The Grouping Methods and Rank Estimator, Based on Ranked Set sampling, for the linear Error in Variable Models

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1 P Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 The Groupng Methods and Rank Estmator, Based on Ranked Set samplng, for the lnear Error n Varable Models Ahmed A. M. Al-Radadeh PColleague of General Studes, Ajman Unverst of Scence and technolog, AL Fujarah, UAE Ua.alradadeh@ajman.ac.aeU30T Abstract The methods of estmatng the parameters of Error-In-Varables (EIV) models were extremel dscussed n the lterature. Wald (940), Bartlett (94) and others started the dscusson n ths area as well as Madansk (959). In ths paper the ranked set samplng method s used to choose a sample and analze t usng EIV model n order to estmate the parameters. The estmatons are made b usng the groupng methods and the ranked estmator. Moreover a smulaton stud has been appled n order to compare these methods. Kewords: Errors n-varables, Groupng Methods, Ranked Set Samplng, Ranked Set Samplng, Smple Random Samplng.. Introducton Statstcal methods are useful tools for assessng the relatonshps between contnuous varables collected for the appled studes. However, t s also mportant to dstngush these statstcal methods. Whle the are smlar mathematcall, ther purposes are dfferent. Regresson analss has undoubtedl been one of the most wdel used technques n appled statstcs. It s well known that the smple lnear regresson model assumes that the errors can happened just n the explanator varable but practcall, errors can happened n both varables. Here we are nterested n the measurement error model where both the response and the explanator varable are subject to error, consder two varables X and Y has the smple lnear relatonshp: = α + βx+ ε () where s n vector and x s n vector, x could be random or functonal and ε s an error term of sze n, wth mean zero and constant varance σ j such that cov( ε, ε j) = 0 j and cov( εx, ) = 0 Moreover, the unknown parameters of ths relatonshp are α and β denoted b ntercept and slope respectvel. The detals of ths model can be found n man text books s. t. (Darper and Smth, 980; Montegomr and Pech, 98; Neer et al, 996). More general, Chen and Vanness (999), extend the smple lnear relatonshp to the so-called Measurement Error Model (ME) or error- n- varable model EIV. In ths models, consder two latent varables η and ξ wth the followng relatonshp 96

2 P vsual P elements P set. Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 η= α + β. ξ () Both varable can t be measure drectl nstead one use manfest varable Y and X to measure latent varables wth error = η+ ε (3) x = ξ+ δ (4) The model can be smplfed as = α + β( x δ) + ε (5) Nothng that δ and ε are the error terms, have the followng assumptons; Kendall and Stuart (979);. E( ε) = E( δ) = 0. cov( ε, ε j) = cov( δ, δ j) = 0 j 3. var( ε) = σ, var( δ) = σ ε δ There are man methodologes to estmate the unknown parameters α and β for model (3)-(5) suggested n the lteratures, one of the most popular s the groupng methods whch was proposed b Wald (940) and Bartlett (949), alternatve methods of estmaton of EIV model have been suggested b Thel (950), Madansk (959), Kendall and Stuart (979), Fuller (987), Chen and Vanness (999), and Al-Nasser et al (005).all the proposed estmaton methods was done based on selectng Smple Random Sample from the populaton of the stud, whch s one of the most common technques for obtanng such data. Other more structured samplng desgns, such as stratfed samplng or probablt samplng, are also avalable to help make sure that the obtaned data collecton provdes a good representaton of the populaton of nterest. An such addtonal structure of ths tpe revolves around how the sample data themselves should be collected n order to provde nformaton s of the larger populaton. Wth an of these methods, once the sample tems have been chosen the desred measurement(s) s collected from each of the selected tems. The concept of ranked set samplng s a recent development that enables one to provde more structure to the collected sample tems. Ths approach to data collecton was frst proposed b McIntre (95) for stuatons where takng the actual measurements for sample observatons s dffcult (e.g., costl, destructve, tme-consumng), but mechansms for ether nformall or formall rankng a set of sample unts s relatvel eas For dscussons of some of the settngs where ranked set samplng technques have found applcaton. The RSS procedure conssts of drawng m random samples, each of sze m, from the populaton. The m unts of each set are ranked vsuall or b a neglgble cost method. th th Then m measurements are obtaned b quantfng the P order statstc from the P The whole process ma be repeated n tmes untl a total of nmp have been drawn from the populaton but onl nm elements have been measured. The nm measured observatons are referred to as the RSS. The procedure conssts of drawng m random samples, each of sze m, from the populaton. Takahas and Wakmoto (968), provded the statstcal theor for RSS procedure. Assumng the samplng s from an absolutel contnuous dstrbuton, the showed that the mean of the RSS, ˆµ, s the unbased estmator for the populaton mean, µ, and has hgher effcenc than the mean of SRS, ˆµ. In order to reduce the tme and the cost, the use of the RSS n order to 97

