Confidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation
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1 Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology North Bagkok, Thailad Abstract--Motivated by the recet work of Herbert, Haye, Macaskill ad Walter [Iterval estimatio for the differece of two idepedet variaces. Commuicatios i tatistics, imulatio ad Computatio, 40: , 0.], we ivestigate, i this paper, the ew cofidece iterval for the differece betwee two ormal populatio stadard deviatios based o the simple cofidece iterval of Doer ad Zou [Closed-form cofidece itervals for fuctios of the ormal mea ad stadard deviatio, -, 00.]. For a sigle cofidece iterval for a stadard deviatio, we derived aalytic expressios to fid the coverage probability ad its expected legth compared with the stadard cofidece iterval. Mote Carlo simulatio results for the differece of stadard deviatios are give to compare proposed cofidece itervals. Keywords-- Coverage probability, expected legth, variaces I. INTRODUCTION Recetly, Herbert et al. [] have argued that methods ad aalyzes for costructig the cofidece iterval estimatio of the differece betwee variaces have ot bee described. They proposed a simple aalytical method to costruct a cofidece iterval for the differece betwee two idepedet samples. It is see that their proposed cofidece iterval works well whe observatios are highly skewed ad leptokurtic. Cojbasica ad Tomovica [] showed that the oparametric cofidece iterval of the populatio variace of two-sample problem based o t-statistic combied with bootstrap techiques performs very well whe data are from expoetial families. Phoyiem ad Niwitpog [5] proposed the ew geeralized cofidece iterval for the differece betwee the ormal variaces.their proposed iterval performs very well compared to the existig cofidece iterval. Related works of the cofidece iterval for the differece betwee two variaces, the reader is referred to the refereces cited i the above papers. I this paper, we emphasize oly data are from the ormal distributio ad we propose the ew cofidece iterval for the sigle ad the differece of two ormal populatio stadard deviatios, based o the simple cofidece iterval of []. For the cofidece iterval for a stadard deviatio, we derived aalytic expressios to fid coverage probability ad expected legth. We also preset the Mote Carlo simulatio results for the cofidece itervals for the differece of stadard deviatios. II. CONFIDENCE INTERVAL FOR THE DIFFERENCE OF TWO NORMAL POPULATION TANDARD DEVIATION Let ad be radom samples from two idepedet ormal distributios with meas, ad stadard deviatios,, respectively. The sample meas ad sample variaces for ad are, respectively, deoted as X, Y, ad whe m X X i, Y m Yi, m ( X i X ) ad ( Yi Y ). We are m i i iterested i cofidece itervals for ad. i i IN: Page
2 A. Cofidece Iterval for The igle tadard Deviatio It is well kow that for a give sample variace X which is the is the Chi-square distributed with a real umber, deviatio: where 0,. X X ( i ) i degrees of freedom. For give We ca costruct the cofidece iterval for a stadard P,, P,, P,,. 00 % cofidece iterval for is A From (), a CI,.,, 00 % cofidece iterval for is therefore CI,, /,. () () B. Cofidece Iterval for The tadard Deviatio with Kow a Coefficiet of Variatio Cosider / where is a coefficiet of variatio, we the have. For a 00 % cofidece iterval for, whe a coefficiet of variatio is kow, ca be easily solved which is CI X c0 /, X c0 / () where c 0 is a 00( / )% percetile of a stadard t-distributio with degrees of freedom. We ow derive coverage probabilities ad expected legths of cofidece itervals CI compared to CI. Theorem The coverage probability ad the expected legth of are respectively E[ ( W ) ( W )] ad ( c c ) / ()( (/ ) / (( ) / ) where IN: Page
3 W c, W c c c () ( (/ ) / (() / )). c, c, c ad,, Proof: Followig [4], the coverage probability of CI is P,, P,, P,, P,, P c c, c, c,, P c c c c P, c () ( (/ ) / (() / )) c c c c c P Z c c, if { W Z W} E[ I{ W Z W }( )], I{ W Z W }( ) 0, otherwise E E I where Z ~ N (0; ). [ [ { W Z W }( )] ] E[ ( W ) ( W )] Note that, Var( ) E( ) (E( )) ( / () (/ ) / (() / ) ( ( / ())( (/ ) / (() / ) ). The expected legth for CI is therefore E CI ( c c ) E( ) ( c c ) / ()( (/ ) / (() / ). IN: Page
