A Note on Test of Homogeneity Against Umbrella Scale Alternative Based on U-Statistics

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1 J Stat Appl Pro No () 93 NSP Journal of Statstcs Applcatons & Probablty --- An Internatonal NSP Natural Scences Publshng Cor A Note on Test of Homogenety Aganst Umbrella Scale Alternatve Based on U-Statstcs Anl Gaur Department of Statstcs Panjab Unversty Chandgarh Inda Emal Address: anlgaur8@gmalcom Receved July7 ; Revsed September 8 ; Accepted October Abstract: A fundamental problem encountered n statstcs s that of testng the equalty of scale parameters aganst umbrella alternatve wth at least one strct nequalty In ths artcle a nonparametrc test based on U-statstc by consderng the subsample mnma and maxma for several sample scale problem aganst umbrella alternatve when peak of the umbrella s known s proposed The proposed test have the advantage of not requrng the several dstrbuton functons to have a common medan but rather any common quantle of order q q (not necessarly ½) whch s assumed to be known Ptman effcency ndcate that the proposed test s equvalent to the test B proposed by Gaur Mahajan and Arora () Keywords: Umbrella orderng U-statstc Test for scale Asymptotc Relatve Effcency Introducton Let X n ; k be ndependent random samples of sze n from absolutely contnuous cumulatve dstrbuton x functons F ( F k We assume that these dstrbuton functons have as the common quantle of order q ( q ) e F ( ) q for k Wthout loss of generalty we assume that the common known quantle F ( q) Fk ( q) of order q s zero for the pre-specfed q It s also assumed that F ( k are dentcal n all respects except possbly ther scale parameters The problem of testng the null hypothess of homogenety of scale parameters H : k aganst smple ordered alternatve hypothess H A : k wth at least one strct nequalty has receved consderable attenton n the lterature For some earler work on ths problem one may refer to Rao [6] Shanubhogue [8] Kusum and Baga [] Gll and Dhawan [9] Sngh and Gll [] among others Most of the tests except Kusum and Baga [] test requre the assumpton that the common quantle of dfferent dstrbutons s of order q = ½ e the dstrbutons have the same medan None of these tests s adequate when the common quantle s dfferent from medan The asymmetry of the stuaton s not reflected n the statstcs used n the above tests Kusum and Baga [] consdered a more general verson of ths problem In the tradtonal set up the dsperson of a dstrbuton s evaluated around ts central value (say the medan) However n many applcatons the dsperson needs to be consdered around a noncentral value say a quantle of order q q (not necessarly ½) whch s assumed to be known These procedures have mportant applcatons for problems where the treatments can be assumed to satsfy a smple orderng such as for a sequence of ncreasng dose-level of a drug Umbrella orderng s mportant n dose-response experment (eg see Smpson and Margoln [])

