Interval Estimation for the Ratio of Percentiles from Two Independent Populations.

Transcription

1 Est Teessee Stte Uiversity Digitl Est Teessee Stte Uiversity Electroic Theses d Disserttios Itervl Estimtio for the Rtio of Percetiles from Two Idepedet Popultios. Pius Mthek Muidi Est Teessee Stte Uiversity Follow this d dditiol works t: Recommeded Cittio Muidi, Pius Mthek, "Itervl Estimtio for the Rtio of Percetiles from Two Idepedet Popultios." (008). Electroic Theses d Disserttios. Pper This Thesis - Ope Access is brought to you for free d ope ccess by Digitl Est Teessee Stte Uiversity. It hs bee ccepted for iclusio i Electroic Theses d Disserttios by uthorized dmiistrtor of Digitl Est Teessee Stte Uiversity. For more iformtio, plese cotct dcdmi@etsu.edu.

2 Itervl Estimtio for the Rtio of Percetiles from Two Idepedet Popultios A thesis preseted to the fculty of the Deprtmet of Mthemtics Est Teessee Stte Uiversity I prtil fulfillmet of the requiremets for the degree Mster of Sciece i Mthemticl Scieces by Pius Muidi August 008 Robert Price, Ph.D., Chir Robert Grder, Ph.D. Edith Seier, Ph.D. Yli Liu, Ph.D Keywords: cofidece itervls (CI), rtio of percetiles, percetiles.

3 ABSTRACT Itervl Estimtio for the Rtio of Percetiles from two Idepedet Popultios by Pius Muidi Percetiles re used everydy i descriptive sttistics d dt lysis. I rel life, my qutities re ormlly distributed d orml percetiles re ofte used to describe those qutities. I life scieces, distributios like expoetil, uiform, Weibull d my others re used to model rtes, clims, pesios etc. The eed to compre two or more idepedet popultios c rise i dt lysis. The rtio of percetiles is just oe of the my wys of comprig popultios. This thesis costructs lrge smple cofidece itervl for the rtio of percetiles whose uderlyig distributios re kow. A simultio study is coducted to evlute the coverge probbility of the proposed itervl method. The distributios tht re cosidered i this thesis re the orml, uiform d expoetil distributios.

4 Copyright by Pius Muidi 008 3

5 DEDICATION To my wife Sus. 4

6 ACKNOWLEDGMENTS I wish to ckowledge ll the grdute studets d stff members of the Deprtmet of Mthemtics t Est Teessee Stte Uiversity. I would like to thk Dr. Edith Seier d Dr. Yli Liu for their iput d support. Specil thks goes to Dr. Robert Grder who hs mde my study t ETSU successful d ejoyble. Most importtly I wish to ckowledge Dr. Robert Price, my dvisor, who hs gretly cotributed to my iterest d kowledge i sttistics d without whom this project would hve bee lmost impossible. Lstly I wish to thk my wife d my fmily for their morl d ficil support. 5

7 CONTENTS ABSTRACT DEDICATION ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES INTRODUCTION Mximum Likelihood Estimtors (MLEs) NORMAL DISTRIBUTION Ubised Estimtors of µ d σ Approximte Cofidece Itervl for the Rtio of Percetiles from Two Idepedet Norml Distributios Simultio Results EXPONENTIAL DISTRIBUTION Ubised Estimtor of θ Approximte Cofidece Itervl for the Rtio of Percetiles from Two Idepedet Expoetil Distributios Simultio Results UNIFORM DISTRIBUTION Ubised Estimtors of d b Approximte Cofidece Itervl for the Rtio of Percetiles from Two Idepedet Uiform Distributios Simultio Results

8 5 CONCLUSION Results Limittios BIBLIOGRAPHY APPENDICES Appedix A: R Code for the CI of Norml Percetiles Rtio Appedix B: R Code for the CI of Expoetil Percetiles Rtio.. 50 Appedix C: R Code for the CI of Uiform Percetiles Rtio VITA

9 LIST OF TABLES 1 Empiricl Coverge Rtes of 90%, 95% d 99% Cofidece Itervls for the Rtio of Percetiles from Two Norml Popultios Empiricl Coverge Rtes of 90%, 95% d 99% Cofidece Itervls for the Rtio of Percetiles from Two Expoetil popultios Empiricl Coverge Rtes of 90%, 95% d 99% Cofidece Itervls for the Rtio of Percetiles from Two Uiform popultios

10 LIST OF FIGURES 1 Coverge Rtes vs Smple Size for Rtio of Norml Percetiles t p Coverge Rtes vs Smple Size for Rtio of Expoetil Percetiles t p Coverge Rtes vs Smple Size for Rtio of Uiform Percetiles t p

11 1 INTRODUCTION Percetiles or qutiles re very commo i sttistics. Percetiles re used to determie the vlue of observtio with give percetge below it. Percetiles re used s mesure of cetrl tedecy s well s mesure of spred. The medi, which is oe of the most used mesures of cetrl tedecy, refers to the 50th percetile. The lower d upper qurtiles which refer to the 5th d the 75th percetiles, respectively, re commoly used s mesures of dispersio. The five-umber summry used i mkig the fmous box plot is bsiclly summry of specific percetiles of distributio mely 1st, 5th, 50th, 75th d 100th. Qutiles re lso used i mesurig relibility by determiig the time to filure, survivl d hzrd fuctios [3]. Percetiles re lso used i hydrology d i sttisticl process cotrol [3]. The eed to compre two distributios rises d the resercher will require some prticulr prmeter s poit of compriso. I my istces reserchers re compelled to use the me s the poit of compriso with the ssumptio tht the me is the most relible prmeter for describig the popultio. This is ot lwys the cse d sometimes the medi my be more relible whe the distributio is strogly skewed [7]. The resercher my lso be iterested i other percetiles of the popultio d if there re two idepedet popultios compriso of the percetiles my be of vlue. Approximte distributio-free itervls for both the differece d rtio of medis hve bee developed [7]. We cosider itervl estimtio for the rtio of percetiles whe the uderlyig distributios re kow. 10

