Vol:7, No:9, 03 Cofidece Itervals for the Coefficiets of Variatio with Bouded Parameters Jeerapa Sappakitkamjor, Sa-aat Niwitpog Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:9, 03 waset.org/publicatio/678 Abstract I may practical applicatios i various areas, such as egieerig, sciece ad social sciece, it is kow that there exist bouds o the values of ukow parameters. For example, values of some measuremets for cotrollig machies i a idustrial process, weight or height of subjects, blood pressures of patiets ad retiremet ages of public servats. Whe iterval estimatio is cosidered i a situatio where the parameter to be estimated is bouded, it has bee argued that the classical Neyma procedure for settig cofidece itervals is usatisfactory. This is due to the fact that the iformatio regardig the restrictio is simply igored. It is, therefore, of sigificat iterest to costruct cofidece itervals for the parameters that iclude the additioal iformatio o parameter values beig bouded to ehace the accuracy of the iterval estimatio. Therefore i this paper, we propose a ew cofidece iterval for the coefficiet of variace where the populatio mea ad stadard deviatio are bouded. The proposed iterval is evaluated i terms of coverage probability ad expected legth via Mote Carlo simulatio. Keywords Bouded parameters, coefficiet of variatio, cofidece iterval, Mote Carlo simulatio. I. INTRODUCTION HE coefficiet of variace (CV) has bee oe of the most Twidely used statistical measures of the relative dispersio sice it was itroduced by Karl Pearso i 896 []. This is due to its importat property such that it is a dimesioless (uit-free) measure of variatio ad also its ability that ca be used to compare several variables or populatios with differet uits of measuremet [], [3]. As a result, it has bee frequetly used i umerous fields of kowledge. Here are some examples of the use of the CV. I sciece, the CV is ofte used as a measure of precisio of measuremet, ad also used to compare the precisio of laboratory experimets or techiques. I egieerig, it is commoly used to evaluate the variability of stregth of buildig materials. It is defied as a referece parameter for measuremets i cliical diagostics i medicie, ad treated as a measure of risk to retur i fiace [3] [5]. Recet applicatios of the CV i busiess, climatology ad other fields are briefly reviewed i [6]. I most situatios, the CV of a populatio is practically ukow. Therefore the sample CV is required to estimate the ukow value. However, for statistical iferece purpose ad Jeerapa Sappakitkamjor is with the Departmet of Applied Statistics, Kig Mogkut s Uiversity of Techology North Bagkok, Bagkok 0800 Thailad (phoe: +66 555-000 Ext. 499; fax: +66 585-605; e-mail: jsj@ kmutb.ac.th). Sa-aat Niwitpog is with the Departmet of Applied Statistics, Kig Mogkut s Uiversity of Techology North Bagkok, Bagkok 0800 Thailad (e-mail: sw@kmutb.ac.th). to make best use of the sample CV, it is ecessary to costruct a cofidece iterval for the populatio CV. This is simply due to the fact that a cofidece iterval provides much more iformatio about the parameter of iterest tha does a poit estimate. As discussed i [7], the cofidece iterval is more iformative tha is the poit estimates itself, because it cotais all plausible values for the estimate of the ukow parameter with a specified level of cofidece. I additio, the width of the cofidece iterval shows how accurate we believe our estimate is, i.e., the smaller width, the more precise our estimate of the parameter. I geeral, cofidece itervals of scale parameters, whe parameter space is restricted, have received little attetio. The developmet i this area is cocetrated o locatio parameters (the populatio mea ad the differece of two meas) as preseted i [8] []. Although there have bee a umber of the cofidece