Cloud Publcatos Iteratoal Joural of Advaced Mathematcs ad Statstcs Volume, pp. -6, Artcle ID Sc- Research Artcle Ope Access Bootstrap Method for Testg of Equalty of Several Coeffcets of Varato Dr. Navee Kumar Boroju ad Dr. M. Krsha Reddy, Departmet of Statstcs, Osmaa Uversty, Hyderabad, Adhra Pradesh, Ida Correspodece should be addressed to Navee Kumar Boroju, abyrozu@gmal.com Publcato Date: 3 December Artcle L: http://scetfc.cloud-jourals.com/dex.php/ijams/artcle/vew/sc- Copyrght Navee Kumar Boroju ad M. Krsha Reddy. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal wor s properly cted. Abstract The coeffcet of varato has bee foud to be very useful ut less measure of relatve cosstecy of sample data may areas such as chemcal expermets, face, surace rs assessmet, medcal studes, etc., a ch-square test s used for testg the equalty of several coeffcets of varato the lterature. Ths ch-square test demostrates oly the statstcal sgfcace of coeffcets of varato. I ths paper, a bootstrap graphcal method s developed as a alteratve to the ch-square test to test the hypothess o equalty of several coeffcets of varato. A example s gve to demostrate the advatage of bootstrap graphcal procedure over the chsquare test from decso mag pot of vew. Keywords Coeffcet of Varato, Ch-Square Test, Bootstrap Method. Itroducto Coeffcet of varato s used such problems where we wat to compare the varablty of two or more tha two groups. The seres for whch the coeffcet of varato s greater s sad to be more varable or coversely less cosstet, less uform, less stable or less homogeeous. O the other had, the seres for whch coeffcet of varato s less s sad to be less varable or more cosstet, more uform, more stable or more homogeeous. The coeffcet of varato s depedet of ut of measuremet ad has bee foud to be a very useful measurg of relatve cosstecy of sample data may stuatos. For example, the coeffcet of varato s useful measure rs assessmet as a measure of the heterogeety of surace portfolos. Coeffcet of varato s also used comparg the characterstcs such as tesle stregths, weghts of materals, etc. the processg type of dustres. Statstcal ferece based o data resamplg has draw a great deal of atteto recet years. The ma goal s to uderstad a collecto of deas cocerg the o-parametrc estmato of bas, varace ad more geeral measures of errors. The ma dea about these resamplg methods s ot to assume much about the uderlyg populato dstrbuto ad stead tres to get the formato about the populato from the data tself varous types of resamplg leads to varous types of
methods le the jacfe ad the bootstrap. Bootstrap method (Efro, 979) use the relatoshp betwee the sample ad resamples draw from the sample, to approxmate the relatoshp betwee the populato ad samples draw from t. Wth the bootstrap method, the basc sample s treated as the populato ad a Mote Carlo style procedure s coducted o t. Ths s doe by radomly drawg a large umber of resamples of sze from ths orgal sample wth replacemet. Both bootstrap ad tradtoal parametrc ferece see to acheve the same goal usg lmted formato to estmate the samplg dstrbuto of the chose estmatorˆ. The estmate wll be used to mae fereces about a populato parameter. The ey dfferece betwee these feretal approaches s how they obta ths samplg dstrbuto whereas tradtoal parametrc ferece utlzes a pror assumptos about the shape of dstrbuto of ˆ. The o-parametrc bootstrap s dstrbuto free whch meas that t s ot depedet o a partcular class of dstrbutos. Wth the bootstrap method, the etre samplg dstrbuto of ˆ s estmated by relyg o the fact that the samplg dstrbuto s a good estmate of the populato dstrbuto. I secto 3, bootstrap method appled to testg of equalty of several coeffcets of varato s explaed [, ].. Testg of Equalty of Several Coeffcets of Varato Let X j,,,,, j,,, represet depedet radom samples of sze ad we assume that X N, j ~ for,,...,. Sce the samples are draw from ormal populatos wth dfferet meas ad dfferet varaces, the coeffcet of varato s a useful characterstc to measure the relatve varablty the ormal populatos. Here, we are terested testg the ull hypothess. H... : (Uow), where alteratve hypothess that at least two coeffcets of varato are uequal. agast the Ch-square test s used for testg H the lterature [3, 4]. Ths test demostrates oly the statstcal sgfcace of the coeffcets of varato beg compared. Ch-square test for testg H, Mller ad Feltz (997) suggested a test statstc ad t s gve by m ~ (Uder H ) (.) (.5 c c c Where ) c xj j m,, s xj x We reject the ull hypothess, H f x, c ad c j x, s c Iteratoal Joural of Advaced Mathematcs ad Statstcs
