On Distribution Function Estimation Using Double Ranked Set Samples With Application

Transcription

1 Journal of Modern Appled Statstcal Methods Volume Issue Artcle -- On Dstrbuton Functon Estmaton Usng Double Raned Set Samples Wth Applcaton Wald A. Abu-Dayyeh Yarmou Unversty, Irbd Jordan, Han M. Samaw Sultan Qaboos Unversty, Sultanate of Oman Lara A. Ban-Han Yarmou Unversty, Irbd Jordan Follow ths and addtonal wors at: Part of the Appled Statstcs Commons, Socal and Behavoral Scences Commons, and the Statstcal Theory Commons Recommended Ctaton Abu-Dayyeh, Wald A.; Samaw, Han M.; and Ban-Han, Lara A. () "On Dstrbuton Functon Estmaton Usng Double Raned Set Samples Wth Applcaton," Journal of Modern Appled Statstcal Methods: Vol. : Iss., Artcle. DOI:.7/jmasm/6 Avalable at: Ths Regular Artcle s brought to you for free and open access by the Open Access Journals at DgtalCommons@WayneState. It has been accepted for ncluson n Journal of Modern Appled Statstcal Methods by an authorzed edtor of DgtalCommons@WayneState.

2 Journal of Modern Appled Statstcal Methods Copyrght JMASM, Inc. Fall, Vol., No, - 97//$. On Dstrbuton Functon Estmaton Usng Double Raned Set Samples Wth Applcaton Wald A. Abu-Dayyeh Department of Statstcs Yarmou Unversty, Irbd Jordan Han M. Samaw Dept. of Mathematcs & Statstcs Sultan Qaboos Unversty Al-Khod, Sultanate of Oman Lara A. Ban-Han Department of Statstcs Yarmou Unversty, Irbd Jordan As a varaton of raned set samplng (RSS); double raned set samplng (DRSS) was ntroduced by Al-Saleh and Al-Kadr (), and t has been used only for estmatng the mean of the populaton. In ths paper DRSS wll be used for estmatng the dstrbuton functon (cdf). The effcency of the proposed estmators wll be obtaned when ranng s perfect. Some nference on the dstrbuton functon wll be drawn based on Kolomgrov-Smrnov statstc. It wll be shown that usng DRSS wll ncrease the effcency n ths case. Key words: Double raned set sample, dstrbuton functon estmaton, Kolomgrov-Smrnov, raned set. Introducton In some practcal stuatons, collectng unts from the populaton s not too costly comparng wth quantfcaton of the samplng unts. A large number of those unts may be dentfed to represent the populaton of nterest and yet only a carefully selected subsample s to be quantfed. Ths potental for observatonal economy was recognzed for estmatng the mean pasture and forge by McIntyre (9). He proposed a method, later called raned set samplng (RSS) by Halls and Dell (966), currently under actve nvestgaton. RSS procedure can be descrbed as follows: Identfy a group of samplng unts randomly from the target populaton. Then, randomly partton the group nto dsjont subsets each havng a pre-assgned szer r, n the most practcal stuatons, the sze r wll be, or. Then, ran each subset by a sutable method of ranng such as pror nformaton, vsual nspecton or by the expermenter hmself. In terms of samplng notaton, The contact person for ths artcle s Han M. Samaw. Emal hm at hsamaw@squ.edu.om. where X j() denotes the -th ordered statstc n the j-th set. Then the -th ordered statstc from the -th subset wll be quantfed, =,, r. Then X (), X ( ),..., X rr () wll be obtaned. The whole process can be repeated -tmes, to get a RSS of sze n = r. The resultng sample s called the balanced raned set sample (RSS). Through all the paper, only balanced RSS wll be used. Al-Saleh and Al-Kadr () extended RSS to double ran set sample (DRSS). DRSS can be descrbed as follows:. Identfy r elements from the target populaton and dvde these elements randomly nto r subsets each of sze r elements.. Use usual RSS procedure to obtan r RSS each of sze r.. Apply agan the RSS procedure n Step, on the r RSS s. We may repeat steps, and -tmes to obtan DRSS sample of sze n = r. In DRSS, ranng n the second stage s easer than ranng n the frst stage, (see Al-Saleh and Al-Kadr, ). Moreover, an up-to-date annotated bblography for RSS can be found n Kaur et al., (99) and Patl et al. (999). Stoes and Sager (9) estmate the dstrbuton functons, F(x) say, for a random varable X by the emprcal cdf (F*) based on the RSS, whch wll be gven n