12 We begi by defiig few of the termiologies used i this thesis. A percetile or qutile is the vlue of vrible below which certi percetge of observtios fll. A percetile c lso be defied s oe of the 99 poit scores tht divide rked distributio ito groups, ech of which coti 1/100 of the scores. A Prmeter is umber tht describes the popultio. I sttisticl prctice, the vlue of prmeter is usully ot kow becuse it is difficult to exmie the etire popultio. A Sttistic is umber tht c be computed from smple without mkig use of y ukow prmeters. I prctice, we ofte use sttistic to estimte ukow prmeter. If the rdom vribles X 1, X,..., X re rrged i scedig order of mgitude d the writte s X (1), X (),..., X (), we cll X (i) the ith order sttistic [] (i 1,,..., ). If E[u(X 1, X,..., X )] θ, the sttistic u(x 1, X,..., X ) is clled ubised estimtor of θ [5]. Theorem 1.1 [6] Drw simple rdom smple of size from y popultio with me µ d stdrd devitio σ. Whe is lrge, the smplig distributio of the smple me x is pproximtely orml with me µ d stdrd devitio σ/. This is referred to s the cetrl limit theorem. Theorem 1. [4] Let X be sequece of rdom vribles such tht (X θ) D N(0, σ ). Suppose the fuctio f(x) is differetible t θ d f (θ) 0. The (f(x ) f(θ)) D N(0, σ (f (θ)) ) where D deotes coverge i distributio. This is referred to s the Delt Method. 11

13 Theorem 1.3 [4] Let {X }be sequece of p-dimesiol vectors. Suppose (X µ 0 ) D N p (0, Σ). Let g be trsformtio g(x) (g 1 (x),..., g k (x)) such tht 1 k p d the k p mtrix of prtil derivtives, [ ] dgi B dµ j i 1,..., k; j 1,..., p, re cotiuous d do ot vish i eighborhood of µ 0. Let B 0 B t µ 0. The (g(x ) g(µ 0 )) D N k (0, B 0 ΣB 0 ). This is extesio of the Delt Method. 1.1 Mximum Likelihood Estimtors (MLEs) Let X 1, X,..., X be rdom smple from distributio tht depeds o oe or more ukow prmeters θ 1, θ,..., θ m with p.m.f. or p.d.f. deoted by f(x; θ 1, θ,..., θ m ). Suppose tht θ 1, θ,..., θ m is restricted to give prmeter spce Ω. The the joit p.m.f. or p.d.f. of X 1, X,..., X mely L(θ 1, θ,..., θm) f(x 1 ; θ 1,..., θm)f(x ; θ 1,..., θ m )... f(x ; θ 1,..., θ m ), (θ 1, θ,..., θ m ) Ω, whe regrded s fuctio of θ 1, θ,..., θ m is clled the likelihood fuctio. Sy [u 1 (X 1, X,..., X ), u (X 1, X,..., X ),... u m (X 1, X,..., X )] is tht m-tuple i Ω tht mximizes L(θ 1, θ,..., θ m ). The 1

14 ˆθ 1 u 1 (X 1, X,..., X ) ˆθ u (X 1, X,..., X ). ˆθ m u 1 (X 1, X,..., X ) re mximum likelihood estimtors of θ 1, θ,..., θ m respectively; d the correspodig vlues of these sttistics, mely u 1 (X 1, X,..., X ), u (X 1, X,..., X ),... u m (X 1, X,..., X ) re clled mximum likelihood estimtes [5]. I my prcticl cses these estimtors (d estimtes) re uique. For my pplictios there is just oe ukow prmeter. I these cses, the likelihood fuctio is give by L(θ) f(x i ; θ). i1 Let X 1, X,..., X be cotiuous rdom vribles from popultio whose uderlyig distributio is kow d prmeters ukow. Let k p deote the 100pth percetile. k p c be expressed i terms of the popultio prmeters. By wy of mximum likelihood estimtio, we c pproximte the ukow popultio prmeters usig sttistics obtied from smple drw from the popultio whose uderlyig distributio is kow. The k p c be pproximted usig those estimtes. 13

15 Let k p,x be the 100pth percetile from the first popultio (X) d let k p,y be the 100pth percetile from the secod popultio (Y ). Deote the percetile rtio s k p rtio. The k p rtio is give by k p,x /k p,y. We ext discuss pproximte (1 α)100% cofidece itervl for the k p rtio. To costruct cofidece itervl for k p rtio we will estimte the turl log of k p rtio d the vrice of the estimted turl log of k p rtio d the expoetite the ed poits of the itervl. This is bsed o the delt method. 14

16 NORMAL DISTRIBUTION Let X 1, X,..., X be rdom smple from orml popultio with me µ d vrice σ. The the p.d.f. of X is give by f(x) 1 σ π e (x µ) /σ, < x <. (1) Let k p deote the 100pth percetile, the k p µ + Z p σ () where Z p deotes the 100pth percetile of the stdrd orml distributio N(0, 1) [1]. Sice µ d σ re ukow, we eed to estimte k p. This c be doe usig the miimum vrice ubised estimtors of µ d σ..1 Ubised Estimtors of µ d σ Let X 1, X,..., X be rdom smple from orml popultio N(µ, σ ) d let X deote the smple me d S deote the smple vrice where X 1 X i, (3) i1 S 1 1 (X i X), (4) i1 d S 1 1 (X i X) (i 1,,..., ). (5) i1 X is the mximum likelihood estimtor of µ d by the cetrl limit theorem X is ubised estimtor of µ d E( X) µ. 15