3 percentle. and such the Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 estmates EIV model was proposed b Al-Radadeh et al. (009), the use the dea of the L ranked set samplng (LRSS) wth Wald- tpe estmator to reduce the cost and ncrease the effcenc of the EIV model In ths paper we rentroduce the use of the RSS and the LRSS wth Wald- tpe estmaton method and wth Thel s estmaton method, n order to have less error s when we estmate the EIV parameter s; moreover we made a comparson between the estmaton methods tself to see whch estmaton would gve a better estmaton for the EIV model parameters n two cases, when we have outler cases n the data, and the other wthout havng an outlers n the data. Ths paper dvded nto sx secton, the next secton talk about the estmaton methods that used n order to estmate the EIV model parameter, secton four descrbe the LRSS for the bvarate data, n sectons fve we present the smulaton stud, and the conclusons s presented n the last secton.. Groupng and Ranked Methods Groupng methods frst proposed b Wald (940) whch s smpler than the other methods and hence/or otherwse ma easl be appled n man practcal cases. The groupng estmators are well known as wald-tpe estmators. Suppose that we have two varables x,, and n pars{( x, ), =,,,n}, then the groupng methods can be descrbed b the followng steps: - Order the n pars ( x, ) b the magntude of xrr; where =,..., n. - Select proportons PRR PRR that P + P, place the frst nprr pars n group one (GRR), and the last nprr pars n another group (GR3R), and dscardng GRR mddle group of observatons; that s to sa: Place G f x x p ' ( x, ) n G f x p < x x p (4) G3 f x > x p where P ˆ β = P x p s the prr G3 G 3 Y X np ( ) np ( ) The slope can be estmated or formulated as follows: P P G G Y np ( ) X np ( ), =,,n (5) Consequentl, the ntercept estmator wll be: ˆ α = Y βx ˆ (6) Notng that when PRR=PRR=/ then the groupng method named b two groups (Wald,940), and when PRR=PRR=/3 then the method s called three groups, (Nar and Shrvastava,94) and (Bartlett,949). Thel (950) proposed the ranked estmator, whch the man dea s to fnd all possble slopes formulated between an two observatons. Then, the medan of all slopes wll be the estmate. The estmates are gven as follows: 98

