ˆ U ˆ ˆ L r S S S ( n ) S L Y t / n n / n S S ( n ) S S U Y t / n n / n. (5) 7. Confdence Interval for the Rato of Lognormal Mean wth Retrcted Mean It ea to ee that the rato of mean a b and ettng p p b a. Hence the rato of mean bounded. From ecton (5.) the generalzed confdence nterval for the rato of lognormal mean wth retrcted mean max p l mn p u GSR gc p gc p [ l u ] T ( / ) T ( / ). gc p gc p p p From ecton (5.) the mple confdence nterval for the rato of lognormal mean wth retrcted mean max p L mn p U R z z L U. z z rov p From ecton (5.3) the mple confdence nterval for the rato of lognormal mean wth retrcted mean baed on eparate confdence nterval of Zhou and Gua (997) max p L mn p U. cox cox In the next ecton we compare thee confdence nterval va coverage probablte and ther average length ung Monte Carlo mulaton 8. Smulaton Stude In th ecton we tart from the confdence nterval between lognormal mean wth retrcted populaton mean. We emploed Monte Carlo mulaton to etmate the coverage probablt and the average length for each confdence nterval:. We wrote GD GDR program n R(veron 3.0.3) and ettng the followng; n = at the level of gnfcance 0.05 wth 0000 mulaton run. For ( 6 6) we et ( ) = ( ) ( ) (0 0) ( ) ( ). For (0 6) we et ( ) (0..5.8) (3 3) (5.8 0.) and. All reult are n Table From Table there no dfference between coverage probablte of each confdence nterval and thee coverage probablte are cloe to or above 0.95 for almot cae. The rato of and greater than GD GDR parameter are cloe to the boundar of whch

5 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () Table. Coverage probablt and the rato of average length wdth of confdence nterval ( 6 6) GD GDR n n ( ) GD GDR GD GDR 0 ( ) ( ) (00) ( ) ( ) ( ) ( ) (00) ( ) ( ) ( ) ( ) (00) ( ) ( ) ( ) ( ) (00) ( ) ( ) ( ) ( ) (00) ( ) ( ) how that confdence nterval GDR and are horter than ther counterpart confdence nterval wherea the have the ame coverage probablte. From Table the rato of confdence nterval GD ( 6 6) hown that the GDR average length wdth of confdence nterval are lghtl horter than the confdence nterval GD GDR for mall ample ze other cae are almot the ame. From Table 3 all coverage probablte and average length wdth of confdence nterval GD GDR are not dfferent from Table Table 4 how that average length wdth of each nterval of (0 6) are GD GDR not dfferent. The econd part of mulaton reult the confdence nterval for the rato of lognormal mean wth retrcted mean are alo GD GDR condered. For ( 6 6) we et ( ) ( 5) (0 0) ( 5) ( ). For (0 6) we et ( ) (5) (3 3) ( ). All reult are n Table 5-8. From Table 5 we found that coverage probablte of confdence nterval GS GSR R are not dfferent ( 6 6). The are cloe to or above the nomnal level of 0.95 for almot cae. The rato of average length of and GS GSR greater than the parameter value of are cloe to the boundar of ( 6 6) for whch n th cae are ( ) and ( ); a a reult confdence nterval GSR and R are horter than ther counterpart confdence nterval:. GS From Table 6 column 3 and 4 we found that a confdence nterval horter than other confdence nterval wherea confdence nterval GS horter than confdence nterval GSR for mall ample

6 36 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () Table. The rato of average length wdth of confdence nterval GD GDR ( 6 6) n n ( ) GD GDR 0 ( ) ( ) (00) ( ) ( ) ( ) ( ) (00) ( ) ( ) ( ) ( ) (00) ( ) ( ) ( ) ( ) (00) ( ) ( ) ( ) ( ) (00) ( ) ( ) ze. In other cae there are no dfference between thee confdence nterval. From Table 7 there are no dfference of coverage probablte between thee confdence nterval (0 6). GS GSR R All coverage probablte are cloe to or above the nomnal level of 0.95 for almot cae. The rato of average length wdth of confdence nterval GS GSR and greater than parameter R value of are cloe to the boundar of ( 6 6) whch are (5.90.5) and (0.55.9). A a reult confdence nterval GSR and R are horter than ther counterpart confdence nterval:. GS Smlarl to the reult n Table 4 and 6 Table 8 how from column 3-4 that confdence nterval horter than confdence nterval. Alo confdence nterval GS horter than confdence nterval GS for mall ample ze. In addton the ame reult appear from column 6-7 hence confdence nterval horter than confdence nterval and there GSR R no dfference between confdence nterval R and GSR but confdence nterval R eaer to compute than confdence nterval GSR whch baed on computatonal approach. 9. Concluon In th paper we propoe new confdence nterval for the dfference between lognormal mean and the rato of lognormal mean wth retrcted populaton mean. For the confdence nterval for the dfference between lognormal mean the mple confdence nterval baed on Zou et al. (009) and Donner and Zou (00) outperform other confdence nterval for mall ample ze wherea for other cae one can ue both the generalzed confdence nterval