17 Lemm.1 [5] Let X 1, X,..., X be rdom smple of size from orml distributio N(µ, σ ). The the distributio of ( 1)S /σ is χ ( 1), where χ ( 1) is Chi-squre distributio with -1 degrees of freedom. Propositio. S is ubised estimtor of σ. Proof We c use Lemm.1 to show tht S is ubised estimtor of σ. By Lemm.1, the distributio of ( 1)S /σ is χ ( 1). Therefore ( ) ( 1)S E σ E ( χ ( 1) ) 1 σ E(S ) 1 E(S ) σ. However, S is ot ubised estimtor of σ. Thus, we eed to fid costt C tht c be used to remove the bisess such tht E(CS) CE(S) σ. We kow from Lemm.1 ( 1)S σ χ ( 1). So, 1S σ χ ( 1). Next we eed to fid the p.d.f. of Y χ ( 1). Suppose f(x) d g(y) re p.d.f. s of χ ( 1) d χ ( 1) respectively. The f(x) 1 Γ( 1 1 ) x 1 1 e x, 0 x <. (6) By chge of vrible techique we hve y x x y d dx dy y. Thus g(y) 1 Γ( 1 1 ) (y ) 1 1 e (y ) y, 0 y <. (7) 16

18 Now, E(Y ) y 1 Γ( 1 (y ) 1 1 ) 1 Γ( 1 1 Γ( 1 1 ) Lettig t y, dt ydy d we obti, (y ) 1 1 e y ydy 1 ) (y ) 3 e y ydy (y ) e y ydy. Hece, E(Y ) Γ( 1 1 Γ( 1 1 Γ( 1 1 ) Γ( ) Γ( 1 1 ) 1 ) 1 ) 0 t e t dt t 1 e t dt t 1 e t dt t 1 e t 0 Γ( ) dt. }{{} 1 Sice Y χ ( 1) 1S, E(Y ) 1E(S) σ σ E(Y ) Γ( ) Γ( 1 ). (8) E(S) σe(y ) 1 d therefore, σ Γ( ) 1 σ 1 σ Γ( 1) Γ( ) Γ( 1 ) 1. 1 Γ( ) Γ( ) 17 (9)

19 Hece, we hve 1 1 Γ( ) Γ( ) E(S) σ C 1 1 Γ( ) Γ( ) (10) d E(CS) σ. (11) Thus ubised estimtor of k p is ˆk p X + Z p CS [1]. (1) To compute cofidece itervls we eed the vrice d the estimted vrice of the sttistic of iterest. I this cse we require the vrice of ˆk p. Therefore V r(ˆk p ) V r( X + Z p CS) V r( X) + (CZ p ) V r(s) σ + [ C Zp E(S ) (E(S)) ] ] σ + C Zp [σ σ by propositio. d (11) C σ σ (1 + C Z p(1 1C ) ) Thus the estimted vrice of ˆk p deoted ( 1 + Z p (C 1) ) [1]. (13) V r(ˆk p ) is ( V S ( r(ˆk p ) 1 + Z p (C 1) )). (14) 18

20 . Approximte Cofidece Itervl for the Rtio of Percetiles from Two Idepedet Norml Distributios I this sectio we re goig to develop method of computig the (1 α)100% cofidece itervls for the rtio of two orml percetiles. These two percetiles must come from two idepedet orml distributios d they must be the sme 100pth percetile. For exmple, if we use the 45th percetile from the first distributio the we must use the 45th percetile from the secod distributio. The mes, vrices, d smple sizes of the two smples eed ot be equl. Theorem.3 Let X 1, X,..., X be rdom smple of size from orml popultio X with me µ x d vrice σ x d let Y 1, Y,..., Y m be rdom smple of size m from other orml popultio Y with me µ y d vrice σ y where X d Y re idepedet. Let k p,x d k p,y be the 100pth percetiles from popultios X d Y, respectively. The pproximte (1 α)100% cofidece itervl for the rtio of the percetiles k p,x d k p,y is give by ˆk p,x exp ± Z V r(ˆk p,x ) (1 α ˆk ) + p,y ˆk p,x V r(ˆk p,y ) ˆk p,y. (15) Proof We kow tht k p,x µ x + Z p σ x d k p,y µ y + Z p σ y (16) d the ubised estimtors of k p,x d k p,y re ˆk p,x X + Z p C x S x d ˆkp,y Ȳ + Z pc y S y (17) 19

21 respectively, where 1 1 Γ( ) C x Γ( ) d C y The vrices of these ubised estimtors ˆk p,x d ˆk p,y re V r(ˆk p,x ) σ x m 1 m 1 Γ( ) Γ( m). (18) ( 1 + Z p (Cx 1) ) d V r(ˆk p,y ) σ y ( 1 + mz m p (Cy 1) ) (19) d their respective estimted vrices re V r(ˆk p,x ) S ( x 1 + Z p (Cx 1) ) d The estimted k p rtio deoted by ˆk p rtio is give by V r(ˆk p,y ) S y ( 1 + mz m p (Cy 1) ). (0) ˆk p rtio ˆk p,x ˆk p,y. (1) Now we eed to fid the vrice of the ˆk p rtio. This c be simplified by itroducig turl logrithms d usig the delt method to compute the vrice. ) l(ˆk p rtio) l ( ˆkp,x ˆk p,y l(ˆk p,x ) l(ˆk p,y ). () The ext few equtios employ the delt method to compute the vrice of l(ˆk p rtio). [ d l(ˆkp rtio) dˆk p,x V r(l(ˆk p rtio)) d l(ˆk p rtio) ] [ V r(ˆk p,x ) 0 dˆk p,y 0 V r(ˆk p,y ) ] d l(ˆk p rtio) dˆk p,x d l(ˆk p rtio) dˆk p,y The vrice covrice mtrix hs zero etries becuse X d Y re idepedet d their covrice is zero. [ ] [ ] [ ] 1 1 V r(l(ˆk p rtio)) ˆk 1 V r(ˆk p,x ) 0 ˆk p,x p,x ˆk p,y 0 V r(ˆk p,y ) 1. (3) ˆk p,y 0.

