Four combination methods of independent tests for testing a simple hypothesis versus one-sided alternative are considered viz. Fisher, the logistic, the sum of P-values and the inverse normal method in case of logistic distribution. These methods are compared via local power in the presence of nuisance parameters for some values of α using simple random sample

Abu-Dayyeh, Walid A.

Al-Momani, M. M. A.

Al-Rawi, Z. R.

Digital Commons@Wayne State University

Journal of Modern Applied StatisticalMethodsVolume 4 | Issue 1 Article 85-1-2005Local Power For Combining Independent Tests inThe Presence of Nuisance Parameters For TheLogistic DistributionWalid A. Abu-DayyehYarmouk University, Irbid Jordan, abudayyehw@yahoo.comZ. R. Al-RawiYarmouk University Irbid-JordanM. M. A. Al-MomaniYarmouk University Irbid-JordanFollow this and additional works at: http://digitalcommons.wayne.edu/jmasmPart of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and theStatistical Theory CommonsThis Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted forinclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState.Recommended CitationAbu-Dayyeh, Walid A.; Al-Rawi, Z. R.; and Al-Momani, M. M. A. (2005) "Local Power For Combining Independent Tests in ThePresence of Nuisance Parameters For The Logistic Distribution," Journal of Modern Applied Statistical Methods: Vol. 4 : Iss. 1 , Article 8.DOI: 10.22237/jmasm/1114906080Available at: http://digitalcommons.wayne.edu/jmasm/vol4/iss1/8Journal of Modern Applied Statistical Methods   Copyright © 2005 JMASM, Inc. May, 2005, Vol. 4, No.1, 75-80                                                                                                                                 1538 – 9472/05/$95.00 75 Local Power For Combining Independent Tests in The Presence of Nuisance Parameters For The Logistic Distribution  W. A. Abu-Dayyeh      Z. R. Al-Rawi      M. MA. Al-Momani Department of Statistics, Faculty of Science Yarmouk University Irbid-Jordan   Four combination methods of  independent tests for testing a simple hypothesis versus one-sided alternative are considered viz. Fisher, the logistic, the sum of P-values and the inverse normal method in case of logistic distribution. These methods are compared via local power in the presence of nuisance parameters for some values of α using simple random sample.  Key words: combination method; independent tests; logistic distribution; local power; simple random sample; nuisance parameter.   Introduction  Combining independent tests of hypotheses is an important and popular statistical practice. Usually, data about a certain phenomena comes from different sources in different times, so we want to combine these data to study such phenomena. Many authors have considered the problem of combining (n) independent tests of hypotheses. For simple null hypotheses, Little and Folks (1971), studied four methods for combining a finite number of independent tests. They found that the Fisher method is better than the other three methods via Bahadur efficiency. Again, Little and Folks (1973) studied all methods of combining a finite number of independent tests and thy found that the Fisher's method is optimal under some mild conditions.    W. A. Abu-Dayyeh, Department of Mathematical Sciences, Dhahran,   Saudi Arabia M. MA. Al-Momani, Department of Mathematical Sciences, Dhahran,   Saudi Arabia Z. R. Al-Rawi Chairman, Department of Statistics Yarmouk University, Irbid, Jordan For correspondence regarding this article, send E-mail to alrawiz@yu.edu.jo. This work was carried out with financial support from the Yarmouk University Research Council.    