Hypothesis Testing on the Parameters of Exponential, Pareto and Uniform Distributions Based on Extreme Ranked Set Sampling

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1 Mddle-East Joural of Scetfc Research (9): 39-36, ISSN IDOSI Publcatos, DOI:.589/dos.mejsr Hypothess Testg o the Parameters of Expoetal, Pareto ad Uform Dstrbutos Based o Extreme Raed Set Samplg R. Shah Departmet of Statstcs, Scece ad Research Brach, Islamc Azad Uversty, Tehra, Ira Abstract: The problem of hypothess testg of the locato parameter of the two-parameter expoetal dstrbuto ad scale parameter of the pareto ad uform dstrbuto are cosdered. We costruct the test statstcs based o Raed Set Samplg () ad Extreme () ad compared ther powers to the power of Uformly Most Powerful (UMP) test by umercal computato. The results show that the test based o s more powerful tha the oe based o ad UMP test. Key words: Expoetal dstrbuto extreme raed set samplg hypothess testg pareto dstrbuto uform dstrbuto raed set samplg INTRODUCTION I samplg stuatos where the uts drow from a populato are dffcult or expesve to quatfy but ca be easly raed, the Raed Set Samplg () procedure, that was frst suggested by McItre [] ad the Extreme Raed Set Samplg () procedure, that was frst suggested by Samaw et al. [], provde effcet methods to estmate the populato parameters. Some researchers used to obta the tests, whch were most powerful tha the usual Uformly Most Powerful (UMP) tests based o Smple Radom Samples (). Amog them, Abu-Dayyeh ad Muttla [3] ad Muttla ad Abu-Dayyeh [4] costructed ew tests usg method for testg parameters of expoetal, uform ad ormal dstrbutos. They showed that these tests outperform substatally agast the usual UMP ad Lelhood Rato (LR) tests. Wag ad Tseg [5] troduced based tests for testg populato mea of ormal ad expoetal dstrbuto, whch were more powerful tha UMP ad UMP Ubased (UMPU) tests. Tseg ad Wu [6] costructed the smlar tests based o Meda (M) ad showed that the resultg tests were most powerful tha the tests based o. For testg scale parameters of the expoetal ad rectagular dstrbutos, some test procedure volvg the M data were troduced by Hossa ad Muttla [7]. They showed that for the scale parameter of expoetal dstrbuto, the M-based tests were more powerful tha -based tests ad for rectagular dstrbuto, the -based tests were much better tha the M-based tests. Correspodg Author: R. Shah, Departmet of Statstcs, Scece ad Research Brach, Islamc Azad Uversty, Tehra, Ira. 39 I ths paper we troduce some test statstcs based o ad procedures for testg locato parameter of the two-parameter expoetal dstrbuto ad scale parameter of the pareto ad uform dstrbutos, whch are more powerful tha the UMP test based o. To ths ed, Secto, we troduce the tests based o ad methods for testg locato parameter of the two-parameter expoetal dstrbuto ad compare them wth UMP test. I Sectos 3 ad 4, we troduce ew tests based o ad methods for testg scale parameter of the pareto ad uform dstrbutos ad compare them wth UMP tests. A cocluso s gve Secto 5. Testg the locato parameter of two-parameter expoetal dstrbuto Let X, X,, X be a radom sample from the two-parameter expoetal dstrbuto wth probablty desty fucto (p.d.f.) (x µ ) σ f = e I ( µ+, ), µ + (, ), σ> σ We wat to test H : µ=µ vs H : µ µ (.) It s well ow that the UMP test of sze α for testg (.) s gve by m{x} <µ or m{x} > c (, α) = otherwse UMPT( x ) (.)

