Comparing two Quantiles: the Burr Type X and Weibull Cases

Transcription

1 IOSR Journal of Mathematcs (IOSR-JM) e-issn: , -ISSN: X. Volume, Issue 5 Ver. VII (Se. - Oct.06), PP Comarng two Quantles: the Burr Tye X and Webull Cases Mohammed A. Shayb, Alakbar Montazer Haghgh, Deartment of Mathematcs Prare Vew A&M Unversty Prare Vew, Abstract: Interval estmaton and hyothess testng for the dfference between two quantles are nvestgated, usng smulaton, n ths artcle. The underlyng dstrbutons that wll be consdered are the Webull and Burr- Tye-X. The estmaton rocedures wll be based on the generalzed confdence nterval rocedure, whle the hyothess testng wll be based on the generalzed -values rocedure. Smulaton wll be carred to check on the accuracy of both rocedures. I. Introducton A statstcal comarson between two oulatons based on ther mean, varances or roortons s a common ractce n the lterature. Ths s carred out to check on the suerorty of one oulaton over the other. In ths artcle we wll make such a comarson between two oulatons based on ther quantles. In robablty and statstcs, the quantle functon of the robablty dstrbuton of a random varable secfes, for a gven robablty, the value whch the random varable wll be at, or below, wth that robablty. A comarson between two quantles for the Normal and Exonental dstrbutons had been done; see Guo and Krshnamoorthy, (005). The quantle functon s one way of rescrbng a robablty dstrbuton. It s an alternatve to the robablty densty or mass functon, to the cumulatve dstrbuton functon, and to the characterstc functon. The quantle functon of a robablty dstrbuton s the nversef of ts cumulatve dstrbuton functon (cdf) F. Assumng a contnuous and strctly monotonc dstrbuton functon, the quartle functon returns the value below, whch the random varable drawn from the gven dstrbuton would fall 00 ercent of the tme. That s, t returns the value of x such that If the robablty dstrbuton s dscrete rather than contnuous then there may be gas between values n the doman of ts cdf, whle f the cdf s only weakly monotonc there may be "flat sots" n ts range. In ether case, the quantle functon s (.), (.) for a robablty 0 < <, and the quantle functon returns the mnmum value of x for whch the revous robablty statement holds. The Q(), or, (Hogg and Crag, 995) s the quantle of order, and thus 0.5 s the medan of the dstrbuton. Consder two ndeendent random varables X and Y. Let x and y denote the th quantle of X and Y resectvely. That s, x = nf{x: P(X x) } and y = nf{y: P(Y y) }. (.3) The roblem of nterest here s to make a statstcal nference about DOI: / Page x - y based on samles of szes m and n observatons on X and Y, resectvely. In ths artcle, we used smulaton for comarng the quantles of Burr Tye-X, and the quantles of Webull dstrbutons. The generalzed -value has been ntroduced by Tsu and Weerahand (989), and the generalzed confdence nterval by Weerahand (993). Usng ths aroach, we wll gve an nferental rocedure for the dfference between two Burr-Tye-X quantles n the followng secton. The erformance of the rocedure wll be evaluated numercally through smulaton. In Secton 3 we resent the webulldstrbuton sand the dervaton of ts quantles for dfferent values of the nvolved arameters. In secton 4 we resent the smulaton results on the dfference for x - y, when the underlyng dstrbutons are taken to be Burr-Tye-X and Webull wth dfferent arameters. Secton 5 wll contan the conclusons and recommendatons.

2 Comarng Two Quantles: The Burr Tye X And Webull Cases II. Burr-Tye-X Dstrbuton Burr (94) ntroduced dfferent forms of cumulatve dstrbuton functons for modelng lfetme data, or survval data. Out of those dstrbutons, Burr-Tye-X and Burr-Tye-XII have receved the maxmum attenton. Several authors have consdered dfferent asects of these two dstrbutons. In ths artcle we wll resent the generalzed nferental rocedures for the dfference between the quantles of two Burr- Tye-X dstrbutons. It s to be noted that the robablty densty functon (df) of the Burr-Tye-X dstrbuton s gven as follows: x x f ( x) xe ( e ), x 0, 0. (.) Moreover, for the one-arameter Burr Tye-X dstrbuton, the cumulatve dstrbuton functon F s gven by ( ) ex( ) 0, 0. F x x x, (.) For any gven 0 < <, the th quantle s the ostve root of F(x) =,.e. x ( ex( x )), or / ln[/ ( )] (.3) Thus f X ~ f(x ), =,, then the th quantle of X can be exressed as / ln[/ ( )], =,. (.4) Theorem. Let X~ Burr-Tye-X ( ), wth df gven by (.), then the random varable ln wll have the one-arameter exonental dstrbuton wth mean / U e x have the followng df hu Proof: u/ e, u 0, 0, otherwse. (.5) By uttng y = u(x) = x = x ln( e ) ln e ln wy ( ) y e x On dfferentatng (.6) wth resect to y, we have y e w'( y) { ln( )} y ( e ), we can see that (.6) y / e (.7) By usng the transformaton formula namely g(y) f(w(y)). w (y), we reach at the df of y as Gven by g(y) = e y, 0 <, wth /. Thus the roof of Theorem. s comlete. Theorem. Because, =,, are ostve, testng H0 : vs. H : H : / c vs. H : / c, (.8) 0 DOI: / Page,.e., U wll s equvalent to