22 Hece V r(l(ˆk p rtio)) V r(ˆk p,x ) ˆk p,x + V r(ˆk p,y ) ˆk p,y (4) d the estimted vrice of the l(ˆk p rtio) is V r l(ˆk p rtio) V r(ˆk p,x ) + ˆk p,x V r(ˆk p,y ) ˆk p,y. (5) The (1 α)100% cofidece itervl for the l(k p rtio) c be computed s l(ˆk p,x ) l(ˆk p,y ) ± Z (1 α ) V r(ˆk p,x ) ˆk p,x + V r(ˆk p,y ) ˆk p,y (6) d expoetitig the expressio we hve the (1 α)100% cofidece itervl for the k p rtio s ˆk p,x exp ± Z (1 α ˆk p,y ) V r(ˆk p,x ) ˆk p,x + V r(ˆk p,y ) ˆk p,y..3 Simultio Results To test this method, simultio ws doe. This simultio ivolved rdom geertio of two ormlly distributed smples usig the R sttisticl softwre [8]. The prmeters µ d σ used to geerte first smple were fixed t 30 d, respectively, d those of the secod smple were fixed t 0 d 3 respectively. The simultio process ivolved rus for ech differet set of smple sizes d the resultig empiricl coverges rtes were recorded. T ble 1 o ext pge shows some of the simultio results. The empiricl coverge c be see to coverge to 0.95 i F igure 1 1

23 Tble 1: Empiricl Coverge Rtes of 90%, 95% d 99% Cofidece Itervls for the Rtio of Percetiles from Two Norml Popultios. percetiles m 90% 95% 99% p p p p

24 I F igure 1, the vlues 1 5 o the x xis hve bee llocted to sets of smples sizes i icresig order. O the y xis we hve the empiricl coverge rtes for 95% cofidece itervl t p For the first cse where < m we hve 1,, 3, 4 d 5 correspodig to to the (10, 50), (50, 100), (100, 00), (150, 50) d (00, 500) smple size combitios. For the secod cse where m we hve 1,, 3, 4 d 5 correspodig to the (10, 10), (50, 50), (100, 100), (00, 00) d (500, 500) smple size combitios. For the third cse where > m we hve 1,, 3, 4 d 5 correspodig to the (50, 10), (100, 50), (00, 100), (50, 150) d (500, 00) smple size combitios. Figure 1: Coverge Rtes vs Smple Size for Rtio of Norml Percetiles t p

25 3 EXPONENTIAL DISTRIBUTION Let X 1, X,..., X be rdom smple from expoetil popultio with me θ d vrice θ. The the p.d.f. of X is give by f(x) 1 θ e x/θ, 0 x <. (7) Sice this is cotiuous distributio the cumultive distributio fuctio (c.d.f) c be used to determie the vlue tht lies o give percetile F (x) P (X k p ) d solvig for k p we hve kp 0 1 x θ e θ dx 1 e kp θ p (8) k p θ l(1 p) (9) where k p is the 100pth percetile. Sice θ is ot kow we eed to fid ubised estimtor of θ tht c be used to estimte k p. 3.1 Ubised Estimtor of θ Let X 1, X,..., X be rdom smple from expoetil popultio with me θ d let X deote the smple me. X 1 X i is ubised estimtor of θ [5]. Also by the Cetrl Limit Theorem the smple me is ubised estimtor of the popultio me d E( X) θ. Therefore the estimted k p deoted ˆk p is give by i1 ˆk p X l(1 p). (30) 4

26 Next we eed to fid the vrice of ˆk p d the estimted vrice of ˆk p deoted by V r( ˆk p ). V r( ˆk p ) V r( X l(1 p)) (l(1 p)) V r( X) (l(1 p)) σ (l(1 p)) θ. (31) The bove follows from the fct tht X is ormlly distributed with me µ d vrice σ d for the expoetil distributio σ θ d so V r( ˆk p ) (l(1 p)) X. (3) 3. Approximte Cofidece Itervl for the Rtio of Percetiles from Two Idepedet Expoetil Distributios I this sectio, we re goig to develop method of computig the (1 α)100% cofidece itervls for the rtio of two expoetil percetiles. These two percetiles must come from two idepedet expoetil distributios d they must be the sme 100pth percetile. For exmple, if we use the 45th percetile from the first distributio the we must use the 45th percetile from the secod distributio. The mes, vrices, d smple sizes of the two smples eed ot be equl. Theorem 3.1 Let X 1, X,..., X be rdom smple of size from expoetil popultio X with me θ x d vrice θx d let Y 1, Y,..., Y m be rdom smple of size m from other expoetil popultio Y with me θ y d vrice θy where 5

27 X d Y re idepedet. Let k p,x d k p,y be the 100pth percetiles from popultios X d Y respectively. The pproximte (1 α)100% cofidece itervl for the rtio of the percetiles k p,x d k p,y is give by ( ) ˆk p,x 1 exp ± Z (1 α ˆk ) p,y + 1 m. (33) Proof We kow tht k p,x θ x l(1 p) d k p,y θ y l(1 p) (34) d the ubised estimtors of k p,x d k p,y re ˆk p,x X l(1 p) d ˆkp,y Ȳ l(1 p). (35) respectively. The ubised estimtor of k p rtio deoted by ˆk p rtio is give by ˆk p rtio ˆk p,x ˆk p,y X l(1 p) Ȳ l(1 p) X Ȳ. (36) Now we eed to fid the vrice of the ˆk p rtio. This c be doe by itroducig turl logrithms d usig the delt method to compute the vrice. l(ˆk p rtio) l ( ) X Ȳ l( X) l(ȳ ). (37) The ext few equtios employ the delt method to compute the vrice of l(ˆk p rtio). V r(l(ˆk p rtio)) 6

28 [ d l(ˆk p rtio) d X ] [ d l(ˆk p rtio) V r( X) 0 dȳ 0 V r(ȳ ) ] [ ] d l(ˆk p rtio). d X d l(ˆk p rtio) dȳ The vrice covrice mtrix hs zero etries becuse X d Y re idepedet d their covrice is zero. Hece V r(l(ˆk p rtio)) [ 1 X 1 Ȳ ] [ σ x 0 σy 0 m ] [ 1 X 1 Ȳ ]. (38) V r(l(ˆk p rtio)) σ x X + σ y mȳ, (39) V r(l(ˆk p rtio)) d the estimted vrice of the l(ˆk p rtio) is θ x X + θ y (40) mȳ V r(l(ˆk p rtio)) X X + Ȳ mȳ, (41) V r l(ˆk p rtio) m. (4) The (1 α)100% cofidece itervl for the l(k p rtio) c be computed s l( X) l(ȳ ) ± Z (1 α ) m (43) d expoetitig the expressio we hve the (1 α)100% cofidece itervl for the k p rtio s ( ) X 1 Ȳ exp ± Z (1 α ) + 1 m. 7