2 Mddle-East J. Sc. Res., (9): 39-36, where x = (x,,x ). Wthout lost of geeralty we may tae µ = ad σ =. The c (, α ) = l( α) The power fucto of ths test s gve by ( µ ) = P(m{x} UMPT µ < ) + P(m{x} µ > l( α)) α µ< = α <µ< α µ> l( α) ( µ ) ( )e ( µ ) e l( ) (.3) To obta the test based o, let X, X, X ; X, X, X ; ; X, X, X be the groups of depedet radom varables, whch have the same p.d.f. ƒ(.). Let X (), X (), X () be the order statstcs of the radom sample X, X, X the -th group (I =,,,). The X (), X (), X () deotes the raed set sample, where X () s the -th order statstcs the -th group. To smplfy the otato, X () wll be deoted by Y throughout ths paper. To test the hypothess (.) based o the, we propose the followg test m{y} < or m{y} > c (, α) = otherwse () x (.4) To fd the value of C (,α) we must solve the equato ( µ ) = P(m{Y} ) + P(m{Y} > c (, α )) µ µ P(Y µ > ) + P(Y µ > c ) µ< = > <µ< µ> c. P(Y µ c ) c (.6) To costruct the test based o, we choose from the -th group X, X, X, the frst order statstcs,.e., X (), =,,. Therefore, X (), X (), X () deotes the extreme raed set sample. To smplfy the otato, X () wll be deoted by Z throughout ths paper. To test the hypothess (.) based o the, we propose the followg test m{z} < orm{z} > c (, α) = otherwse (.7) To fd the value of c (,α) we must solve the equato α= P (m{z} > c (, α )) = P (Z > c (, α) µ= µ= = + + (z) (z) (c ) e dz e dz { e } (.8) = = = c c Therefore, c = lα. The power fucto of the s gve by α= P (m{y} > c (, α )) = P (Y > c (, α) µ= µ= = = +! (y µ ) (y µ ) + e.e dy c ( )!( )! (.5) The, the power fucto of ths test s ( µ ) = P (m{ Z } ) + P (m{ Z } > c (, α )) µ µ ( µ ) (c µ ) { } { } (c µ ) { } + µ< e e = e <µ< c µ> c (.9) Table : Crtcal values of the sze-α, ad -based tests for the locato parameter of two parameter expoetal dstrbuto α =. α =.5 α = c c RRS c c c RRS c c c RRS c

3 Mddle-East J. Sc. Res., (9): 39-36, Fg. : ( ) µ, ( ) UMPT µ ad ( ) µ for α =.5 ad = 3,4,,8 The values of c, c ad c for dfferet values of α ad for = 3,4,,8 are calculated ad are gve Table (Computatos doe by umercal tegrato usg computer software Maple ). I Fg., plots of the, ad UMPT for = 3,4,,8 ad α =.5 are gve. These 3 fgures show that for all, s more powerful tha ad UMPT. Also, for = 3,4,,8, s more powerful tha UMPT. To get more quattatve comparso, let MRI deote the maxmum rate of the mprovemet, whch s defed by

4 Mddle-East J. Sc. Res., (9): 39-36, Table : The comparso betwee µ ad * ( ) µ ad * ( ) the maxmum rate of mprovemet (α =.5) α = µ * * * ( µ ) ( µ ) φ MRI ( µ ) ( µ ) * * MRI = * ( µ ) % where µ * = µ * (), for gve, s the value such that * * ( µ ) ( ) µ = max ( ( µ ) ( ) µ ). The comparso betwee ad terms of MRI, for = 3,4,,8, s preseted Table. Note that all values of MRI are above 37%; whch dcate that the mprovemet of over s substatal, hece practcally sgfcat. TESTING THE SCALE PARAMETER OF THE PARETO DISTRIBUTION Let X, X,, X be a radom sample from the pareto dstrbuto wth p.d.f. We wat to test β βθ f = I ( θ+, ), θ>, β> β+ x H : θ=θ vs H : θ θ (3.) wthout loss of geeralty, we assume that θ = ad β =. It ca be easly show that the UMP test of sze α for testg (3.) s gve by To test the hypothess (3.) based o the we propose the followg test m{y} < or m{y} > (, α) = otherwse (3.4) To fd the value of (,α) we must solve the equato α= P (m{y} > (, α )) = P (Y > (, α) θ= θ= + +! =. dy ( )!( )! y y! = ( ) + + (3.5) ( )!( )! = + + ( ) Furthermore the power fucto of ths test ca be wrtte as θ > + θ > θ< P(Y ) P(Y ) = > <θ< θ> P(Y θ ) (3.6) To obta the test based o method, from the samplg method gve Secto, we have the sample S = {Z, Z, Z }. Now for testg the hypothess (3.) wth sze α, we propose the followg test m{z} < or m{z} > (, α) = (3.7) otherwse where where m{x} < or m{x } > UMPT = otherwse gve by (3.) =. The power fucto of ths test s α θ ( α) θ< θ> = αθ <θ< UMPT (3.3) Let Y = X () be the -th order statstcs the -th group ( =,,,), whch was deoted Secto. 3 α= P (m{z} > (, α )) = P (Z > (, α) θ= θ= { z dz} = = ( ) Therefore =. Also the power fucto of α s gve by The values of, ad for dfferet values of α ad for = 3,4,,8 are calculated ad are gve Table 3 (Computatos doe by umercal tegrato usg computer software Maple ).