2 94 Anl Gaur: A Note on Test of Homogenety Aganst Umbrella Scale Alternatve In case where mode of acton of a drug s related to ts toxc effects eg n case of lfe savng therapy of heart falure lfe savng dgtalzaton therapy of heart falure umbrella behavor s antcpated and careful dosage plannng s requred There has been substantal work n testng equalty of locaton parameters aganst umbrella alternatve wth at least one strct nequalty but very lttle work for testng of scale parameters For detaled references one may refer Mack and Wolfe [3] Chen and Wolfe [5] Chen and Wolfe [6] Chen [4] Kosseler [] and Abebe and Sngh [] Shetty and Bhat [9] Bhat and Patl [3] and Bhat [] proposed test statstcs based on lnear combnaton of two sample U-statstcs for testng homogenety of locaton parameters aganst umbrella alternatve wth at least one strct nequalty Sngh and Lu [] proposed a test statstcs for homogenety of scale parameters aganst umbrella alternatve wth at least one strct nequalty based sotonc estmator of scale parameter They also provded one-sded smultaneous confdence ntervals for all the ordered parwse scale ratos and crtcal ponts for two parameter exponental probablty dstrbuton Recently Gaur et al [8] provded three test statstcs based on weghted lnear combnaton of two-sample U-statstcs for testng homogenety of scale parameters aganst umbrella alternatve wth at least one strct nequalty when the peak of the umbrella h s known In ths paper a new test based on subsample mnma and maxma for testng homogenety of scale parameters aganst umbrella alternatve wth at least one strct nequalty s proposed when peak of the umbrella h s known It s found that the proposed test s equvalent to the test B proposed by Gaur et al [8] for heavy-taled dstrbutons when the common quantle s not the medan but of some order q (q not necessarly equal to ½) Stuatons where two populatons may have a common quantle of order other than q = ½ arse n many real lfe examples In partcular ths assumpton appears to be qute realstc as n examples of automatc can fllng mechansms and models of wage dstrbuton as ponted by Deshpande and Kusum [7] Ths artcle s organzed as follows The new proposed test for testng homogenety of scale parameters aganst umbrella alternatve wth at least one strct nequalty and ts dstrbuton s gven n secton Secton 3 s devoted to the optmal choce of weghts Effcacy results and Ptman asymptotc relatve effcency are gven n secton 4 The Proposed Test and ts Dstrbuton Let X ; k be k ndependent random samples of sze n from absolutely n x contnuous cumulatve dstrbuton functons F ( F k We assume that these dstrbuton functons have zero as the common quantle of order q ( q ) e F ( ) q for k It s also assumed that F ( k are dentcal n all respects except possbly ther scale parameters The hypothess whch s of nterest n ths paper could be formally stated as follows: H : k aganst the umbrella alternatve H : h h h k k wth at least one strct nequalty and h the peak of the umbrella s known

3 Anl Gaur: A Note on Test of Homogenety Aganst Umbrella Scale Alternatve 95 Frst we propose a two-sample U-statstc n the context of a two-sample scale problem where the assumpton of the common quantle of order q ( q ) s made and then extend t to the k-sample problem consdered here Now for j defne the kernel ( X ; X ) j j f max( X or max( X f max( X or max( X otherwse j j ) mn( X ) mn( X ) mn( X ) mn( X j j ) and X ) and X ) and X ) and X j j j j The two-sample U-statstc correspondng to the kernel j s Uj ( X X X ; X X j X ) j ( X X X ; X X j X ) () n n j c 3 3 n where c denotes the summaton extended over all possble n j combnatons of X Xn 3 3 and X X j jn j The statstc U j s obvously a U-statstc (Lehman []) correspondng to the kernel j It can be seen that the kernel takes on a non-zero value only when both the X s and X j s have the same sgn Under H the observatons from the th populaton are expected to be smaller (larger) than those from the (+) th populaton therefore U ( U ) h ( h h k ) s expected to take large values Motvated by the fact that under H U ( U ) h ( h h k ) s expected to take large values we propose a class of test statstcs U k based on subsample extreme of sze three as T T T () where h k T a U h T a U for testng H aganst H where a a a k are sutable chosen real postve constants It may be noted that for each set of values a a a k we get a dstnct member of ths class of test statstc A large value of T leads to the rejecton of H aganst H When a = ( k ) we obtan Mack-Wolfe verson of T as h k T U U (3) M h

4 96 Anl Gaur: A Note on Test of Homogenety Aganst Umbrella Scale Alternatve The Dstrbuton of T Clearly E T h where k a a h and [ F ( q] [ F ( ] ( 3 [ q F ( ] F ( ( [ F ( q] [ F ( ] ( 3 [ q F ( ] F ( ( [ F ( q] [ F ( ] ( 3 [ q F ( ] F ( ( [ F ( q] [ F ( ] ( 3 [ q F ( ] F ( ( Under H we have F () Fj () whch mples ( ) h ( h h k ) Hence under H E ( T ) E ( T M ) (4) Usng the results of Lehman [] and Pur [4] the proof of the followng theorem follows from the transformaton theorem (see Serflng [7] page ) mmedately Theorem Let N k n The asymptotc null dstrbuton of N T as n p p k s normal wth mean and varance where N N n such a way that Var ( A) (5) ' ' Var ( A ) a a Var( A ) a a (( j )) (( j )) ' ' a a a a a a a h (6) a h h k for j h- p p for j ; h- j p (7) for j -; 3h- p otherwse