4 Theorem The coverage probability ad the expected legth of are c0 respectively, E[( W ) ( W )] ad / ( )( (/ ) / (( ) / ). Proof: Followig [4], the coverage probability of CI is 0 / 0 / / / / / P X c / X c / 0 0 P X c X c P X c X c 0 0 X c0 X c0 P c c c / / X c0 X c0 P Z,c ( / ())( (/ ) / (() / ) c c, if { W Z W} X c0 / E[ I{ W Z W }( )], I{ W Z W }( ), W, 0, otherwise c X c 0 / E[ E[ I{ W Z W }( )] ], W c E[( W ) ( W )] The expected legth of CI is therefore 0 / 0 / E c0 / E CI E X c X c c0 E( ) c0 / ()( (/ ) / (() / ). C. Cofidece Itervals for the Differece betwee Normal tadard Deviatios Doer ad Zou [] proposed the cofidece iterval for the differece betwee which is CI L U, ˆ ˆ ˆ ˆ, where L l u U ˆ ˆ u ˆ ˆ l. Plug i ˆ ˆ, c4, c5, c / ()( (/ ) / (() / ), c / (m)( (m/ ) / ((m) / ), 4 5 IN: Page 4
5 ( ) ( ) ( m ) ( m ) ( l, u ),, ( l, u), /,( ) /,( ) /,( m) /,( m) have a ew cofidece iterval for. i to CI L U,, we Aother cofidece iterval for is CI L U,, 4 where ˆ ˆ ˆ ˆ, U u l4 L l u 4 ( l, u ) X c /, X c /, ( l, u ) Y c / m, Y c / m. 4 4 ˆ ˆ ˆ ˆ, III. IMULATION REULT I this sectio, usig the R program (versio.0), we compare our proposed cofidece itervals via Mote Carlo simulatio. We set the sample m 0, 0, 50,00, 00,(, ) (0.05, 0.05), (0.05, 0.0),(0.0, 0.0),(0.0, 0.0), (0.0, 0.0),(0.0, 0.0),(0.0, 0.0),(0.0, 0.40),(0.40, 0.40),(0.40, 0.50),(0.50, 0.50),(0.50, 0.60), (0.60,0.60),(0.60,0.70),(0.70,0.70),(0.70,0.80),(0.80,0.80),(0.80,0.90),(0.90,0.90),(0.90,.00). The coverage probabilities ad the expected legths for cofidece itervals CI L, U to CI L, U are reported i Table with 0, 000 simulatio rus. 4 TABLE COVERAGE PROBABILITY AND EXPECTED LENGTH FOR CONFIDENCE CI, CI WITH INTERVAL 4 M COV COV LENGH LENGTH compared IN: Page 5
6 COV stads for the coverage probability for CI COV stads for the coverage probability for CI 4 LENGTH stads for the expected legth of CI LENGTH stads for the expected legth of CI 4 TABLE (CONTINUE) m COV COV LENGH LENGTH IN: Page 6
7 TABLE (CONTINUE) m COV COV LENGH LENGTH COV stads for the coverage probability for CI COV stads for the coverage probability for CI 4 LENGTH stads for the expected legth of CI LENGTH stads for the expected legth of CI 4 IN: Page 7
8 To compare cofidece itervals, we prefer cofidece iterval which has coverage probability at least 0.95 ad has shortest expected legth. From Table, we prefer the cofidece iterval CI L U CI L, U whe =0, 0,50, 00, 00 ad to the cofidece iterval, 4 small values of coefficiets of variatio i.e., 0.5. CI L, U which has a coverage otherwise we choose the cofidece iterval 4 probability better tha the cofidece iterval CI L U., IV. CONCLUION I this paper, we drive aalytics expressios to fid the coverage probability ad the expected legth for the cofidece iterval for a stadard deviatio CI compared to CI. For the cofidece iterval for the differece betwee stadard deviatios, we compared CI to CI, 4 Mote Carlo simulatios are carried out. It tured out that CI performed better tha CI 4 for small ad moderate values of coefficiets of variatio, i.e., 0.5. The large values of the quatities,, resultig to the better coverage probabilities of CI 4 ad also a larger of expected legths of CI 4. ACKNOWLEDGEMENT We are appreciated the fudig from Faculty of Applied cieces, Kig Mogkut s Uiversity of Techology North Bagkok. REFERENCE [] V. Cojbasica, A. Tomovica, Noparametric cofidece Itervals for populatio variace of oe sample ad the differece of variaces of two samples, Computatioal tatistics & Data Aalysis, vol. 5, pp , 007. [] R.D. Herbert, P. Haye, P. Macaskill, D. Walter, Iterval estimatio for the differece of two idepedet variaces, Commuicatios i tatistics: imulatio ad Computatio, vol. 40, pp , 0. [] A. Doer, G.Y. Zou, Closed-form cofidece itervals for fuctios of the ormal mea ad stadard deviatio, tat Methods Med Res, pp. -, 00. [4]. Niwitpog,. Niwitpog, Cofidece iterval for the differece oftwo ormal populatio meas with a kow ratio of variaces, Applied Mathematical cieces, vol. 4, pp , 00. [5] W. Phoyiem,. Niwitpog, Geeralized cofidece iterval For the differece betwee ormal populatio variaces, Far East Joural of Mathematical cieces, vol. 69, pp. 99-0, 0. IN: Page 8
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