3 ON DISTRIBUTION FUNCTION ESTIMATION USING DOUBLE RANKED SET Secton. They ponted out that, F* s an unbased for F and s more effcent than the emprcal dstrbuton functon of a SRS ( Fˆ) of sze n wth [ F ] m Var(F* ) = F, where r = F = F = I (, r + ) (.) () F for perfect ranng, and I F (,r + ) s the ncomplete beta rato functon. Basc Settng of DRSS Let Y,..., Yr be a DRSS, and assume that Y ~ g (y) wth df, mean and varance are: * * G (y),µ and σ, respectvely. Al-Saleh and Al-Kadr () showed that: r () f(y) = g (y), (.) r = r () F(y) = G (y), (.) r = r * () µ = µ, (.) r = (v) r r * * σ = σ + ( µ µ ), (.) r = = where f, F, µ and σ are the pdf, cdf, mean and varance of the populaton. In ths paper, we wll consder the problem of estmatng the dstrbuton functon F usng DRSS. In Secton, the emprcal cdf estmator based on DRSS ( FˆDR) wll be consdered. The effcency between the DRSS estmator and those estmators based on SRS and RSS wll be obtaned when ranng s perfect. In Secton the Kolmogrov-Smrnov statstc wll be studed based on a DRSS. Also, a confdence nterval of F wll be constructed usng the Kolmogrov-Smrnov statstc based on DRSS. Estmatng The Dstrbuton Functons Usng DRSS In ths Secton the dstrbuton functon wll be estmated usng the DRSS, n the cases where ranng s perfect and when ranng s mperfect. The suggested estmator wll be compared wth the cdf estmators based on SRS and RSS va ther varances. Defnton and Some Basc Results For the l-th cycle, let {Y, Y,..., Y }, l l rl l =,,, be a DRSS of sze r, and assume that Y has the probablty densty functon (pdf) g (y) and the cdf G (y). Note that g (y) s the densty of the -th ordered statstc of a RSS wth denstes f, f,..., f and dstrbuton functons () () (r) F, F,..., F respectvely. Then () () (r) r j r G (y) = F F (L) = = = + (L) j Sj L L j (.) where the set S conssts of all permutatons (,,..., ) r of,,, r for whch <... < j and <... < (see Al-Saleh and Al-Kadr, j + r ). Let Fˆ DR, Fˆ and F* be the edf s (emprcal dstrbuton functons) of DRSS, SRS and RSS from the populaton wth cdf F, then: r FˆDR = I[Yj t] (.) r j= = r Fˆ = I[X t] r = r F * I[X t] r ()j j = = (.) = (.) respectvely, where I(.) s the ndcator functon. Then, we have the followng results. a) E[ FˆDR] = F