5 Anl Gaur: A Note on Test of Homogenety Aganst Umbrella Scale Alternatve 97 for j hh k- p p for j ; hh k- j p (8) for j -; h h k- p otherwse Cov A A ) a a Cov( T T ) (( a a ) / p ) (9) ( h h h h h h h h h where 3 [ q ( q) ] 3 In case all the sample szes are equal e p = p = = p k = k then substtutng (6) (7) (8) and (9) n (5) we have h h k k k a aa a aa aha h h h k k k a aa ahah () Smlarly the asymptotc null dstrbuton of N [ TM E( TM )] s normal wth mean zero and varance M 6k Practcal mplementaton of ths procedure may requre an estmator for the common quantle under the null hypothess of order F ) F ( ) We suggest to use a pooled estmator of ( k F ( q) Fk ( q) for a gven (predetermned values of) q To acheve ths we pool all the observatons X n ; k nto a sngle vector Z and estmate by obtanng the q- th quantle of Z Remark The correspondng two-sample problem s under consderaton n dfferent paper by the author 3 Optmal Choce of Weghts Under the sequence of Ptman alternatves the square of the effcacy of test T s gven by k I a ( ) e T (3) where I 8 x F ( [ q F( ] f ( dx x[ F( q] [ F( ] f ( dx

6 98 Anl Gaur: A Note on Test of Homogenety Aganst Umbrella Scale Alternatve For effcency comparsons we consder the equal sample sze and equally spaced alternatves of the type for h (3) (h ) for h k Makng use of the results due to Rao [5] (page 6) for determnng optmal weghts we obtan the * optmal weghts a for whch T has maxmum effcency For odd k and h=(k+)/ a * ( /k)( k( h-) -( h-)) for h- (( k ) /k) ( k( -h) -( h-)) for h k- (33) The respectve square of the effcacy of tests T wth optmal choce of weghts n (33) s gven by 4 ( k k 3) / 4k * I e ( T) (34) And the square of the effcacy of tests T M s gven by e ( T I( k ) ) (35) 6k * M 4 Asymptotc Relatve Effcency We compute the Ptman asymptotc relatve effcency (ARE) of the proposed tests wth Gaur-Mahajan- Arora B-test (see Gaur et al [8]) The effcency of the new proposed test s gven n (34) and (35) The effcacy of B s gven by 4 * J ( k k 3) / 4k e ( B) e ( B J( k ) ) 6k * M where q q J 57 and 95 7 x F ( [ q F( ] f ( dx x[ F( q] [ F( ] f ( dx Then the asymptotc relatve effcency (ARE) of T (T M ) wth respect to B (B M ) test can be computed from the rato of the Ptman effcaces and we notce that ARE(T B) = ARE(T M B M ) = ( k k 3) Also ARE (T T M ) = 8k( k ) et al [8] 4 and the values of ARE for dfferent values of k can be found n Gaur