5 Mddle-East J. Sc. Res., (9): 39-36, Table 3: Crtcal values of the sze-α, ad -based tests for the scale parameter of pareto dstrbuto α =. α =.5 α = Fg. : ( ) θ, ( ) UMPT θ ad ( ) θ for α =. ad = 3,4,,8 33

6 Mddle-East J. Sc. Res., (9): 39-36, () θ = θ θ + θ< θ <θ< θ> I Fg., plots of the, (3.8) ad for UMPT = 3,4,,8 ad α =. are gve. These fgures show that for all, s more powerful tha ad UMPT. Also, for = 3,4,,8, s more powerful tha UMPT. Smlar to the case of expoetal dstrbuto, we compute the MRI of wth respect to ad observe that all values of MRI are above 8%, whch dcate that the mprovemet of over s substatal, hece practcally sgfcat. TESTING THE SCALE PARAMETER OF THE UNIFORM DISTRIBUTION Let X, X,.,X be a radom sample from the uform dstrbuto wth probablty desty fucto To test the hypothess (4.) based o the, Abu-Dayyeh ad Muttla [3] proposed the followg test max{y} > or max{y} < l (, α) 3 = otherwse (4.5) where Y = X () s the -th order statstcs the -th group ( =,,,) ad l s the soluto of the followg equato l! α= [ y ]. [ y] dy ( )!( )!! + = ( ). l (4.6) ( )!( )! = + Also the power fucto of ths test s gve by θ l ( θ ) = P(Y l ) l <θ 3 θ θ + θ θ> P(Y ) P(Y l ) (4.7) We wat to test f = I (,) θ, θ> θ (4.) H : θ=θ vs H : θ θ (4.) It s well ow that the UMP test of sze α for testg (4.) s gve by max{x} >θ or max{x} < l (, α) UMPT3 = (4.3) otherwse wthout lost of geeralty we tae θ =. The l (, α ) = α. The power fucto of ths test s gve by To obta the test based o, we choose from the -th group X, X,..., X, the largest order statstcs X, =,...,. Therefore () X, X,..., X deotes the extreme raed set ( ) ( ) ( ) sample. To smplfy the otato, X () wll be deoted by W throughout ths paper. To test the hypothess (4.) based o the we propose the followg test max{w} > or max{w} < l (, α) 3 = (4.8) otherwse To fd the value of l (,α) we must solve the equato θ = > + < α ( ) P(max{x} UMPT3 θ ) P(max{x} θ ) θ< α θ ( α) θ> θ = α α<θ< (4.4) 3 s gve by 34 α= P (max{w} < l (, α )) = P (W < l (, α) θ= θ= l = = w dw ( l ) (4.9) whch reduce to l = α. The power fucto of the

7 Mddle-East J. Sc. Res., (9): 39-36, Table 4: Crtcal values of the sze-α, ad -based tests for a scale parameter of the uform dstrbuto α =. α =.5 α = l l l l l l l l l Fg. 3: ( ) θ, ( ) UMPT3 θ ad ( ) 3 θ for α =. ad = 3,4,,8 3 35

8 Mddle-East J. Sc. Res., (9): 39-36, ( θ ) = P(max{W} ) + P(max{W} l (, α )) 3 θ θ θ< α θ ( α) θ> θ = α α<θ< (4.) The values of l, l ad l, for dfferet values of α ad for = 3,4,,8 are calculated ad are gve Table 4 (Computatos doe by umercal tegrato usg computer software Maple ). UMPT3 I Fg. 3, plots of the, 3 3 ad for = 3,4,,8 ad α =. are gve. From these fgures we observe that for all, 3 s more powerful tha 3 ad UMPT3. Also, for = 3,4,,8, 3 s more powerful tha UMPT3. Smlarly, we compute the MRI of 3 wth respect to 3 ad observe a substatal mprovemet. CONCLUSION Accordg to the results obtaed prevous sectos, we observe that the test troduced based o method o locato parameter of the twoparameter expoetal dstrbuto ad scale parameters of the pareto ad uform dstrbutos are more powerful tha -based ad UMP tests. Also, based tests are more powerful tha UMP test. So, stuatos where the measuremet of survey varable s costly ad/or tme-cosumg but rag of sample tems relatg to the survey varable ca easly doe, the test based o s recommeded. REFERENCES. McItyre, G.A., 95. A method for ubased selectve samplg, usg raed sets. Austral. J. Agrcult. Res., 3: Samaw, H., W. Abu-Dayyeh ad S. Ahmed, 996. Extreme raed set samplg. Bomatrcal Joural, 3: Abu-Dayyeh, W. ad H.A. Muttla, 996. Usg raed set samplg for hypothess tests o the scale parameters of expoetal ad uform dstrbutuos. Pasta Joural of Statstcs, : Muttla, H.A. ad W. Abu-Dayyeh, 998. Testg some hypotheses about the ormal dstrbuto usg raed set sample: A more powerful test. Joural of Iformato ad Optmzato Sceces, 9: Wag, M.F. ad Y.L. Tseg,. Raed-setsample based better tests for a ormal mea. J. Chese Statst. Assoc., 4 (3): Tseg, Y.L. ad S.W. Wu, 7. Raed-setsample-based tests for ormal ad expoetal meas. Commu. Statst. Smul. Computat., 36: Hossa, S.S. ad H.A. Muttla, 6. Hypothess tests o the scale parameter usg meda raed set samplg. Statstca, ao LXVI,