Brown, Cohen and Strawderman (1976) have shown that such all tests form a complete class. Abu-Dayyeh and Bataineh (1992) showed that the Fisher's method is strictly dominated by the sum of P-values method via Exact Bahadur Slop in case of combining an infinite number of independent shifted exponential tests when the sample size remains finite. Also, Abu-Dayyeh (1992) showed that under certain conditions that the local limit of the ratio of the Exact Bahadur efficiency of two tests equivalent to the Pitman efficiency between the two tests where these tests are based on sum of iid r.v’s. Again Abu-Dayyeh and El-Masri (1994) studied the problem of combining (n) independent tests as     (n →  ∞ ) in case of triangular distribution using six methods viz. sum of P-values, inverse normal, logistic, Fisher, minimum of P-values and maximum of P-values. They showed that the sum of P-values is better than all other methods. Abu-Dayyeh (1997) extended the definition of the local power of tests to the case of having nuisance parameters. He derived the local power for any symmetric test in the case of a bivariate normal distribution with known correlation coefficient, and then he applied it to the combination methods.  Specific Problem Suppose there is (n) simple hypotheses:    H0(i) : θi = θ0i         vs     H1(i) : θi > θ0i   i=1,2,…,n                                                                      (1) ABU-DAYYEH, AL-RAWI, & AL-MOMANI 76 Where θ0i is known for i=1,2,…,n and H0(i) is rejected for sufficiently large values of some continuous real valued test statistic T(i) , i=1,2,…,n and we want to combine the (n) hypotheses into one hypothesis as follows:            H0: (θ1, θ2 , …, θn ) = (θ01 , θ02 , …, θ0n  )     vs           H1: θi ≥ θ0i for all i, and θi > θ0i for some i, i=1,2, …, n                                          (2)                          Many methods have been used for combining several tests of hypotheses into one overall test.  Among these methods are the non-parametric (omnibus) methods that combine the P-values of the different tests.  The P-value of the i-th hypothesis is given by:             )(1)()(0)(0)( tFtTPPii HiHi−=≥=           (3)  where FH0(i)(t) is the cdf of T (i)under H0(i).  Note that Pi ~ U(0,1) under H0(i).                  Considered in this article is the case of ii θγθ =∗ , where 0,...,, 21 ≥rθθθ fixed constants and γ  is the unknown parameter. Then ( ) ( ) ( )rTTT ,...,, 21  are independent r.v’s such that for ri ,..,2,1= and we want to test   0:0 =γH             vs 0:1 >γH  `                                       (4)  and therefore considered is the problem of combining a finite number of independent tests by looking at the Local Power of tests which is defined for a test φ by:            ( ) ( ) 0P EinfL =γθγθ ϕγ∂∂=ϕ                     (5)                       where   ( ) riir ,...,2,1,0,,...,,,0 21 =≥=≥ θθθθθγ , in case of logistic distribution. Compared (5) for the four methods of combining tests for the location family of distributions when 2=r and 3=r . These methods are: Fisher, logistic, the sum of  p-values and the inverse normal methods.    Methodology  Now we will find expressions for the Local Power of the four combination methods of tests then compare them via the Local Power.  