4 ABU-DAYYEH, SAMAWI & BANI-HANI r b) var( FˆDR) = G [ G ], r = (.) (see the Appendx for the prove of these results.) Also, we show n the Appendx that / [ Fˆ DR E(FˆDR)] [var(fˆdr)] converges n dstrbuton to a standard normal random varable as when r and t are held fxed. Moreover, t can be shown that an unbased estmator of var [FˆDR] s gven by var[fˆdr] = r Ĝ [ Ĝ ], ( )r = (.6) where Ĝ = I[Yj t] s the edf based j= on all of the -th judgment order statstc and hence t can be shown also that / [ Fˆ DR E(FˆDR)]/[var[FˆDR]] converges n dstrbuton to a standard normal random varable as when r and t are held fxed. (See the Appendx for the prove of the above results.) Therefore, when s large for a specfed value t, an approxmate (-α)% confdence nterval for F s Fˆ DR ± Zα / var[fˆ DR] (.7) Fnally, as a specal case when r =, t can be shown that var[ FˆDR] var[fˆ(t )] and var[ FˆDR] var[f * ]. (See the Appendx Lemma for the prove of ths results.) mprovement n precson that results when estmatng F by Fˆ DR rather than by Fˆ or F*. Now, the relatve precson (RP) of the double raned set to the smple random samplng estmator and to raned set sample estmator, are defned by: var[ Fˆ ( t )] RP ( t) = var[ Fˆ DR ( t )] RP = = F[ F] = r F G = var[f*] var[fˆ DR] rf rf r F = r = G () r (.) (.9) Table and Table show the value of RP (F (p)) and RP (F (p)) respectvely, for some values of p and r =,,,. It can be notced that both of RP and RP are monotone ncreasng from p = to p =., to acheve ther maxmum at p =.. Also, they are symmetrc about p =.. Table and Table show that the gan n effcency from DRSS for estmaton of F s substantal when the ranng can be done perfectly. Fˆ DR Effcency of The edf s used for mang pontwse estmates of F, as well as for mang nference concernng the overall populaton dstrbuton. In ths secton, we wll examne the magntude of the

5 ON DISTRIBUTION FUNCTION ESTIMATION USING DOUBLE RANKED SET 6 Table. RP (F (p)) when ranng of X s perfect. P r Table. RP (F (p)) when ranng of X s perfect. P R Inference on the dstrbuton functon Because the dstrbuton functon F can be estmated more effcently from a double raned set sample than from a SRS and a RSS, t s suffces to note that the statstcs based on an estmate of F, such as the Kolmogrov-Smrnov statstc, would be mproved n some sense as well. In partcular, we observe that the null dstrbuton of the statstc D sup [ FˆDR F ] t smaller than D* sup [ F* F ] t than sup [ Fˆ F ] = s stochastcally = and smaller D = t when D, D* and D are all based on the same number of measured observatons. We mean that * * * H ( r) (d) H (r) (d) and H ( r) (d) H (r) (d) wth strct nequalty for some d, where * H ( r) (d) = p(d d] * * H ( r) (d) = p(d d) and H r (d) p(d d] =. Where D, D*, and D are calculated from a SRS, a RSS and a DRSS of sze r respectvely. Ths mples that (-α)% of D, whch be denoted by C α, wll always be less than or equal to correspondng percentle of the statstcs D and D*, denoted by C α and * C α respectvely. A confdence band for F based on D s FˆDR± Cα, (.) s narrower than the correspondng band based on D and D*. In ths secton, the smulatons whch we done, s true for some fnte values of r and n the case of perfect judgment ranng. To fnd the table of crtcal values of D (C α ) we draw a double raned set samplng (Y ' s) of sze n from unform dstrbuton wth parameters,. Then all elements n the sample wll be ran ( X( ) 's). Now for =, D = max max F (Y ), max F (Y ), n n () n () n where F =. (X() ) X()

6 7 ABU-DAYYEH, SAMAWI & BANI-HANI untl we get, The prevous procedure wll be repeated D * * 's wll be raned to fnd P(D C α ) = α where [( α)] D,D,..., D. Also, C α such that,,.e., the =, where [d] s the C = D α () greatest nterge of d. Now, Table reports the crtcal values C α for the test statstc D for α =.,. and. for r =,,, and =,,,. The table shows that DRSS can result n a substantal decrease n wdth of the smultaneous confdence band. The amount of the mprovement can be descrbed by the quanttes, C R = α α (.) C α * C R = α α (.) C α Because R α and R α are the square of the rato of confdence-band wdths, then they can be nterpreted as a measure of relatve precson. The ratos R α and R α are computed from the entres of Table ( C α ), Table (C * α ) (from Stoes and Sager; 9) and the Table of crtcal values for the Kolmogrove- Smrnov statstc D (from Gbbons and Charabort (99)). Table gves the values of R α and R α at r =,, and =,,. These values are comparable wth those of Table and Table. So, R α and R α ndcate the same thng whch gven by Rp and Rp, when ranng of X s perfect.