29 3.3 Simultio Results To test this method, simultios were doe. Two rdom smples were geerted usig the R sttisticl softwre [8]. The first smple ws geerted from expoetil distributio with me fixed t 0.1 d the secod smple ws geerted from expoetil distributio with me fixed t The simultio process ivolved rus for ech differet set of smple sizes d the resultig empiricl coverge rtes were recorded. T ble o the followig pge shows the empiricl coverge rtes for few sets of smple sizes. The empiricl coverge c be see to coverge to 0.95 i F igure. 8

30 Tble : Empiricl Coverge Rtes of 90%, 95% d 99% Cofidece Itervls for the Rtio of Percetiles from Two Expoetil popultios. percetiles m 90% 95% 99% p p p p

31 I F igure, the vlues 1 5 o the x xis hve bee llocted to sets of smples sizes i icresig order. O the y xis we hve the empiricl coverge rtes for 95% cofidece itervl t p For the first cse where < m we hve 1,, 3, 4 d 5 correspodig to to the (10, 50), (50, 100), (100, 00), (150, 50) d (00, 500) smple size combitios. For the secod cse where m we hve 1,, 3, 4 d 5 correspodig to to the (10, 10), (50, 50), (100, 100), (00, 00) d (500, 500) smple size combitios. For the third cse where > m we hve 1,, 3, 4 d 5 correspodig to to the (50, 10), (100, 50), (00, 100), (50, 150) d (500, 00) smple size combitios. Figure : Coverge Rtes vs Smple Size for Rtio of Expoetil Percetiles t p

32 4 UNIFORM DISTRIBUTION Let X 1, X,..., X be rdom smple from uiform popultio with miimum d mximum b. The the p.d.f. of X is give by f(x) 1 b x b. (44) Just like y other cotiuous distributio, the cumultive distributio fuctio (c.d.f) c be esily mipulted to come up with formul for computig the percetiles F (x) P (X k p ) kp 1 b dx k p b p. (45) Solvig for k p we hve k p + (b )p (46) where k p deotes the 100pth percetile. For us to estimte the ukow prmeters d b, we eed to fid ubised estimtors of both prmeters. 4.1 Ubised Estimtors of d b Sice d b re the popultio miimum d popultio mximum, the first d lst order sttistics re the most likelihood estimtors of d b respectively. Order sttistics re observtios of rdom smple rrged, or ordered i mgitude from the smllest to the lrgest [5]. Let X 1, X,..., X be rdom smple from uiform popultio with miimum d mximum b d let X (1), X (),..., X () be the order sttistics ssocited with the rdom smple where X (1) d X () re the 31

33 miimum d mximum of X i, respectively. The X (1) d X () re the mximum likelihood estimtors of d b, respectively. Though X (1) d X () re ot exctly ubised estimtors of d b they re symptotic ubised estimtors. Tht is lim E(X (1) ) d lim E(X () ) b. This is show below. The p.d.f. of the rth order sttistics deoted f r (x) is give by f r (x)! (r 1)!1!( 1)! [F (x)]r 1 [1 F (x)] r f(x) < x < b 1 r [5].(47) Substitutig r1 d r for the first d lst order sttistics we hve f 1 (x) [1 F (x)] 1 f(x) (b x) 1 (b ) < x < b (48) d f (x) [F (x)] 1 f(x) (x ) 1 (b ) < x < b [5]. (49) Now we eed to fid E(X (1) ) d E(X () ). E(X (1) ) b (b x) 1 x dx (b ) (b ) b x(b x) 1 dx. Applyig itegrtio by prts we hve u x, du dx, dv (b x) 1, v (b x) E(X (1) ) [ x(b x) (b ) [ x(b x) (b x) (b x)+1 (b ) ( + 1) [ ] x(b x) b (b x)+1 (b ) (b ) ( + 1) ] b ] b dx + b + 1. (50) 3

34 E(X () ) b (x ) 1 x dx (b ) (b ) b x(x ) 1 dx. Applyig itegrtio by prts we hve u x, du dx, dv (x ) 1, v (x ) E(X () ) [ x(x ) (b ) [ x(x ) (b ) [ x(x ) (x ) (x )+1 ( + 1) (x )+1 (b ) (b ) ( + 1) ] b dx b b + 1. (51) From bove lim E(X (1) ) d lim E(X () ) b d the two re the best miimum vrice ubised estimtors of d b respectively. Now we c estimte k p deoted ˆk p s follows, ] b ] b ˆk p X (1) + (X () X (1) )p (1 p)x (1) + px (). (5) Sice we re buildig cofidece itervls we eed the vrice of ˆk p. V r( ˆk p ) (1 p) V r(x (1) ) + p V r(x () ) + p(1 p)cov(x (1) X () ). (53) The lst term of (53) exists becuse X (1) d X () re ot idepedet. We begi by computig the vrice of X (1) s V r(x (1) ) E(X (1)) (E(X (1) )) 33

35 where E(X (1)) b (b x) 1 x dx (itegrtio by prts) (b ) (b ) b x (b x) 1 dx u x, du xdx, dv (b x) 1, v (b x) E(X (1)) [ x (b x) (b ) x(b x) ] b dx s x, ds, dt (b x), t (b x) E(X (1)) [ x (b x) + 1 [ x(b x) +1 + (b ) ]] b (b x) +1 dx E(X (1)) [ [ 1 x(b x) x (b x) +1 (b ) (b ) (b ) ( + 1)( + ) (b x)+ ]] b ( + 1)( + ) (54) d therefore ( V r(x (1) ) (b ) (b ) ( + 1)( + ) + b ) + 1 (b ) ( + 1) ( + ). (55) Next we compute the vrice of X () s V r(x () ) E(X ()) (E(X () )) 34

