On the population median estimation using quartile double ranked set sampling

Transcription

1 On te population edian estiation using quartile double ranked set sapling Aer Ibrai Al-Oari Faculty of Science Departent of Mateatics Al al-bayt University Mafraq Jordan Loai M. Al-Zubi Faculty of Science Departent of Mateatics Al al-bayt University Mafraq Jordan Aad Kazale Faculty of Science Departent of Mateatics Al al-bayt University Mafraq Jordan Abstract In tis article quartile double ranked set sapling (QDRSS) etod is considered for estiating te population edian. Te saple edian based on QDRSS is suggested as an estiator of te population edian. Te QDRSS is copared wit te siple rando sapling (SRS) ranked set sapling (RSS) and quartile ranked set sapling (QRSS) etods for estiating te population edian. To verify tis etod a real data exaple is applied. It turns out tat for te syetric distributions considered in tis study te QDRSS estiators are unbiased estiators of te population edian and are larger tan teir counterparts using SRS RSS and QRSS based on te sae saple size of easured units. For asyetric distributions QDRSS is biased. It is ore efficient tan te SRS and te QRSS for all saples of size wile it is ore efficient tan RSS if 4. Keywords: Siple rando sapling; Quartile ranked set sapling; Ranked set sapling; Quartile double ranked set sapling; Median.. Introduction Ranked set sapling was first suggested by McIntyre (95) as a cost efficient sapling procedure wen copared to te coonly used siple rando sapling in situations were visual ordering of set units can be done easily but te exact easureent of te units is difficult and expensive. McIntyre (95) found tat te RSS is ore efficient tan SRS for estiating te population ean. Let be a rando variable wit a probability density function (pdf) f( x ) and a cuulative distribution function (cdf) Fx ( ) wit ean and variance. Also let f ( ) ( i : ) x be te pdf of te it order statistic of a rando saple of size i i... i for i.... Ten te pdf of ( i : ) is given by F( x) d i i ( i : )( ) ( ) B( i i ) dx 0 f x u u f x Pak.j.stat.oper.res. Vol.I No.4 05 pp53-54

2 Aer Ibrai Al-Oari Loai M. Al-Zubi Aad Kazale were ( ) ( ) 0 0 B u u du wit ean ( ) ( i: ) xf( i: ) x dx and variance 0 ( ) ( i: ) x ( i: ) f( i: ) x dx David and Nagaraja (003). Takaasi and Wakioto (968) provided te necessary ateatical teory of RSS. Tey sowed tat f ( x) f( i : )( x) ( i : ) and ( i: ) ( i: ). i i i i Muttlak (997) suggested edian ranked set sapling for estiating te population ean. Al-Sale and Al-Kadiri (000) considered double ranked set sapling (DRSS) etod for estiating te population ean and tey sowed tat te ranking at te second stage is easier tan te ranking at te first stage. 3 Te double ranked set sapling etod can be described as follows: Randoly identify units fro te target population and divide te randoly into sets eac of size. Te procedure of ranked set sapling is applied to tese sets to obtain ranked set saples eac of size again reapply te ranked set sapling procedure on te ranked set saples to obtain a DRSS of size. Al-Sale and Al-Oari (00) generalized te DRSS to ultistage ranked set sapling to increase te efficiency of te estiators for specific value of te saple size. Muttlak (003) proposed quartile ranked set sapling (QRSS) for estiating te population ean. Al- Oari and Al-Sale (009) suggested quartile double ranked set sapling (QDRSS) for estiating te population ean. Al-Oari (00) suggested an estiator of te population edian using double robust extree ranked set sapling. Entropy estiation and goodnessof-fit tests for te inverse Gaussian and Laplace distributions using paired ranked set sapling etod is suggested by Al-Oari and Haq (05). Biradar and Santosa (05) proposed estiation of te population ean using paired ranked set sapling. Santos and Barrios (05) considered predictive accuracy of logistic regression odel using ranked set saples. Confidence intervals and ypotesis tests for a population ean using ranked set sapling are considered by Stella et al. (05). For ore about RSS and its odifications see Sina et al. (006) Ozturk and Jozani (04) Hatefi et al. (04) Saawi and Al-Sale (04) Bouza (00) and Tiwari and Pandey (03).. Estiation of te population edian.. Using SRS Let... be a rando saple of size fro a distribution wit pdf f( x ) cdf Fx ( ) ean edian and variance. Te SRS estiator of te population edian fro a saple of size at te t cycle... n is defined as 45