7 ON DISTRIBUTION FUNCTION ESTIMATION USING DOUBLE RANKED SET Table. Crtcal values of D ( C α ) r= r= r= r= α:

8 9 ABU-DAYYEH, SAMAWI & BANI-HANI Table. The values of R α and R α R α r= r= R α α: r=

9 ON DISTRIBUTION FUNCTION ESTIMATION USING DOUBLE RANKED SET References Al-Saleh, & Al-Kadr (). Double raned set samplng. Statstcs and Probablty Letters. (), -. Gbbons, J. D., & Charabort, S. (99). Nonparametrc statstcal nference. Marcel Deer, Inc. New Yor, Basel, Hong Kong. Halls, L. K., & Dell, T. R. (966). Tral of raned set samplng for forage yelds. Forest Scence,, -6. Kaur, A. Patl, G. P., Snha, A. K. and Tale, C. (99). Raned set samplng. An annotated bblography. Envronmental and Ecologcal Statstcs,, -. McIntyre, G. A. (9). A method for unbased selectve samplng usng raned set. Australan Journal of Agrcultural Research,, -9. Patl G. P., A. K. Snha and Tlle C. (999). Raned set samplng: A Bblography. Envronmental Ecologcal Statstcs, 6, 9-9. Stoes, S. L., & Sager, T. (9). Characterzaton of a raned set sample wth applcaton to estmatng dstrbuton functons. Journal of Amercan Statstcal Assocaton,, 7-. Appendx Proposton. Fˆ DR s an unbased estmator of F. a) E[ FˆDR] = F b) r var( FˆDR) = G[ G]. r = Proof: From the defnton of a DRSS the proof wll follow smply by usng (.) and (.). Proposton. / [ Fˆ DR E(FˆDR)] [var(fˆdr)] converges n dstrbuton to a standard normal random varable as when r and t are held fxed. Proof: Ths follows from rewrtng Fˆ DR as Fˆ DR = U j, where j= r I[Yj t] U j =, then U s = r j ' are d, therefore the proof follows drectely from the Central Lmt Theorem. Lemma. (a) var[fˆ DR] var [FˆDR]. where: var[fˆdr] = ( )r = s an unbased estmator of r Ĝ [ Ĝ ] and Ĝ = I[Yj t] s the edf j= based on all of the -th judgment order statstc. (b) / [ Fˆ DR E(FˆDR)]/[var[FˆDR]] converges n dstrbuton to a standard normal random varable as when r and t are held fxed. Proof: (a) E[var[FˆDR)] = r [E(Ĝ ) E[Ĝ ] ( )r = because E(Ĝ ) = G and E (Ĝ ) = var( I[Yj t]) + [G ] j=

10 ABU-DAYYEH, SAMAWI & BANI-HANI = var(i[y t]) + G G [ G ] G = + G + ( )G Then =. E[var(FDR)] ˆ = r G G + ( )G ( )r = r = r = G [ G ] = var[fˆ DR] Part (b) can be shown by notng that: v a r(fˆ DR) p as, and var (Fˆ DR) p because Ĝ G. Furthermore, by Lemma when s large for a specfed value t, an approxmate (-α)% confdence nterval for F s: FˆDR ± Zα / var[fˆdr] Lemma. : For the specal case when r =, (a) var[ FˆDR] var[fˆ(t )] (b) FˆDR] var[f * ] var[. Proof: Let = and F = F F = F F, F = F, G = F F + F, and G = F F. F F Then var[fˆ ] = var[ Fˆ * ( t )] F + F F + F =, and 7 6 var[ Fˆ DR ( t )] [ F + F F F + F F + F] =. F ( F) Then var[fˆ DR] = var[fˆ] var[fˆ(t )] (F F F + F + ) F [ F ] [ F ][ F + ] Also, var[ Fˆ DR ( t )] = var[ F * ( t )] var[ F * ( t )], F.