36 where E(X ()) b (x ) 1 x dx (itegrtio by prts) (b ) (b ) b x (x ) 1 dx u x, du xdx, dv (x ) 1, v (x ) E(X ()) [ x (x ) (b ) x(x ) ] b dx s x, ds, dt (x ), t (x ) E(X ()) [ x (x ) 1 [ x(x ) +1 1 (b ) ]] b (x ) +1 dx E(X ()) [ [ 1 x(x ) x (x ) +1 + (b ) + 1 b b(b ) (b ) ( + 1)( + ) (x )+ ]] b ( + 1)( + ) (56) d so ( V r(x () ) b b(b ) (b ) ( + 1)( + ) b b ) + 1 (b ) ( + 1) ( + ). (57) Lstly we eed to compute the covrice of X (1) d X () s Cov(X (1) X () ) E(X (1) X () ) E(X (1) )E(X () ). (58) 35

37 The joit p.d.f of X (1) d X () is give by f 1, (x (1), x () ) Let X (1) x d X () y 1 (b ) ( 1)(x () x (1) ) < x (1) x () < b []. (59) E(X (1) X () ) 1 (b ) ( 1) (b ) ( 1) (b ) b y b y b [ y y xy( 1)(y x) dxdy xy(y x) dxdy ] x(y x) dx } {{ } 1 dy. First we itegrte 1 usig itegrtio by prts (1) y x(y x) dx x(y x) [ x(y x) 1 1 (y ) y u x, du dx, dv (y x), v (y x) 1 dx 1 (y x) ( 1) (y ) ( 1). ] y (y x) 1 1 Pluggig (1) bck to our equtio d usig itegrtio by prts we hve E(X (1) X () ) ( 1) (b ) b [ ] (y ) 1 (y ) y + dy 1 ( 1) 1 b (b ) y(y ) 1 dy + y(y ) dy }{{}}{{} b 3 36

38 where () b y(y ) 1 dy, [ y(y ) ] (y ) b dy [ ] y(y ) b (y )+1 ( + 1) b(b ) (b ) u y, du dy, dv (y ) 1, v (y ) d (3) b y(y ) dy, [ y(y ) +1 ] (y ) +1 b dx [ ] y(y ) +1 b (y )+ + 1 ( + 1)( + ) b(b ) u y, du dy, dv (y ), v (b )+ ( + 1)( + ). (y ) Puttig () d (3) together we hve 1 [([b(b ) (b ) hece E(X (1) X () ) ] [ (b )+1 b(b ) E(X (1) X () ) ])] (b )+ ( + 1)( + ) )] 1 [(b(b ) (b )+1 b(b )+1 (b )+ + (b ) ( + 1)( + ) (b ) E(X (1) X () ) b + 1 b + + b(b ) + 1 (b ) ( + 1)( + ) (b ) + 1 (b ) ( + 1)( + ). (60) 37

39 Now we hve d Cov(X (1) X () ) E(X (1) X () ) E(X (1) )E(X () ) (b ) b (b ) ( + 1)( + ) (b ) ( + 1) ( + ) ( + b ) ( b b ) (61) V r( ˆk p ) (1 p) V r(x (1) ) + p V r(x () ) + p(1 p)cov(x (1) X () ) ( ) ( ) (b ) (1 p) (b ) + p ( + 1) ( + ) ( + 1) ( + ) ( ) (b ) +p(1 p) ( + 1) ( + ) V r( ˆk p ) ( ) (b ) ((1 p) + p + p(1 p) ) (6) ( + 1) ( + ) d the estimted vrice of ˆk p deoted V r( ˆk p ) is ( ) V r( ˆk (X() X (1) ) ((1 p ) p) + p + p(1 p) ). (63) ( + 1) ( + ) 4. Approximte Cofidece Itervl for the Rtio of Percetiles from Two Idepedet Uiform Distributios I this sectio, we re goig to develop method of computig the (1 α)100% cofidece itervls for the rtio of two uiform percetiles. Just like the previous two cses, these two percetiles must come from two idepedet expoetil distributios d they must be the sme 100pth percetile. For exmple, if we use the 45th percetile from the first distributio the we must use the 45th percetile from the 38

40 secod distributio. The popultio prmeters d smple sizes of the two smples eed ot be equl. Theorem 4.1 Let X 1, X,..., X be rdom smple of size from uiform popultio X with miimum d mximum b d let Y 1, Y,..., Y m be other rdom smple of size m from other expoetil popultio Y with miimum c d mximum d where X d Y re idepedet. Let k p,x d k p,y be the 100pth percetiles from popultios X d Y respectively. The pproximte (1 α)100% cofidece itervl for the rtio of the percetiles k p,x d k p,y deoted kp,x k p,y ˆk p,x exp ± Z (1 α ˆk ) p,y V r(ˆk p,x ) ˆk p,x + V r(ˆk p,y ) ˆk p,y is give by. (64) Proof We kow tht k p,x + (b )p d k p,y c + (d c)p (65) d the respective ubised estimtors of k p,x d k p,y re ˆk p,x (1 p)x (1) + px () d ˆk p,y (1 p)y (1) + py (m). (66) The vrices of these ubised estimtors ˆk p,x d ˆk p,y re ( ) (b ) ((1 V r(ˆk p,x ) p) + p + p(1 p) ) ( + 1) ( + ) d ( ) (d c) (m(1 V r(ˆk p,y ) p) + mp + p(1 p) ) (67) (m + 1) (m + ) 39