itervals for the CV, as appeared i recet literature [3], [4], the cofidece itervals for the CV with restricted parameters have ot much bee doe. It is, therefore, of sigificat iterest to costruct cofidece itervals for the scale parameters. I this study we choose the CV as a parameter of our iterest because of its widespread use i describig the variatio withi a data set. Moreover, amog scale parameters, the CV is a more iformative quatity tha others. As oted i [5], the CV is preferred to the variace or stadard deviatio i various fields of iterest, especially i biological ad medical research. The rest of the paper is orgaized as follows. Sectio II provides cofidece itervals for the CV obtaied by several existig methods whe data are ormally distributed. The cofidece itervals for the CV whe the populatio mea ad stadard deviatio are bouded are proposed i Sectio III. Sectio IV presets the results of simulatio studies ad reports o the performace of the proposed cofidece itervals. Fially, Sectio V gives a coclusio with a few remarks. II. CONFIDENCE INTERVAL FOR COEFFICIENT OF VARIATION FROM NORMAL DISTRIBUTION I this sectio we review some of the existig methods for costructig cofidece itervals of the CV whe data are ormally distributed. The populatio CV (deoted as τ ) is defied as a ratio of the populatio stadard deviatio (σ ) to ), i.e., the populatio mea ( μ, μ 0 σ τ = () μ Iteratioal Scholarly ad Scietific Research & Iovatio 7(9) 03 46 scholar.waset.org/307-689/678
Vol:7, No:9, 03 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:9, 03 waset.org/publicatio/678 Let x, x, x3,..., x be a idepedetly ad idetically distributed (iid) radom sample of size from a ormal distributio, N( μ, σ ). The sample mea ( x ) ad sample variace ( s ) are the ubiased estimates of μ ad σ, respectively. Therefore the typical sample estimate of τ is give as i i cv = s x where x = = x ad s = i = ( i ). x x To costruct a cofidece iterval of the CV, there are several methods available. I this study we cosider 6 methods amely Miller s, s, s, two ew methods proposed by Mahmoudvad, ad Hassai [] ad the Method of Variace Estimates Recovery () [6]. The lower ad upper cofidece limits of the 00( α )% cofidece iterval for τ from each method are obtaied by the followigs. A. Miller s Cofidece Iterval Miller [] proposed a cofidece iterval based o the sample CV that approximates a asymptotic ormal distributio. Miller s method, referred to as Mil, has cofidece limits [ L Mil, U Mil ] give by where z α / / cv LMil = cv z α / + cv / cv UMil = cv + z α / + cv () (3) is the 00( α /)% percetile of the stadard ormal distributio. B. s Cofidece Iterval [7] developed a cofidece iterval for ormal CV by usig the approximatio method. s method, referred to as McK, has cofidece limits[ L Mck, U Mck ] give by / u u LMck = cv cv + / u u U Mck = cv cv + where u = χ( ), ad u α / = χ are respectively ( ), α / the 00( α /) ad 00( α /) percetile of the chi-square distributio with ( ) degrees of freedom. (4) C. s Cofidece Iterval [8] modified a cofidece iterval proposed by to obtai a early exact iterval. s method, referred to as Va, provides cofidece limits slightly differet from those obtaied by s i (4); s cofidece iterval [ L Va, U Va ] is give by / u + u LVa = cv cv + / u + u U Va = cv cv + D. Mahmoudvad, ad Hassai s Cofidece Itervals Mahmoudvad, ad Hassai [] itroduced two ew cofidece itervals for the CV whe data are ormally distributed. They are respectively referred to, i this study, as M&H(I) s ad M&H(II) s. M&H(I) s cofidece iterval: by usig the ormal approximatio, its cofidece limits [ LH& M( I), U H& M( I) ] are give by LH& M( I) = cv c + z α / c UH& M( I) = cv c z α / c / where c = ( / ( )) [( Γ( / ) / ( Γ(( ) / )]. M&H(II) s cofidece iterval: Mahmoudv ad, ad Hassai itroduced a ew approximate poit