Lemma 1    Let 21, XX  be independent r.v’s such that ( )1,~ ii LogisticX θγ for 2,1=i . Then  A(1)     ( )( ) ( )21F0F2,1 KE θ+θ=ϕγ∂∂=γθθγ ,where ∫−⎟⎠⎞⎜⎝⎛−=−acF dyyyyeK132 21 , 2cea =  and ( ) ( )2 1,4 αχ −=c   A(2)    ( )( ) ( )21L0yL2,1 KE θ+θ=ϕγ∂∂=θθγ , where ( )( )( )∫∞−+−−−=13112 dyyeyyyKcL , and c satisfies the following ( )( )21111ccece−−−+−=−α .  A(3)    ( )( ) ( )21S0S2,1 KE θ+θ=ϕγ∂∂=γθθγ ,where  ( )6232 ccKS−= , and α2=c .   A(4)   LOCAL POWER FOR COMBINING INDEPENDENT TESTS 77 ( )( ) ( )21N0N2,1 KE θ+θ=ϕγ∂∂=γθθγ ,where  ∫−⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛Φ−−Φ−−= −aN dyyyycK131 211 , ( )ca −Φ=1and ( )α−Φ= − 12 1c .   Proofs of the previous lemma are similar to proofs of lemma 2, so we will not write it.    Lemma 2   Let 321 ,, XXX  be independent r.v’s such that ( )1,~ ii LogisticX θγ for 3,2,1=i . Then   B(1)   ( )( )( )1 2 33, ,0 11 2 3               whereF F iiFE KKγ θ θ θγϕ θγθ θ θ= =∂=∂= + +∑,  ( ) dyyyycyeKa cF 312 2ln211 −⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−+−=∫−, 2cea = , and ( ) ( )2 1,6 αχ −=c .  `B(2)    , where  ( )( )( )( )( )∫ ∫∞∞−+−−−−−−=1 1321111211 dvduvuevuvvuKcL , and c satisfies the following:  ( )( )( )( )∫ ∫∞∞−+−−−−=−1 1221111111 dvduvuevuvucα      B(3)   ( ) ( )( )1 2 3, ,031 2 31whereSS i SiEK Kγ θ θ θγϕγθ θ θ θ==∂∂= = + +∑,           ( )1223 ccKS−=  and 3 6α=c .  B(4)    ( ) ( )( )1 2 33, ,0 11 2 3N N iiNE KKγ θ θ θγϕ θγθ θ θ= =∂=∂= + +∑, where           ( ) ( )( )( ) ( )1 11 11 1 2Na bKc u v v dudv− −=− −Φ − −Φ −Φ −∫∫  ( )ca −Φ= , ( )( )vcb 1−Φ−−Φ= , and ( )α−Φ= − 13 1c .     Now, we will prove just B(1), because the proof of the others can be done in the same way.   Proof of B(1):  ( ) ( ) ( )1 2 33, ,1,F F i i iiE f x dx∞ ∞ ∞γ θ θ θ=−∞ −∞ −∞ϕ = φ − γθ∏∫ ∫ ∫ where ( )iixf γθ− is the fdp .. of  ( ) 3,2,11, =iforLogistic iγθ    It easy to show that: ( ) ( )( ) ( )1 2 3, ,311 1FF i i iiEf x dxγ θ θ θ ϕφ γθ∞ ∞ ∞=−∞ −∞ −∞= − − −∏∫ ∫ ∫ ABU-DAYYEH, AL-RAWI, & AL-MOMANI 78 so, ( ) ( )( ) ( )1 2 3, ,311 1FF i i iiEf x dxγ θ θ θ ϕγφ γθγ∞ ∞ ∞=−∞ −∞ −∞∂∂⎡ ⎤∂= − − −⎢ ⎥∂⎣ ⎦∏∫ ∫ ∫  ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )1 2 3, ,0\ \1 2 3 1 2 3\1 2 31 2 31F FEf x f x f x f x f x f xf x f x f xdx dx dxγ θ θ θγϕ ϕγ∞ ∞ ∞=−∞ −∞ −∞∂= − − ×∂⎧ ⎫+⎪ ⎪⎨ ⎬+⎪ ⎪⎩ ⎭∫ ∫ ∫ such that when   ( ) ( )( )2\30 , 1, 2,3.1i iix xiixe ef x for ie− −−θ −γ = = =+ By symmetric of ix  we have   ( )( ) F31ii0F3,2,1 KE ⎟⎠⎞⎜⎝⎛∑θ=ϕγ∂∂==γθθθγ , where  ( ) ( ) ( )( )( )3 31 21 2 321 2 32 2 311 1 1Fx xx xFx x xKe ee e dxdx dxe e eϕ− −∞ ∞ ∞− −− − −−∞−∞−∞=−− −+ + +∫ ∫ ∫ where ( )⎪⎩⎪⎨⎧≤−=−∑=wocpiiF.,0ln2,1131ϕ , 3,2,1,11=+= iepixi  ( ) ( ) ( ) cppp ≤−−− 321 ln2ln2ln2 implies that ( )( )⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−++≤ 111ln3221 xxceeex , also ( ) ( ) cpp ≤−− 32 ln2ln2 and ( ) cp ≤− 3ln2 implies that ( )⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−+≤ 11ln322 xceex  and ⎟⎟⎠⎞⎜⎜⎝⎛−≤ 1ln 23cex respectively.   Let⎟⎟⎠⎞⎜⎜⎝⎛−= 1ln 2cea , ( )⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−+= 11ln32xceeb and let ( )( )⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−++= 111ln322xxceeed , then we will get  ( ) ( )( )( )3 31 21 2 321 2 32 2 31 1 1x xx xa b dx x xFe ee edx dx dxe e eK− −− −− − −−∞ −∞ −∞−−+ + +=∫ ∫ ∫.   Let ( )∫∞− −−+=dxxdxeeI 121111, then put 11 xeu −+= to get that deI−+=111 , so , ( )( )111 3221 ++−= − xxc eeeI .                           ( )( )( ) ( )( )( )3 32322 3221 22 31 1 11 1x xxa bcxxx xFe eee e e dxdxe eK− −−−− −−∞ −∞−− − + ++ +∴ =∫ ∫ Also, let ( ) ( )( )∫∞−−−−⎟⎟⎠⎞⎜⎜⎝⎛++−+=bxxcxxdxeeeeeI 2222 11113222 LOCAL POWER FOR COMBINING INDEPENDENT TESTS 79        ( )( ) ( )223222221111b xxbcxxe dxee e dxe−−−∞−−−∞=+− ++∫∫        ( )( ) ( )33 3221 11 ln 12cxcx xe ece e e−−= − +⎛ ⎞− + − +⎜ ⎟⎝ ⎠       ( ) ( )⎟⎠⎞⎜⎝⎛+−++−=−1ln2111 332 xxcecee  ( ) ( ) ( )3 33 332231 1 1 ln 121Fa x xcx xxKe e ce e e dxe− −−−−∞∴ =− ⎡ ⎤⎛ ⎞− − + + − +⎜ ⎟⎢ ⎥+⎝ ⎠⎣ ⎦∫. Finally put   31 xey += we get ( ) dyyyycyeKd cF∫−⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−+−=−132 2ln211 , 2ced =   ( ) ( )3 30 01 12 ln 1 2 ln ,i ii iP p c P p c= =α= − ≥ = − − ≤⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑ because ( ) ( )2631~ln2 χ∑=−iip  under 0H ,  then ( ) ( )2 1,6 αχ −=c , which completes the proof.    Also, here for the logistic distribution we will compare the Local Power for the previous four tests numerically. So from tables (1) and (2) when 01.0=α and 2=r the sum of p-values method is the best method followed by the inverse normal method, the logistic method and Fisher method respectively, but for all of the other values of α and r the inverse normal method is the best method followed by the sum of p-values method followed by logistic method and the worst method is Fisher method.   References  Abu-Dayyeh, W. A. (1989).   Bahadur exact slope, pitman efficiency  and local power combining independent tests. Ph.D. Thesis, University of Illinois at Urbana – Champaign. Abu-Dayyeh, W. A. (1992). Exact bahadur efficiency. Pakistan Journal of Statistics, 8 (2),  53-61.         Abu-Dayyeh, W. A., &  Bataineh (1992). Comparing the exact Bahadur of the Fisher and sum of P-values methods in case of shifted exponential distribution. Mu’tah slopes.  Journal for Research and Studies, 8, 119-130. Abu-Dayyeh, W. A., & El-Masri. (1994). Combining independent tests of  triangular distribution. Statistics & Probability Letters, 21, 195-202. Abu-Dayyeh, W. A. (1997). Local power of tests in the prescience of nuisance parameters with an application. The Egyptian Statistical Journal ISSR, 41, 1-9.    Little, R. C., & Folks, L. J. (1971). Asymptotic optimality of Fisher’s method of combining independent tests. Journal of the American Statistical Association, 66, 802-806. Little, R. C., & Folks, L. J. (1973). Asymptotic optimality of Fisher’s method of combining Independent Tests II. Journal of the  American Statistical Association, 68, 193-194.              ABU-DAYYEH, AL-RAWI, & AL-MOMANI 80                                      The following tables explain the term AK  where { }NSLFA ,,,∈ for the logistic distributions.  Table (1): Local power for the logistic distribution when ( )2=r   α  FK  LK  SK  NK  0.010 0.0073833607 0.0081457298 0.0090571910 0.0089064740 0.025 0.0174059352 0.0192749938 0.0212732200 0.0214554551 0.050 0.0326662436 0.0361783939 0.0394590744 0.0415197403   Table (2): Local power for the logistic distribution when ( )3=r   α  FK  LK  SK  NK  0.010 0.0062419188 0.0071070250 0.0080425662 0.0083424342 0.025 0.0144747833 0.0165023359 0.0183583839 0.0199610766 0.050 0.0267771426 0.0304639648 0.0332641762 0.0381565019  

Local Power For Combining Independent Tests in The Presence of Nuisance Parameters For The Logistic Distribution

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Local Power For Combining Independent Tests in The Presence of Nuisance Parameters For The Logistic Distribution

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