4 R Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 ( ˆj β ) ˆ β = medan (7) Where ˆ j βj =, =,...,n-, j=,...,n (8) x x j and ˆ α = medan ˆ βx, =,..., n (9) ( ) 3. L Ranked Set Samplng for Bvarate Data There are varetes of technques b whch the sample ma be selected. For each technque, rough estmates of the sample sze can be made from knowledge of the degree of the precson desred; the related costs and tme nvolved for each technque are also compared before makng a decson (Cochran, 997). The smplest form of random samplng and the most used samplng methods n the practcal studes s called Smple Random Samplng, n whch all the unts n the populaton have equal chance to be chosen, ths samplng method does not attempt to reduce the effect of the data varaton or the error estmaton. McIntre (95) ntroduced a samplng method known as ranked set samplng (RSS). Ths technque was ntroduced as an effcent alternatve method to smple random samplng method. Al-Nasser (007) proposed a robust procedure based on LRSS for detectng outlers, whch s a generalzaton for man tpes of ranked set samplng that ntroduced n the lterature for estmatng the populaton mean. Later, Al-Nasser and Al-Radadeh (008) used LRSS to ft smple lnear regresson. Assume that (X, Y) s a bvarate random vector such that varable Y s dffcult to be measured, but the concomtant varable X, whch s collected wth Y, s easer to measure. Then one can follow the steps below n order to select LRSS: STEP: Randoml draw m ndependent sets each contanng m bvarate samplng unts. STEP: Rank the unts wthn each sample wth respect to the X ' sb vsual nspecton or an other cheap method. STEP3: Select LRSS coeffcent, k = [ mp] such that 0 p < 0.5, and [ Z ] the largest nteger value less than or equal to Z. STEP4: For each of the frst ( k + ) ranked samples; select the unt wth rank k + and measure the value that correspondng to x ( k + ) and denote t b ( k + ). th STEP5: For j = k+,..., m k, the unt wth rank j n the j ranked sample s selected and measures the value that corresponds. STEP6: The procedure contnued untl ( m k) th unt selected from the each of the last ( m k) th ranked samples, wth respect to the frst characterstc and measure the correspond value. 99

5 Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 It can be noted that, for an sample sze, when K = 0, then ths steps wll lead to the novel RSS, n McIntre s procedure. Also for a sample of sze n wth K =, then the selected sample wll gves a sample as n the Medan Ranked Set Samplng (MRSS), addng for that the percentle Ranked Set Samplng (PRSS) can be also consdered as specal case of the LRSS scheme. 4. Groupng and Ranked Estmators Based on Ranked Data: In ths secton we ntroduce the two groups, three groups, and Thel s estmates usng the LRSS. In order to ft the EIV model, let: X( k+ ) j k+ x L() j= X() j k+ m k ; j =,,.., r X( m k) j m k m (0) be LRSS of X varable, and Y[ k+ ] j k+ L[] j= Y[] j k+ m k ; j =,,..., r Y[ m k] j m k m () Groupng estmaton methods n EIV models, t classfes the observatons nto a fnte number of groups and the centres of gravtes (average) of these groups n a scatter dagram are joned together b a lne to fnd the slope. Wald (940) proposed groupng estmators n whch he dvdes the observatons nto two equal groups after sortng them the correspondng observed values obtaned from the response varable Y. the two groups estmates can be dentfed as ˆ L L βlrss. = g x x L L ˆ α = L ˆ β L x LRSSg LRSS LRSSg LRSS () Smlarl; Bartlett (949) dvdes the observatons nto three equall groups and sorted t accordng to x. the estmates wll not depend on the second group whch wll be removed n order to get a better and more effcent estmates the three groups estmates can be formulated as ˆ β ˆ α L L 3 LRSS. = 3 g x x L3 L = L ˆ β L x LRSS3g LRSS LRSS3g LRSS (3) And accordng to Thel (950) the ranked estmator for the slope and the ntercept based on the ranked are gven b: 00