estimator ˆ τ = cv /( c ) for τ. They showed that the ew estimator ot oly gives smaller variace tha the typical estimator cv but it is also asymptotically ubiased. The M&H(II) s cofidece limits [ L, U ] are give by H& M( II) H& M( II) ˆ τ ˆ τ L ˆ H& M( II) = τ z α / ( c) + c ˆ τ ˆ τ U ˆ H& M( II) = τ + z α / ( c) + c E. s Cofidece Iterval Doer ad Zou [6] preseted closed-form cofidece itervals for fuctios of the ormal mea ad stadard deviatio icludig the coefficiet of variatio. By usig the method of variace estimates recovery or, referred to as MOV, cofidece limits [ L MOV, U MOV ] for the CV based o separate cofidece limits computed for the mea ad stadard deviatio are give by (5) (6) (7) Iteratioal Scholarly ad Scietific Research & Iovatio 7(9) 03 47 scholar.waset.org/307-689/678
Vol:7, No:9, 03 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:9, 03 waset.org/publicatio/678 where LMOV = ( x max{ 0, x + ac( a )}) s / c UMOV = ( x + max{ 0, x + bc( b )}) s / c a = ( )/ u b = ( )/ u c = x zα / s /. III. CONFIDENCE INTERVAL FOR COEFFICIENT OF VARIATION WITH BOUNDED PARAMETERS Followig the idea preseted by Wag [9], we propose cofidece itervals for the CV whe the ukow parameters μ ad σ are bouded. Although, a true value of a parameter of iterest is practically ukow, the parameter space is ofte kow to be restricted ad the bouds of the parameter space are kow. Whe a parameter to be estimated is bouded, it is widely accepted that a cofidece iterval for a parameter θ whe a < θ < b is the cofidece iterval of the itersectio betwee a < θ < b ad [ Lθ, Uθ ], where L θ ad U θ are lower ad upper limits of the cofidece iterval for θ, thus i this situatio the cofidece iterval for θ, deoted as CI θ, is give by ( ) ( ) CIθ = max a, Lθ, mi b, U (9) θ There are four possible outcomes for the cofidece iterval i (9) as follows: a) if a > L θ ad b > U θ the CI θ is reduced to CI = a, U (0) θa b) if a > L θ ad b < U θ the CI θ is reduced to CI = a, b () θb c) if a < L θ ad b > U θ the CI θ is reduced to CI = L, U () θ θc θ θ d) if a < L θ ad b < U θ the CI θ is reduced to CI = L, b (3) θd Whe the populatio mea is bouded, say a < μ < b where 0 < a < b, it is straight forward to show that the populatio variace ad the stadard deviatio are also bouded as follows: θ (8) where σ a < μ < b a < μ < b b < μ < a Xi Xi Xi b < μ < a N N N σb < σ < σa σb < σ < σa Xi a = a ad N σ Xi b = b. N Similarly, the bouded populatio mea ca lead to the bouded populatio CV. a < μ < b > > a μ b < < b μ a σ σ σ < < b μ a Sice σ b < σ < σ, we have a σ σ σ b < < a b μ a σ σ b < τ < a b a Thus the CV is also bouded whe the mea ad stadard deviatio are bouded. Accordig to Wag [9] ad Niwitpog [0], the proposed cofidece iterval for τ with bouded mea ad stadard deviatio based o (9) is give by σ b σ a CI τ = max, L, mi, U b τ a τ (4) Equatio (4) ad cofidece limits from existig methods preseted i Sectio II are the used to obtai cofidece itervals for τ whe the mea ad stadard deviatio are bouded. IV. SIMULATION STUDIES I this study, we examie the performace of the proposed cofidece itervals for the CV of a ormal distributio uder the additioal iformatio that the populatio meas lies i some bouded iterval. I additio, we compare the proposed cofidece itervals to those obtaied from the existig methods i terms of coverage probability ad average legth of the cofidece itervals. Simulatio studies usig differet values of sample size ( = 5, 0, 5, 5, 50, ad 00) ad coefficiets of variatio (CV = 0.05, 0.0, 0.0, 0.33, ad 0.50) are cosidered. Without loss of geerality, the populatio variace is set to, i.e., we cosider a sample take from a populatio that has N( μ, ), where μ is adjusted to get the required CV. Thus μ =, 3, 5, 0, ad 0. Each value of μ is set to lie i a bouded iterval that has two stadard deviatio wide. The 95% cofidece itervals are Iteratioal Scholarly ad Scietific Research & Iovatio 7(9) 03 48 scholar.waset.org/307-689/678