7 Anl Gaur: A Note on Test of Homogenety Aganst Umbrella Scale Alternatve 99 Gaur et al [8] mentoned that the test B whch s based on the medans of subsamples performs better for heavy-taled dstrbuton (when the dfferent dstrbutons have common quantle) Snce asymptotc relatve effcency of the test T wth respect to test B s therefore the test T whch s based on the mnma and maxma of subsamples performs better for heavy-taled dstrbuton and s equvalent to B-test So n case of heavy-taled dstrbutons lke the Double Exponental dstrbuton and Cauchy dstrbuton when the dstrbutons have common quantle then the test T can be used n place of B for testng the homogenety of scale parameters aganst umbrella alternatve wth at least one strct nequalty Remark 4 It can be noted that the proposed test has less varance then B-test of Gaur et al [8] Also the asymptotc varance computaton of the proposed test s smpler than the asymptotc varance computaton of B-test of Gaur et al [8] Acknowledgments The author s thankful to the anonymous referee for the valuable suggestons and comments whch led to the mprovement of the manuscrpt References [] Abebe A Sngh P (8) An exact test for U-shaped alternatves of locaton parameters: The Exponental dstrbuton case Communcatons n Statstcs Theory and Methods [] Bhat SV (9) Some k-sample rank tests for umbrella alternatves Research Journal of Mathematcs and Statstcs () 7-9 [3] Bhat SV Patl AB (8) A class of k-sample dstrbuton-free tests for the umbrella alternatves Internatonal Journal of Agrcultural and Statstcal Scences 4() 53-6 [4] Chen YI (993) Nonparametrc comparsons of umbrella pattern treatment effects wth a control n a one-way layout Communcatons n Statstcs Smulaton and Computaton (3) [5] Chen YI Wolfe DA (99) A study of dstrbuton-free tests for umbrella alternatves Bometrcal Journal [6] Chen YI Wolfe DA (993) Nonparametrc procedures for comparng umbrella pattern treatment effects wth a control n a one-way layout Bometrcs [7] Deshpande JV Kusum K (984) A test for the non-parametrc two sample scale problem Australan Journal of Statstcs 6() 6-4 [8] Gaur A Mahajan KK Arora S () New nonparametrc tests for testng homogenety of scale parameters aganst umbrella alternatve Statstcs and Probablty Letters [9] Gll AN Dhawan AK (999) A one-sded test for testng homogenety of scale parameters aganst ordered alternatves Communcatons n Statstcs Theory and Methods 8() [] Kossler W (6) Some c-sample rank tests of homogenety aganst umbrella alternatves wth unknown peak Journal of Statstcal Computaton and Smulaton 76() [] Kusum K Baga I (988) A new class of dstrbuton free procedures for testng homogenety of scale parameters aganst ordered alternatves Communcatons n Statstcs - Theory and Methods 7(5) [] Lehmann EL (963) Robust estmaton n analyss of varance Annals of Mathematcal Statstcs [3] Mack GA Wolfe DA (98) k-sample rank tests for umbrella alternatves Journal of Amercan Statstcal Assocaton [4] Pur ML (965) Some dstrbuton-free k-sample rank order test of homogenety aganst ordered alternatves Communcaton Pure Appled Mathematcs [5] Rao CR (973) Lnear Statstcal Inference and ts applcatons Wley Eastern Ltd New York [6] Rao KSM (98) Non-parametrc tests for homogenety of scale aganst ordered alternatves Annals of Insttute of statstcal and Mathematcs A(34) [7] Serflng RJ (98) Approxmaton theorems of mathematcal statstcs John Wley New York [8] Shanubhogue A (988) Dstrbuton-free test for homogenety aganst stochastc orderng Metrka

8 Anl Gaur: A Note on Test of Homogenety Aganst Umbrella Scale Alternatve [9] Shetty ID Bhat SV (997) A dstrbuton-free test for umbrella alternatves based on subsample medans Journal of Karnataka Unversty Scence [] Smpson DG Margoln BH (986) Recursve nonparametrc testng for dose-response relatonshps subject to down-turns at hgh dose Bometrka [] Sngh P Gll AN (4) A one-sded test based on sample quas ranges Communcatons n Statstcs Theory and Methods 33(4) [] Sngh P Lu W (6) A test aganst an umbrella ordered alternatve Computatonal Statstcs and Data Analyss

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