6 Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 ˆ β = medan( ˆ β ) (4) j Where ˆ L Lj βj =, < j, =,..., n, j=,..., n x x L L j (5) Then we have ˆ ˆ β = medan βj (6) ˆ α = medan( ˆ βx) And t s eas to a proof that the proposed estmators based on the ranked data are unbased estmators. 5. Smulaton stud We made computer smulaton to gan nsght n the propertes of the EIV model parameters estmates usng the ranked data. The smulaton was performed wth r = 5, 0 and m = 3, 4, 5 and 6 for the SRS, RSS, and LRSS wth coeffcent ( k = ) data sets. Usng 0,000 replcates, estmates of the mean square errors and the relatve effcenc were computed for each method. The model parameters are ntalsed asα =, β =. The error termsε, ξ and δ are generated from Normal dstrbuton, the tables summarze the followng smulated statstcs: MSE (.). RE (.) = SRS. (7) MSE... MSE ( mod el ) 5. Experment (.) ( ) = 0000 ( ) n 0000 ˆ ε j j= ;where ˆ ˆ = n ε = (8) In ths experment we consder a smulated data from a normal dstrbuton; then under the smulaton assumpton the relatve effcenc () s evaluated. The samples was taken randoml frstl and then usng SRS, RSS and LRSS methods n order to compare between these samplng schemes the estmaton EIV model mean squared error where computed usng three methods Thel, two group and three group method The results based on the proposed estmaton methods are summarzed n the followng tables. Here nsert table From the results we can see that the use of the LRSS was more effcent and more powerful than the use of the SRS n order to estmate the EIV model parameter usng the groupng methods and Thel s method. Also t seems that the use of the RSS s almost has the same effcenc as the use of the SRS scheme and t s more effcent when we use the three group s method. 0

7 Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 Also n order to compare between the estmaton methods we made a comparson between the relatve effcenc of the MSE for the EIV model usng dfferent estmaton methods (groupng methods and Thel method). We fnd out that the use of the two and the three groups methods wth all the samplng methods gves the same ndcaton about the MSE of the EIV model, n other words the model MSE decreased whle the sample sze ncrease, but n the case of the use of Thel s estmaton method we found that the model MSE s almost staple whle the sample sze s ncreased. Here nsert table 5. Experment : Outler case Statstcs provdes a few tools for dealng wth outlers n the data set. Some of these methods are onl vald for small sample szes, and none of them are overl relable for statstcal analss. The ke s to be careful f ou have too man outlers n our data, t ma be an ndcaton that ou should redo our experment, or choose an alternatve expermental method to collect our data. One of the solutons for the outler problem can be started from the begnnng when ou choose our sample to do the analss, especall when ou have pror nformaton that the populaton ma contan some outlers or extreme values ma create some outler cases, then to avod to have such problems, one can use the LRSS to be sure that there wll be an outler case n the sample. Now, to llustrate the performance of the proposed estmators n the presence of outlers, we carred out ths experment under the same smulaton assumptons. The outlers are generated for each samplng method accordng to the quntles method. We smulate 0,000 samples for each sample sze and calculate the RE for the EIV model parameter and for the regresson model as well. The results for the EIV model parameters and average of the regresson model are represented n the next tables. Here nsert table 3 The results gve a ver good ndcaton that n the presents of the outler n the populaton t wll cause a problem f we use a sample taken from the populaton usng the SRS method n order to estmate the EIV model parameter but n the other hand we have a powerful samplng scheme whch s the LRSS and RSS whch gves more effcent model estmaton. In other words the LRSS samplng method gves more effcent model estmaton than the use of the SRS usng the groupng and Thel s estmaton method. Also t seems that the use of the RSS n the case of usng three groupng and Thel s method gves more effcent model estmaton as n the case of the use of the SRS scheme. Here nsert table 4 To compare agan between the estmaton methods n the case that the populaton have some outlers the comparson between the EIV models MSE usng dfferent estmaton methods (groupng methods and Thel method) was made. we fnd out that the use of the two and the three groups methods wth all the samplng methods gves the same ndcaton about the EIV model MSE, n other words the model MSE ncreased whle the sample sze ncrease, but n the case of the use of Thel s estmaton method we found that the model MSE s almost staple whle the sample sze s ncreased. 0