3 Comarng Two Quantles: The Burr Tye X And Webull Cases where c = ln / ln, and I =, as defned n Theorem.. In the remanng of the artcle we wll be referrng to the means of the exonental dstrbutons as cted n Theorem. above. Proof: From (.4), we see that f and only f f and only f e e ff ln( e ) ln( e ) ff (/ )ln (/ )ln ff ( )ln ( )ln ff / ln / ln = c. In addton, we can easly see that the above test s equvalent to H : / c vs H : / c. 0 Hence the roof for Theorem. s comlete. Utlzng Theorems. and., we fnd that all we need s to generate samles from exonental dstrbutons wth arameters and X Xn be a samle from F(u ), =,. Defne Y X,, n j. Notce that j resectvely. Let,..., Y and Y are ndeendent wth n ff Y / ~, =,, and hence Y/ dstrbuted as a constant tmes an F random varable, (see Guo, and Krshnamoorthy (005)). Thus t can be seen that the -value for testng (.8) s gven by P ( F cn / ( n y )), n,n y (.9) where F ab, denotes the F dstrbuton wth a degrees of freedom for the numerator, and b degrees of freedom for the denomnator. Thus the null hyothess n (.8) wll be rejected whenever ths -value s less than. To fnd the above -value of the test, and due to the small degrees of freedom that are tabulated for the F- Dstrbuton, we chose to have small samles for the above cases wth szes (n, n ) = (0, 5) and ( 5, 0 ) to go wth the arameters values (, 3) and (3, 5) resectvely. The results of the smulaton are tabulated n Table. From Table, we have for Formula (.9) the followng values: cn y / ( n y), for dfferentvaluesof c, for the frst case we have: (n, n ) = (0, 5), (, 3) C = C =3 cn y / ( n y ) = , and the -value > 0.0, cn y / ( n y ) =.75, and the -value:.05 < P < 0.0. Thus endng on the value of c, and on comarng the -value to that of the level of sgnfcance, decson can be made on rejectng H 0 or not. Smlarly, cn y / ( n y ), for dfferentvaluesof c, for the second case we have (n, n ) = (5, 0), (3, 5) C = C = 5 cn y / ( n y ) = 0.467, and the -value > 0.0, cn y / ( n y ) =.0835, and the -value s almost Agan, endng on the value of c, and on comarng the -value to that of the level of sgnfcance, decson can be made on rejectng H 0 or not. Y s DOI: / Page

4 Comarng Two Quantles: The Burr Tye X And Webull Cases III. The Webull Dstrbuton A common alcaton of the Webull dstrbuton s to model the lfetmes of comonents such as bearngs, ceramcs, caactors, and delectrcs. The Webull dstrbuton, n model fttng, s a strong comettor to the gamma dstrbuton. Both the gamma and Webull dstrbutons are skewed, but they are valuable dstrbutons for model fttng. Webull dstrbuton s commonly used as a model for lfe length because of the roertes of ts falure rate functon h( x) ( / ) x, when the df s gven by: x x e, x,, 0, f( x) (3.) 0, otherwse. Ths falure rate functon, for, s a monotoncally ncreasng functon wth no uer bound. Ths roerty gves the edge for the Webull dstrbuton over the gamma dstrbuton, where the falure rate functon s always bounded by/, when the robablty densty functon for the gamma dstrbuton s / ( ) y gy y e, ( ) y > 0; where ( ) e uu du (3.) 0 Another roerty that gves the edge for the Webull dstrbuton over the gamma dstrbuton s by varyng the values of and, a wde varety of curves can be generated. Because of ths, the Webull dstrbuton can be made to ft a wde varety of data sets. In ths aer we wll consder the Webull dstrbuton wth two arameters, namely and, where s the / shae arameter whle s the scale arameter of the dstrbuton. Based on the above form of the df for the Webull dstrbuton, the mean and the varance of the dstrbuton are gven as / E(X) = ( / ), and V(x) = / ( / ) { ( / )} (3.3) We wll use the notaton X ~ Webull (, ) for the random varable X havng a Webull robablty densty functon wth arameters and, as shown above n (3.). We wll also take the shae arameter as n Hogg and Tans (988) to be < < 5. In addton, the cdf for the Webull dstrbuton gven n (3.) s /, and thus the th quantle s the soluton of the equaton F(x) =, namely, Fx ( ) ex x [ ln{/ ( )}] / (3.4) Now, as t was the case n the Burr Tye-X, we have the followng theorem. Theorem 3. If X ~ Webull(, ) as gven n (3.), then ) U X wll have an exonental dstrbuton wth arameter,.e., u/ e, u 0, hu 0, otherwse, DOI: / Page