3. Bootstrap Graphcal Method for Testg of Equalty of Several Coeffcets of Varato Let X j,,, ; j,, represet avalable depedet radom samples of sze ad the coeffcet of varato of the th sample s gve by c s x for =,... Bootstrap graphcal procedure for testg the equalty of several coeffcets of varato s gve the followg steps.. Let Yjb be the b-th bootstrap sample of sze, draw from th avalable sample, where b=, B (=3), =, ad j=,.. Compute yb ad s b, the mea ad stadard devato of b-th bootstrap sample form th avalable sample ad are gve by y b Y jb j ad s b Yjb yb 3. Compute c b, be the coeffcet of varato of b-th bootstrap sample from th avalable sample ad s gve by 4. Compute c s b cb, =, ad b=, B. yb b c b, b=, B. 5. Obta the samplg dstrbuto of coeffcet of varato usg B-bootstrap estmates ad compute the cetral decso le (CDL) as B by SEc c b c B b (UDL) for the comparso of each of the LDL c UDL c Where z z / / SE c SE c c B c b B b j ad the stadard error s gve. The lower decso le (LDL) ad the upper decso le c are gve by z s the -th upper cut off pot of stadard ormal dstrbuto. 6. Plot c agast the decso les. If ay oe of the pots plotted les outsde the respectve decso les, H s rejected at 5% level ad coclude that the coeffcets of varato are ot homogeous. The proposed method s very useful hadlg of small samples of sze less tha 3. Ths method ot oly tests the sgfcat dfferece amog the coeffcets of varato but also detfy the source of heterogeety of coeffcets of varato. Sze of the proposed test s obtaed usg smulato of radom samples from ormal populatos havg wth the equal coeffcet of varatos. Let the populatos, X ~ N,, X ~ N 4, 4, X ~ N 6,9, X ~ N 8,6 ad X ~ N, 5 havg wth the 3 4 5 same coeffcets of varato. The proposed test procedure s performed tmes to compare the populatos wth respect to coeffcets of varato usg the dfferet samples (equal sze) draw. Iteratoal Joural of Advaced Mathematcs ad Statstcs 3
from the above fve populatos. The sze of the test s defed as umber of tmes the test procedure rejectg the ull hypothess of equalty of coeffcets of varatos teratos. That s, Number of tmes the ull hypothess s rejected. The followg table presets the sze of the test for comparg -populato coeffcets of varato based o the samples of sze = 5,, 5,, 5 ad 3. Table : Sze of the Proposed Test \ 5 5 5 3 3.5.4.4... 4.9.8.5.4.. 5....7.5. Power of the test procedure s computed usg smulatg radom samples from ormal populatos. Let the populatos, X ~ N,, X ~ N 4,, ~ 6,9, ~ 4, 4 ~ 5,5 X N X N ad X N, the populatos 3 4 5 are cosdered such a way that these are havg wth the dfferet coeffcets of varato across the populatos. The test procedure s performed tmes by cosderg the dfferet samples from the -populatos. Let be the ype-ii error ad whch s computed as Number of tmes acceptg H. Power of the test s gve by ad s computed for comparso of -populatos based o the samples of sze =5,, 5,, 5 ad 3. The followg table presets the power of the proposed test. Table : Power of the Test \ 5 5 5 3 3.85.85.87.89.9.93 4.8.84.87.86.89.9 5.8.85.88.89.94.93 I the above tables represets the umber of populatos compared ad s the sze of the each sample draw from the -populatos testg of equalty of coeffcets of varato. From the above two tables, t s observed that the sze of the test s decreasg ad the power of the test s creasg as the sample sze creases. The proposed test procedure s explaed wth a umercal example the followg secto. 4. Numercal Example Example 5. from the paper of Tsou (9) s cosdered ad ths example descrbes the umbers of brth 978 o Moday, Thursday, ad Saturday the Uted Kgdom. We use the ew procedure to test whether the coeffcets of varato of the three dfferet dates are the same [5]. Let c represets the coeffcet of varato of umbers of brth o Moday, c represets the coeffcet of varato of umbers of brth o Thursday ad c 3 represets the coeffcet of varato of umbers of brth o Saturday. For the gve data c =.649, c =.58, c 3 =.465, =3 ad =5. We obta 5.. test statstc value s 5.579 ad the sgfcat value at 5% level s 995,.5 Iteratoal Joural of Advaced Mathematcs ad Statstcs 4
Sce the test statstc value s less tha the crtcal value, therefore we accept H at 5% level. By applyg the bootstrap procedure explaed Secto 3, the LDL, CDL ad UDL are obtaed as.44,.55 ad.675 respectvely. Prepare a chart as Fgure, wth the above decso les ad plot the pots,,3 c. From the Fgure, we observe that all the pots wth the decso les, hece H s accepted ad we may coclude that the coeffcets of varato of the three dfferet dates are the same. 5. Cocluso Note that H o s accepted by both Ch-Square test ad the bootstrap graphcal method. Whe H o s rejected, ch-square test reveals the statstcally sgfcat dffereces amog the coeffcets of varato beg compared, whle the graphcal method ot oly reveals the statstcally sgfcat dffereces but also detfy the source of heterogeety of coeffcets of varato. Fgure Fgure : Decso Les for the Coeffcets of Varato Iteratoal Joural of Advaced Mathematcs ad Statstcs 5
Refereces [] Efro B. Bootstrap Methods: Aother Loo at the Jacfe. The Aals of Statstcs. 979. 7 () -6. [] Efro B., et al. 994: A Itroducto to the Bootstrap. st Ed. Chapma ad Hall/CRC, New Yor, 456. [3] Fetz C.J., et al. A Asymptotc Test for the Equalty of Coeffcets of Varato from - Populatos. Statstcs Medce. 996. 5 (6) 646-658. [4] Mller G.E., et al. Asymptotc Iferece for Coeffcets of Varato. Commucatos I Statstcs- Theory ad Methods. 997. 6 (3) 75-76. [5] Tsug-Sha Tsoua. A Robust Score Test for Testg Several Coeffcets of Varato wth Uow Uderlyg Dstrbutos. Commucatos Statstcs- Theory ad Methods. 9. 38 (9) 35-36. Iteratoal Joural of Advaced Mathematcs ad Statstcs 6