7 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () Table 3. Coverage probablt and the rato of average length wdth of confdence nterval (0 6) GD GDR n n ( ) GD GDR GD GDR 0 (0.5.8) (33) (5.80.) (0.5.8) (33) (5.80.) (0.5.8) (33) (5.80.) (0.5.8) (33) (5.80.) (0.5.8) (33) (5.80.) Table 4. The rato of average length wdth of confdence nterval GD GDR (0 6) n n ( ) GD GDR 0 (0.5.8) (33) (5.80.) (0.5.8) (33) (5.80.) (0.5.8) (33) (5.80.) (0.5.8) (33) (5.80.) (0.5.8) (33) (5.80.) or confdence nterval but confdence GDR nterval eaer to ue than confdence nterval whch baed on computatonal approach. The GDR confdence nterval of the rato of lognormal mean perform better that other a t provde a horter expected length and th confdence nterval ha the coverage probablt cloe to or above the nomnal level of The reult n th paper are alo extended to contruct the confdence

8 38 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () Table 5. Coverage probablt and the rato of average length wdth of confdence nterval ( 6 6) GD GDR n n ( ) GS GSR GS GSR R R 0 ( ) (-5) (00) (-5) ( ) ( ) (-5) (00) (-5) ( ) ( ) (-5) (00) (-5) ( ) ( ) (-5) (00) (-5) ( ) ( ) (-5) (00) (-5) ( ) nterval for the rato and the dfference between lognormal coeffcent of varaton wth retrcted mean. Acknowledgment Th reearch wa upported b grant number 556 A90007 from Kng Mongkut Unvert of Technolog North Bangkok. We thank the atant managng edtor and the referee for helpful comment that greatl mproved the manucrpt. Reference Chen Y. H. and Zhou X. H. 006b. Interval etmate for the rato and dfference of two lognormal mean. Stattc n Medcen Donner A. and Zou G.Y. 00. Cloed-form confdence nterval for functon of the normal mean and tandard devaton. Stattcal Method n Medcal Reearch Feldman G.J. and Coun R.D Unfed approach to the clacal tattcal anal of mall gnal. Phcal Revew D Gampaol V and Snger J.M Comparon of two normal populaton wth retrcted mean. Computatonal Stattc and Data Anal Mandelkern M. 00. Settng confdence nterval for bounded parameter. Stattcal Scence Nwtpong S. 03a. Confdence nterval for the mean of Lognormal dtrbuton wth retrcted parameter Space. Appled Mathematcal Scence. 7 (4) Nwtpong S. 03b. Confdence nterval for the rato of mean of Lognormal dtrbuton wth retrcted parameter Space. Appled Mathematcal Scence. 7 (04) Nwtpong S. 03c. Confdence nterval for coeffcent of varaton of lognormal dtrbuton wth retrcted parameter pace. Appled Mathematcal Scence. 7 (77-80)

9 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () Table 6. The rato of average length wdth of confdence nterval ( 6 6) GS GSR R n n ( ) (-66) GS GS GSR R GSR R 0 ( ) (-5) (00) (-5) ( ) ( ) (-5) (00) (-5) ( ) ( ) (-5) (00) (-5) ( ) ( ) (-5) (00) (-5) ( ) ( ) (-5) (00) (-5) ( ) Krhnamoorth K. and Mathew T Inference on the mean of lognormal dtrbuton ung Generalzed p- value and generalzed confdence nterval. Journal of Stattcal Plannng and Inference Kumar D. and Gbbon D An upper predcton lmt for the arthmetc mean of a lognormal random varable. Technometrc Kumar M and Trpath M Etmatng a potve normal mean. Stattcal Paper n Roe B.P. and Woodroofe M.B Settng confdence belt. Phcal Revew D Trpath M Kumar S. and Srvatava T Etmatng the Mean of a Lognormal Populaton under Retrcton. Internatonal Journal of Appled Mathematc and Stattc Wang H Modfd P-value of two-ded tet for normal dtrbuton wth retrcted parameter pace. Communcaton of Stattc Theor and Method Wang H Modfed p-value for one-ded tetng n retrcted parameter pace Stattc & Probablt Letter Wang H Confdence nterval for the mean of a normal dtrbuton wth retrcted parameter pace. Journal of Stattcal Computaton and Smulaton. 78(9) Weerahand S Generalzed confdence nterval. Journal of the Amercan Stattcal Aocaton Zhou X.H. and Gao S. 997.Confdence nterval for the lognormal Mean. Stattc n Medecen Zou G.Y. Huo C.Y. and Taleban J Smple confdence nterval for lognormal mean and ther dfference wth envronmental applcaton. Envronmetrc

10 40 S. Nwtpong / Songklanakarn J. Sc. Technol. 37 () Table 7. Coverage probablt and the rato of average length wdth of confdence nterval (0 6) GD GDR n n ( ) GS GSR GS GSR R R 0 (5.90.5) (5) (33) (0.55.9) (5.90.5) (5) (33) (0.55.9) (5.90.5) (5) (33) (0.55.9) (5.90.5) (5) (33) (0.55.9) (5.90.5) (5) (33) (0.55.9) Table 8. The rato of average length wdth of confdence nterval (0 6) GD GDR n n ( ) GS GS GSR R GSR R 0 (5.90.5) (5) (33) (0.55.9) (5.90.5) (5) (33) (0.55.9) (5.90.5) (5) (33) (0.55.9) (5.90.5) (5) (33) (0.55.9) (5.90.5) (5) (33) (0.55.9)