5 Comarng Two Quantles: The Burr Tye X And Webull Cases And ) mu ~ m ) Smlarly, fy ~ Webull(, ), then VY wll have an exonental dstrbuton wth arameter,.e. v) g v nv v/ e, v 0 0, otherwse, ~ n, and v) Proof: V U F. ~ n,m As t was n Theorem. above we can wrte y = u(x) = X α, and we can have X = Y /α = w(y). On dfferentatng wth resect to y, we reach at w (y) = (/α).y (-α)/α. Henceg(y) = f(w(y)). w (y) = (/β ).e -y/β, y = u(x) > 0; and 0 otherwse. Thus the roof of Theorem 3., ). For art ), when Y = X /, we reach at w(y) = (βy/) /α. By dfferentatng wth resect to y, we get w (y) = (/α).[(βy/) /α- ].(β/). By substtuton n g(y) = f(w(y)). w (y), we have g( y) e y e ( r / ) y/ r/ y/ r/. The above functon s a df for a Ch-square wth r = degrees of freedom. Thus the random varable mu wll have a Ch-Square dstrbuton wth m degrees of freedom, based on the sum of m ndeendent Ch-square varables each wth degrees of freedom. Hence the roof of Theorem 3. art ) s comlete. Followng the stes above, n the roof of Theorem 3. arts ) and ), we see that art ) and v) follow verbatm. Therefore, the roofs for arts ) and v) are done. The roof for art v), of Theorem 3., follows by the defnton of the F-Dstrbuton. Hence the roof of Theorem 3. s comlete. Theorem 3. From (3.4) and because, =,, are ostve, testng s equvalent to H : vs. H : 0 H0 : / c vs H : / c, where c = / ln( ). ln( ) Proof: The roof of Theorem 3. follows from the roof of Theorem.. Under H 0, The -value for ths test s W R V 0 ~ F n,m R0 U DOI: / Page

6 Comarng Two Quantles: The Burr Tye X And Webull Cases z = mn [ P H 0 (W>w), P H 0 (W<w)] = mn [- F (w), F (w)], where w s the observed value of the test statstc W and F s the dstrbuton functon of W under H 0. The - value of ths test ndcates how strongly H 0 s suorted by the data. IV. The Smulaton Set-U There are two cases to consder. I. The underlyng dstrbuton s Burr-Tye-X. The followng values wll be used for the smulaton: A. = 0.9 wth = ½, and, and on usng (.4) we fnd the 90 th ercentles corresondngly are gven by:.8869,.5743, and.739. B. = 0.95 wth = ½, and, and on usng (.4) we fnd the 95 th ercentles corresondngly are gven by:.5575,.7308, and II. The underlyng dstrbuton s Webull. The followng values wll be used for the smulaton: A. = 0.9, wth =, and = ½, and, and on usng (3.4) we fnd the 90 th ercentles corresondngly are gven by: ,.3059, and B. = 0.9, wth =, and = ½, and, and on usng (3.4) we fnd the 90 th ercentles corresondngly are gven by:.0759, , and C. = 0.95 wth =, and = ½, and, and on usng (3.4) we fnd the 95 th ercentles corresondngly are gven by: ,.99573, and D. = 0.95 wth =, and = ½, and, and on usng (3.4) we fnd the 95 th ercentles corresondngly are gven by: , , Table below, s based on the smulaton of an exonental dstrbuton, snce there s no software to generate the Burr-Tye-X data, and then usng the transformaton As was dslayed n Theorem.. x ln e u Table has the smulaton that was carred on 30 samles wth samle sze of 30 for each. Agan the table was done on generaton an exonental dstrbuton then usng the transformaton to get the Burr- Tye-X data. The data n each samle was ordered to show the order statstcs, or the quantles. Moreover, The Mn, the Max, and the Avg. for each quantle was found. In the Burr-Tye-X case the value of the arameter was taken to.the overall averages are dslayed n the last row of the table. Table 3 has the smulaton that was carred on 30 samles wth samle sze of 30 for each, usng MINITAB software to generate the data. As t was the case n Table, the same rocedure was done n Table 3. Table 3 has the Webull Smulated data when = and θ = ½,, and resectvely, and lstng the Avg., Mn, and Max of the order statstcs. The overall averages are dslayed n the last row. Table 4 s a coy of Table 3 wth one dfference n ths case that =, wth the same values for θ. The other tables are dslayng the dfferences n the quantles of the Burr-Tye-X, as dslayed n Table, and the choce of one value arameter θ = 0.5, usng the Max, Mn, and Avg of those quantles. Table 5 has thetabulated dfferences between the Burr-Tye-X and the Webull dslayed as B W for the Case of the Webull Dstrbuton when the arameters are = and θ = ½. Table 6 has thetabulated dfferences between the Burr-Tye-X and the Webull dslayed as B W for the Case of the Webull Dstrbuton when the arameters are = and θ = ½. DOI: / Page