41 d their estimted vrices re d ( ) V r(ˆk (X() X (1) ) ((1 p,x ) p) + p + p(1 p) ) ( + 1) ( + ) ( ) V r(ˆk (Y() Y (1) ) (m(1 p,y ) p) + mp + p(1 p) ). (68) (m + 1) (m + ) The estimted k p rtio deoted by ˆk p rtio is give by ˆk p rtio ˆk p,x ˆk p,y. (69) Now we eed to fid the vrice of the ˆk p rtio. This c be simplified by itroducig turl logrithms d usig the delt method to compute the vrice. ) l(ˆk p rtio) l ( ˆkp,x ˆk p,y l(ˆk p,x ) l(ˆk p,y ). (70) The ext few equtios employ the delt method to compute the vrice of l(ˆk p rtio). [ d l(ˆkp rtio) dˆk p,x V r(l(ˆk p rtio)) d l(ˆk p rtio) ] [ V r(ˆk p,x ) 0 dˆk p,y 0 V r(ˆk p,y ) ] d l(ˆk p rtio) dˆk p,x d l(ˆk p rtio) dˆk p,y The vrice covrice mtrix hs zero etries becuse X d Y re idepedet. d their covrice is zero. [ ] [ ] [ ] 1 1 V r(l(ˆk p rtio)) ˆk 1 V r(ˆk p,x ) 0 dˆk p,x p,x dˆk p,y 0 V r(ˆk p,y ) 1 dˆk p,y. (71) Hece V r(l(ˆk p rtio)) V r(ˆk p,x ) ˆk p,x + V r(ˆk p,y ) ˆk p,y (7) 40

42 d the estimted vrice of the l(ˆk p rtio) is V r l(ˆk p rtio) V r(ˆk p,x ) + ˆk p,x V r(ˆk p,y ) ˆk p,y. (73) The (1 α)100% cofidece itervl for the l(k p rtio) c be computed s l(ˆk p,x ) l(ˆk p,y ) ± Z (1 α ) V r(ˆk p,x ) ˆk p,x + V r(ˆk p,y ) ˆk p,y (74) d expoetitig the expressio we hve the (1 α)100% cofidece itervl for the k p rtio s ˆk p,x exp ± Z (1 α ˆk p,y ) V r(ˆk p,x ) ˆk p,x + V r(ˆk p,y ) ˆk p,y. 41

43 4.3 Simultio Results To test this method of buildig cofidece itervls for the rtio of uiform percetiles, simultios were doe. Two rdom rdom smples were geerted usig the R sttisticl softwre [8]. The first rdom smple ws geerted from uiform distributio with miimum d mximum fixed t 1 d 3 respectively d the secod smple ws geerted from uiform distributio with miimum d mximum fixed t d 5 respectively. The simultio process ivolved rus for ech set of smple sizes d the resultig empiricl coverge rtes were recorded. T ble 3 shows some of the empiricl coverges rtes from the simultios. The empiricl coverge c be see to coverge to 0.95 i F igure 3. 4

44 Tble 3: Empiricl Coverge Rtes of 90%, 95% d 99% Cofidece Itervls for the Rtio of Percetiles from Two Uiform popultios. percetiles m 90% 95% 99% p p p p

45 I F igure 3, the vlues 1 5 o the x xis hve bee llocted to sets of smples sizes i icresig order. O the y xis we hve the empiricl coverge rtes for 95% cofidece itervl t p For the first cse where < m we hve 1,, 3, 4 d 5 correspodig to to the (10, 50), (50, 100), (100, 00), (150, 50) d (00, 500) smple size combitios. For the secod cse where m we hve 1,, 3, 4 d 5 correspodig to to the (10, 10), (50, 50), (100, 100), (00, 00) d (500, 500) smple size combitios. For the third cse where > m we hve 1,, 3, 4 d 5 correspodig to to the (50, 10), (100, 50), (00, 100), (50, 150) d (500, 00) smple size combitios. Figure 3: Coverge Rtes vs Smple Size for Rtio of Uiform Percetiles t p

46 5 CONCLUSION The method used to compute pproximte cofidece itervl for rtio of percetiles from the three distributios c be exteded to other distributios. Typiclly, the pproximte coverge probbility for rtio c be improved by first pplyig the delt method to the turl log fuctio. The itervl for the rtio c be foud by expoetitig the ed poits of the itervl foud bsed o the turl log trsformtio. 5.1 Results From the simultios results i T bles 1 3, it c be see tht the empiricl coverge rtes re lrgely depedet o the smple sizes. Obviously, it c be see tht the lrger the smple size the better the coverge. There re few vritios which correspod to whether the smples sizes i questio re equl or ot equl d other vritios my occur s result of chgig the popultio prmeters. Whe the smple sizes re uequl, the coverge c be low if the lrger of the two smple sizes correspoded to the popultio tht hd the lrger vribility. For the uiform distributio, the coverge is slow to coverge to the specified level of cofidece. This my be due to the fct tht the order sttistics re bised estimtors of the prmeters or it my be cused by the kurtosis preset i the uiform distributio. 5. Limittios There re two mi limittios ssocited with this method of computig cofidece itervls. 45

47 First is the fct tht this method oly works for popultios whose uderlyig distributio is kow. If oe iteds to pply this method o y collected dt, the oe must determie wht kid of distributio the dt comes from otherwise the method cot be used. A possible distributio-free method kow s bootstrppig my be solutio. Secodly the method is depedet o smple size d my ot be relible for very smll sizes. For the uiform distributio this method my be oly relible for smple sizes greter th or equl to 50. This method lso relies o the cetrl limit theorem thus the reso for the use of the Z-sttistic. Other mior problems my rise from the difficulty to determie the mximum likelihood estimtors d the ubised estimtors for the popultio prmeters. This c be see i the cse of the uiform distributio where order sttistics were used d this mde the whole process quite legthy d cumbersome. 46

48 BIBLIOGRAPHY [1] S. Chkrborti d J. Li (007), Cofidece itervl estimtio of orml percetile, The Americ Sttistici, Volume 61, No. 4, Published by Americ Sttisticl Associtio Nov. (331). [] H. A. Dvid (1981), Order Sttistics, d editio, Published by Wiley, (10). [3] W. G. Gilchrist (000), Sttisticl Modelig with Qutile Fuctios, Published by Chpm d Hll/CRC, (37, 41, 43). [4] R. V. Hogg, J. W. Mcke d A. T. Crig (005), Itroductio to Mthemticl Sttistics, 6th editio, Published by Perso Pretice Hll, (15-30). [5] R. V. Hogg d E. A. Tis (005), Probbility d Sttisticl Iferece, 7th editio, Published by Perso Pretice Hll, (83, 337, 340, 394, 397). [6] D. S. Moore (001), The Bsic Prctice of Sttistics, 3rd editio, Published by W. H. Freem d Compy (59). [7] R. M. Price d D. G. Boett (00), Distributio free cofidece itervls for differece d rtio of medis, J. Comput. Sttist. Simul., Volume 7(), Published by Tylor d Frcis Ltd, (119). [8] R Developmet Core Tem (008), R: A Lguge d Eviromet for Sttisticl Computig, by R Foudtio for Sttisticl Computig, Vie, Austri. ISBN , URL 47