Vol:7, No:9, 03 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:9, 03 waset.org/publicatio/678 costructed based o the existig methods with ubouded ad bouded parameters. The results via Mote Carlo simulatio with 0,000 rus for each combiatio of ad CV, usig fuctios writte i R, are summarized i Tables I ad II. By detailig the estimated coverage probabilities ad the average legths (i paretheses) for the 95% cofidece itervals based o six methods icludig sample sizes ad the correspodig CV, Tables I ad II preset the simulatio results for the cases of ubouded ad bouded parameters, respectively. I terms of coverage probabilities, the results i Tables I ad II have idetical coverage probabilities for sample size larger tha 5. All methods perform very well as the coverage probabilities exceed the omial level of 95% ad reached 00%. However, the average legths of the proposed itervals from all methods are similar or shorter i legth. Whe = 5 ad CV=0.05, Miller s method ad s method are iferior to the other methods i terms of coverage probabilities. As see i Tables I ad II, these two methods have coverage probabilities lower tha the omial level. This is because; their cofidece itervals are much shorter tha those obtaied by other methods. For CV = 0.50, s itervals become egative ad have otably wider iterval legths tha the other itervals. I most cases, the coverage probabilities of the proposed itervals that iclude the bouds of parameters i Table II ad those of the existig itervals that exclude bouds of parameters i Table I are ot oly higher tha the omial level but also idetical. Thus, i order to access the performace of cofidece itervals, the average legths are compared. For this purpose, the ratios of average legths from ubouded itervals to those from bouded itervals are calculated ad preseted i Table III. If the ratio is greater tha oe, the average legths from ubouded itervals are wider tha those from bouded itervals. It ca be easily see that the ratios are equal or greater tha oe i most cases, i.e., the proposed cofidece itervals have arrower widths. We ca coclude that the proposed cofidece itervals are superior to the itervals obtaied from the existig methods that do ot take the bouds of parameters ito accout. TABLE I COVERAGE PROBABILITY AND AVERAGE LENGTH OF 95% CONFIDENCE INTERVALS FOR CV WITH UNBOUNDED PARAMETERS Method CV 0.05 0.0 0.0 0.33 0.50 5 Miller 0.848 0.977 0.997 (0.065) (0.3) (0.74) (0.508) (0.936) 0.95 0.998 (0.09) (0.8) (0.598) (.84).350) 0.95 0.998 (0.08) (0.0) (0.484) (0.97) (.73) 0.944 0.997 (0.093) (0.86) (0.37) (0.634) 0.987) 0.797 0.96 0.99 0.999 (0.056) (0.3) (0.3) (0.44) 0.76) 0.954 0.998 (0.08) (0.4) (0.58) (.5) (0.074) 0 Miller 0.045) 0.09) 0.89) 0.340) (0.588) 5 Miller 5 Miller 50 Miller 0 0 Miller 0.056) (0.4) 0.45) 0.58) (.09) (0.055) (0.3) (0.38) (0.46) 0.930) (0.05) (0.05) (0.0) (0.35) (0.536) (0.04) (0.086) (0.8) 0.340) (0.6) (0.057) (0.) (0.3) (.083) (-0.97) (0.036) 0.073) (0.5) (0.73) (0.465) 0.04) 0.084) (0.79) (0.343) (0.740) 0.04) (0.084) (0.76) (0.38) (0.645) 0.040) 0.080) 0.6) 0.69) (0.407) 0.035) (0.07) (0.55) (0.30) (0.568) 0.043) (0.095) (0.63) 0.95) 5.95) 0.08) (0.057) (0.7) (0.09) (0.35) (0.030) 0.06) (0.8) (0.36) (0.44) (0.030) (0.06) 0.7) (0.3) (0.48) 0.030) 0.059) 0.9) (0.98) (0.98) (0.08) (0.057) (0.9) (0.66) (0.58) 0.03) 0.074) 0.9) (0.84) (.867) (0.00) (0.040) (0.08) (0.46) 0.44) 0.0) 0.04) 0.086) (0.55) 0.7) 0.0) 0.04) (0.085) (0.54) (0.67) 0.00) (0.040) (0.08) (0.35) 0.03) (0.00) 0.043) 0.04) 0.33) 0.480) 0.03) 0.059) (0.06) (0.794) (8.440) 0.08) (0.058) 0.03) (0.7) (0.09) (0.059) 0.06) (0.8) 0.04) 0.09) 0.059) 0.06) (0.80) 0.08) (0.056) (0.094) (0.4) (0.033) (0.089) (0.5) (0.459) 