7 Comarng Two Quantles: The Burr Tye X And Webull Cases Other tables can be comuted on the other values of the arameters of the Webull dstrbuton and that of the Burr-Tye-X. It s left to the nterested reader to match Table 5 and Table 6. V. Concluson It s clearly understood that both the Burr-Tye-x and the Webull dstrbutons are qute used n lfe testng and falure analyss. As t can be seen from the tabulated quantles values, t s clearly the for the lower quartles,., e. those that are less than the medan, n a samle of sze 30, those quartles are hgher for the Burr- Tye-x than the corresondng Webull quantles. The dstrbuton between X Y, where X and Y are the th quntles based on the Burr-Tye-X and Webull dstrbutons resectvely s yet to be dslayed. On the other hand, we have notced that the Uer Quantles,.e. those that are greater than the medan, for the Webull dstrbuton are greater than ther corresondng quantles for the Burr-Tye-X. The nvestgaton s stll ongong. References []. Tsu, K. W. and Weerahand, S. (989) Generalzed -value n sgnfcance testng of hyotheses n the resence of nusance arameters, J. Amer. Statst. Assoc. 84, []. Weerahand, S. (993) Generalzed confdence nterval. J. Amer. Statst. Assoc. 88, [3]. Guo, H. and Krshnamoorthy, K. (005) Comarson between two quantles: The Normal and Exonental cases, Communcatons n Statstcs-Smulaton and Comutaton, 34,, [4]. Hogg, R. V. and Crag, A. T. (995) Introducton to Mathematcal Statstcs, 5 th Edton, Prentce Hall. DOI: / Page

8 Comarng Two Quantles: The Burr Tye X And Webull Cases Frst Case Second Case Table Fst Samle Second Samle Frst Samle Second Samle =, n = 0 = 3, n = 5 = 3, n = 5 = 5, n = = = n y /(n y ) n y /(n y ) Table Burr-Tye-X EXP BURR- TYPE-X Max Mn Avg Max Mn Avg DOI: / Page

9 Comarng Two Quantles: The Burr Tye X And Webull Cases DOI: / Page

10 Comarng Two Quantles: The Burr Tye X And Webull Cases TABLE 3 WEIBULL Case Alha = 0.5 Beta = Case Alha= Beta = Case Alha= Beta = AVG MIN MAX AVG MIN MAX AVG MIN MAX DOI: / Page

11 Comarng Two Quantles: The Burr Tye X And Webull Cases TABLE 4 WEIBULL Case Alha = 0.5 Beta = Case Alha = Beta = Case Alha = Beta = AVG MIN MAX AVG MIN MAX AVG MIN MAX DOI: / Page

12 Comarng Two Quantles: The Burr Tye X And Webull Cases Table 5 Alha = 0.5 Beta = B - W Avg Mn Max Dff Dff Dff DOI: / Page

13 Comarng Two Quantles: The Burr Tye X And Webull Cases Table 6 Alha = 0.5 Beta = B - W Avg Mn Max Dff Dff Dff DOI: / Page