49 APPENDICES Appedix A: R Code for the CI of Norml Percetiles Rtio [8] cofidece itervl fuctio(1,,mu1,sig1,mu,sig,lph,p){ zqorm(1-lph/) zpqorm(p) #z for the pth percetile #smple1(isert smple 1 here) #smple(isert smple here) smple1rorm(1,mu1,sig1) #rdom orml smple 1 smplerorm(,mu,sig) #rdom orml smple c1 (sqrt((1-1)/)*gmm((1-1)/))/gmm(1/) #costt for removig std dev bis c (sqrt((-1)/)*gmm((-1)/))/gmm(/) #costt for removig std dev bis #lc1 log(sqrt((1-1)/)) + lgmm((1-1)/) - lgmm(1/) # if 1 is lrge (outside gmm fuctio) #c1 exp(lc1) #lc log(sqrt((-1)/)) + lgmm((-1)/) - lgmm(/) # if is lrge (outside gmm fuctio) #c exp(lc) me1me(smple1) #me of smple 1 meme(smple) #me of smple sd1sd(smple1) #stdrd devitio of smple 1 48

50 sdsd(smple) #stdrd devitio of smple kp1me1+(c1*zp*sd1) # MVUE of the pth percetile 1 kpme+(c*zp*sd) # MVUE of the pth percetile vr kp1 ht(sd1^ /1)*(1 + 1*zp^ *(c1^ - 1)) # estimted vrice of kp1 vr kp ht(sd^ /)*(1 + *zp^ *(c^ - 1)) # estimted vrice of kp kp rtiokp1/kp # estimted percetile rtio vr lrtio vr kp1 ht/kp1^ + vr kp ht/kp^ #vrice of the turl log of kp rtio lb kp rtio*exp(-z*sqrt(vr lrtio)) #lower limit of the percetile rtio cofidece itervl ub kp rtio*exp(z*sqrt(vr lrtio)) #upper limit of the percetile rtio cofidece itervl list(c1,c,me1,me,sd1,sd,kp1,kp,vr kp1 ht,vr kp ht,lb,ub) #list ll the sttistics defied bove tht you desire to compute } cofidece itervl(00,100,0,3,30,,.05,.1) # isert your vlues here 49

51 Appedix B: R Code for the CI of Expoetil Percetiles Rtio[8] cofidece itervl fuctio(1,,mu1,mu,lph,p){ zqorm(1-lph/) #100pth percetile of the stdrd orml N(0,1) #smple1(isert smple 1) #smple(isert smple ) smple1rexp(1,mu1) #rdomly geerted expoetil smple 1 smplerexp(,mu) #rdomly geerted expoetil smple me1me(smple1) #me of smple 1 meme(smple) #me of smple kp1 -me1*log(1-p) #100pth percetile of smple 1 kp1 -me*log(1-p) #100pth percetile of smple rtio htkp1/kp #rtio of the two 100pth percetiles vr lrtio 1/1 + 1/ #vrice of the turl log of rtio ht lb rtio ht*exp(-z*sqrt(vr lrtio)) #lower boud of cofidece itervl ub rtio ht*exp(z*sqrt(vr lrtio)) #upper boud of cofidece itervl list(me1,me,kp1,kp,rtio ht,lb,ub) #list ll the sttistics required } cofidece itervl(100,00,1/10,1/0,.05,.1) #Isert your vlues here 50

52 Appedix C: R Code for the CI of Uiform Percetiles Rtio [8] cofidece itervl fuctio(1,,,b,c,d,lph,p){ zqorm(1-lph/) #smple1(isert smple 1 here) #smple(isert smple here) smple1ruif(1,,b) #rdom uiform smple 1 smpleruif(,c,d) #rdom uiform smple xsort(smple1) # sort smple 1 scedig ysort(smple) # sort smple scedig kp1x[1]+(x[1]-x[1])*p # MVUE of the 100pth percetile 1 kpy[1]+(y[]-y[1])*p # MVUE of the 100pth percetile vr kp1(((x[1]-x[1])^ )*((1*(1-p)^ )+(1*p^ )+(*p*(1-p))))/ ((((1+1)^ )* (1+))*(kp1^ )) #vrice of kp1 vr kp(((y[]-y[1])^ )*((*(1-p)^ )+(*p^ )+(*p*(1-p))))/ ((((+1)^ )* (+))*(kp^ )) #vrice of kp kp rtiokp1/kp #estimted percetile rtio vr lrtio (vr kp1)+(vr kp) #vrice of turl log of percetile rtio lb kp rtio*exp(-z*sqrt(vr lrtio)) #lower limit of percetile rtio cofidece itervl ub kp rtio*exp(z*sqrt(vr lrtio)) #upper limit of percetile 51

53 rtio cofidece itervl list(me(smple1),me(smple),kp1,kp,kp rtio,lb,ub) #list desired sttistics } cofidece itervl(00,100,1,3,,5,.05,.1) # isert your vlues here 5

54 VITA PIUS MUINDI Persol Dt: Eductio: Professiol Experiece: Plce of Birth: Mchkos, Key Mritl Sttus: Mrried M.S. Mthemtics (Sttistics), Est Teessee Stte Uiversity, Johso City, Teessee 008 B.A. Mthemtics d Ecoomics, Uiversity of Nirobi, Nirobi, Key 1999 Grdute Assistt, Est Teessee Stte Uiversity, Johso City, Teessee, Techer, Khls Girls Secodry School, Nirobi, Key, Reserch Assistt, Reserch Solutios, Nirobi, Key,