8 Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul Conclusons: One of the man am s of statstcs s to estmate the parameter of nterest wth less errors, less tme, and cost s. In some applcaton were the varable of nterest s expensve or s dffcult to measure accordng to ths problem, the researcher wll spend too much tme and cost s at the same tme for dong such research, latel some of the statstcans the found that the use of the LRSS and the RSS methods s approprate samplng method to solve such these problems. Here we used LRSS and RSS method n order to have samples to estmate the EIV model parameters usng Groupng estmaton methods and Thel s estmaton method. After we dd a smulaton stud we made comparson between the use of the samples whch taken b usng the LRSS or the RSS and wth the samples taken b the SRS we fnd out that the use of the LRSS and RSS schemes wll gves more accurate and more effcent estmators than the use of the SRS scheme.. The second comparson we made between the estmaton methods, n general the smulaton results ndcate that all the estmaton methods wll gves more effcent estmators when we use the LRSS and the RSS nstead of usng the SRS method. Moreover especall the LRSS method approved that t s a robust method to detect the outler from the samples, and t wll provde more precse estmates for the parameters when the data contans some outlers. Also as we saw t can be used as a generalzaton method for man knds of RSS schemes. 03

9 Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 Appendx Table () EIV model relatve effcenc usng dfferent samplng schemes Model Eff for RSS Model Eff for LRSS Ccel Set sze G 3G THEIL G 3G THEIL Table () EIV model mean squared usng dfferent estmaton methods SRS RSS LRSS Ccel Set G 3G THEIL G 3G THEIL G 3G THEIL sze Table (3) EIV model relatve effcenc usng dfferent samplng schemes (outler case) Model Eff for RSS Model Eff for LRSS Ccel Set sze G 3G THEIL G 3G THEIL

10 Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 Table (4) EIV model mean squared usng dfferent estmaton methods (outler case) SRS RSS LRSS Ccel Set sze G 3G THEIL G 3G THEIL G 3G THEIL References Al-Radadeh Ahmed; Amjad D. ALNasser; Enrco Cavolno (008). Effcent Wald-Tpe Estmators for Smple Lnear Measurement Error Model. MTISD 008 conference. Al-Nasser, Amjad D. and Al-Radadeh, Ahmed. (008). Estmaton of Smple Lnear Regresson Model Usng L Ranked Set Samplng. Internatonal Journal of Open Problems n Computer Scence and Mathematcs (IJOPCM). ():8-33. Al-Nasser, Amjad D. (007), L Ranked Set Samplng: A Generalzaton Procedure for Robust Vsual Samplng. Communcaton n Statstcs - Smulaton and Computaton. 36: Bartlett, M. S. (949). Fttng Straght Lne when Both Varables are Subject to Error. Bometrcs. 5:07-. Cheng, C. L. and Van Ness, J. W. (999) Statstcal Regresson wth Measurement Error. Arnold: New York. Fuller, W. A. (987). Measurement Error Models. New York: John Wle. Kendall, M. G. and Stuart, A. (979). The Advanced Theor of Statstcs. Vol., Inference and Relatonshp. Grffn: London. Madansk, A. (959). The Fttng of straght lnes when both varables are subject to error. Journal of the Amercan Statstcal Socet, 55, McIntre, G. A. (95). A Method for Unbased Selectve Samplng Usng Ranked Set Samplng. Australan Journal Agrculture Research. 3: Wald, A. (940). The Fttng of Straght Lnes f Both Varables are Subject to Error. Annals of Mathematcal Statstcs. :

11 Internatonal Journal of Scentfc Engneerng and Appled Scence (IJSEAS) Volume-, Issue-7,Jul 06 Takahash, K. and Wakmoto, K. (968). On Unbased Estmates of the Populaton Mean Based on the Sample Stratfed b Means of Orderng. Annals of the Insttute of Statstcal Mathematcs. 0: -3. Thel, H. A Rank Invarant Method of Lnear and Polnomal Regresson Analss, Indgnatons Math., (950), Nar, R. K. and Shrvastava, P. M. (94). On a Smple Method of Curve Fttng. Sannkha. 6(): -3. Wllam G. Cochran (977). Samplng Technques.3rd Edton, John Wle: New York. 06

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