0.07) (0.050) (0.95) (0.778) 8.79) V. CONCLUSIONS This study looks at the performace of the proposed cofidece iterval for the ormal populatio CV by takig ito accout the bouds of the populatio mea ad stadard deviatio. We cosider six existig methods for costructig cofidece itervals for the ormal populatio CV ad apply them ito two mai situatios, ubouded ad bouded parameter spaces. The importat result from this research is that whe we take ito accout the bouds of the populatio mea ad stadard deviatio, the cofidece iterval obtaied from each method provides a smaller width i most cases. Moreover, the results Iteratioal Scholarly ad Scietific Research & Iovatio 7(9) 03 49 scholar.waset.org/307-689/678
Vol:7, No:9, 03 Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:9, 03 waset.org/publicatio/678 of this study ot oly provide useful isights ito the iterval estimatio whe parameters beig estimated are bouded but also support the argumet by Koopmas et al. [] that with some prior iformatio about the rage of the parameter μ, it is possible to obtai cofidece itervals for τ that have fiite legth with probability oe for all values of μ adσ. A key advatage of addig the bouds of parameters ito accout is that we are certai that the cofidece limits ever go beyod their parameter bouds. As a result, it should be oted that if there is a presece of outliers i a data set, the width of a cofidece iterval obtaied by icludig the bouds of the parameter space will be less effect from observatios with extreme values. TABLE II COVERAGE PROBABILITY AND AVERAGE LENGTH OF 95% CONFIDENCE INTERVALS FOR CV WITH BOUNDED PARAMETERS Method 5 Miller 0 Miller 5 Miller 5 Miller 50 Miller CV 0.05 0.0 0.0 0.33 0.50 0.838 0.975 0.996 0.999 (0.050) (0.0) (0.4) (0.38) (0.645) 0.950 0.997 (0.00) (0.07) (0.455) (0.777) (.036) 0.950 0.997 (0.099) (0.03) (0.47) (0.753) (.037) 0.944 0.997 (0.087) (0.74) (0.35) (0.587) (0.886) 0.78 0.956 0.990 0.997 (0.043) (0.087) (0.77) (0.303) (0.478) 0.95 0.998 (0.00) (0.07) (0.459) (-0.38) (-5.054) (0.04) (0.083) (0.7) (0.306) (0.506) (0.055) (0.) (0.40) (0.49) (0.93) (0.055) (0.) (0.33) (0.449) (0.84) (0.05) (0.0) (0.05) (0.34) (0.59) (0.038) (0.077) (0.54) (0.60) (0.400) (0.056) (0.9) (0.303) (0.776) (-7.477) (0.035) (0.07) (0.46) (0.59) (0.439) (0.04) (0.084) (0.76) (0.338) (0.700) (0.04) (0.083) (0.74) (0.33) (0.67) (0.039) (0.079) (0.58) (0.64) (0.40) (0.033) (0.067) (0.35) (0.7) (0.353) (0.04) (0.094) (0.58) (0.85) (-8.644) (0.08) (0.056) (0.6) (0.07) (0.344) (0.030) (0.06) (0.8) (0.36) (0.436) (0.030) (0.06) (0.7) (0.3) (0.43) (0.09) (0.059) (0.8) (0.98) (0.96) (0.07) (0.054) (0.0) (0.86) (0.88) (0.03) (0.074) (0.7) (0.80) (-.87) (0.00) (0.040) (0.08) (0.46) (0.44) 0 0 Miller (0.0) (0.0) (0.00) (0.09) (0.03) (0.07) (0.04) (0.04) (0.040) (0.039) (0.058) (0.08) (0.09) (0.09) (0.08) (0.08) (0.050) (0.086) (0.085) (0.08) (0.079) (0.06) (0.058) (0.059) (0.059) (0.056) (0.057) (0.95) (0.55) (0.54) (0.35) (0.37) (0.786) (0.03) (0.06) (0.06) (0.094) (0.0) (0.775) (0.7) (0.67) (0.03) (0.9) (-8.303) (0.7) (0.83) (0.8) (0.4) (0.70) (-33.970) TABLE III RATIOS OF AVERAGE LENGTHS FROM UNBOUNDED INTERVALS TO AVERAGE LENGTHS FROM BOUNDED INTERVALS Method CV 0.05 0.0 0.0 0.33 0.50 5 Miller.300.94.80.330.45.090.0.34.54.303.09.084.33.8.3.069.069.060.080.4.30.99.3.399.594.080.08.9-0.957-0.05 0 Miller.098.08.099..6.08.07.0.055.90.08.0.07.05.00.09.04.09.033.05.7.8.308.553.08.05.030.396-0.056 5 Miller.09.08.04.054.059.04.07.05.057.04.0.0.05.09.06.03.09.09.0.06.075.48.330.609.04.0.09.3-0.39 5 Miller.08.009.00.00.0.0.034.008.007.037.056.73.430.799.009.049.004 50 Miller.053.03.36.70.9.07.00-4.504 00 Miller 0.994 0.995 0.994.79.56.08.700.004-0.57 Iteratioal Scholarly ad Scietific Research & Iovatio 7(9) 03 40 scholar.waset.org/307-689/678
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