3 On te Population Median Estiation using Quartile Double Ranked Set Sapling ˆ SRS if is odd if is even. () Based on f ( x ) if is odd te pdf of ( i : ) is given by! f ( x) F( x) F( x) f ( x) ()! and if is even! f ( x) F( x) F( x) f ( x) (3)!! and! f ( x) F( x) F( x) f ( x). (4)!!.. Using RSS Te RSS (McIntyre 95) involves randoly selecting units fro te population. Tese units are randoly allocated into sets eac of size. Te units of eac saple are ranked visually or by any inexpensive etod wit respect to a variable of interest. Fro te first set of units te sallest ranked unit is easured. Fro te second set of units te second sallest ranked unit is easured. Te process is continued until fro te t set of units te largest ranked unit is easured. Te process can be repeated n ties to get a saple of size n fro te initial n units. Let... ;... ; ;... be independent siple rando saples eac of size in te t cycle... n. Let i() i() i( ) be te order statistics of te it saple i i... i for i... Terefore () () ( ) denote te easured RSS units. Te RSS estiator of te population edian fro a saple of size at te t cycle... n is given by RSS () () ( ) ˆ Median.... (5).3. Using QRSS Te QRSS procedure suggested by Muttlak (003) is described as follows: select rando saples eac of size units fro te target population and rank te units witin eac saple wit respect to te variable of interest. If te saple size is even select for easureent fro te first / saples te q ( )t sallest ranked unit and fro te 44

4 Aer Ibrai Al-Oari Loai M. Al-Zubi Aad Kazale second / saples te q ( )t sallest ranked unit were 3 q 0.5 and q were te nearest integers of q ( )t and q ( )t 3 will always be taken. If te saple size is odd select fro te first ( ) / saples te q ( )t sallest ranked unit and fro te oter ( ) / saples te q ( )t 3 sallest ranked unit and fro one saple te edian of tat saple for actual easureent. Te procedure can be repeated n ties if needed to increase te saple size to n units. If te saple size is even at te t cycle... n let be te first quartile i( q ( )) of te it saple i... and i( q 3 ( )) be te tird quartile of te it saple 4 i.... Terefore te easured QRSSE units are ( q ( )) Te QRSSE estiator of te population edian ( q ( )) ( q3 ( )) ( q3 ( )) is given by ˆ QRSSE Median ( q( )) ( q3( )) (6) ( q( )) ( q3( )) If te saple size is odd let i... i i( q3 ( )) be te first quartile of te it saple i( q ( )) be te tird quartile of te it saple easured QDRSSO units are ( q3 ( )) be te edian of te it saple of te rank... ( q ( )) i and let 3 5 i.... Terefore te... ( q ( )). Te estiator of te population edian using QRSSO is defined as 3 ( q3 ( )) ˆ Median QRSSO ( q( )) 3 ( q3( )) ( q( )) ( q3( )).4. Using QDRSS Te quartile double ranked set sapling etod (Al-Oari and Al-Sale 009) can be carried out as follows: 3 Step : Randoly select units fro te target population and allocate te into sets eac of size units. Step : Rank te units witin eac set wit respect to te variable of interest and ten apply te RSS etod on te sets. Tis step yields ranked set saples eac of size. Step 3: Witout doing any actual quantifications apply te QRSS etod on te DRSS sets obtained in Step. Te wole process can be repeated n ties if needed to get (7) 45

5 On te Population Median Estiation using Quartile Double Ranked Set Sapling a saple of size n fro te QDRSS data. For even and odd saple sizes we denote te easured QDRSS units as QDRSSE and QDRSSO respectively. Let us consider te following exaple. Select a rando saple of size 8 so we will 3 select 5 units. Allocate te into 8 sets eac of 64 units. Rank te units witin eac set wit respect to te variable of interest. Let jik be te it unit i...8 in te jt set j...8 in te kt subset k...8. Select te j() i k fro eac subset te processes appears as sown below: () () (7) (8) () () (7) (8) ()8 ()8 (7)8 (8)8 () () (7) (8) Te st set of size 64 units () 8() 8(7) 8(8) 8() 8() 8(7) 8(8) 8()7 8()7 8(7)7 8(8)7 8()8 8()8 8(7)8 8(8)8 Te 8t set of size 64 units Select te it sallest ranked unit fro te it subset i..8 in eac set. Tis step yields 64 units i.e. 8 RSS sets eac of size 8 as follows: () () (3)3 (4)4 (5)5 (6)6 (7)7 (8)8 () () (3)3 (4)4 (5)5 (6)6 (7)7 (8)8 3() 3() 3(3)3 3(4)4 3(5)5 3(6)6 3(7)7 3(8)8 4() 4() 4(3)3 4(4)4 4(5)5 4(6)6 4(7)7 4(8)8 5() 5() 5(3)3 5(4)4 5(5)5 5(6)6 5(7)7 5(8)8 6() 6() 6(3)3 6(4)4 6(5)5 6(6)6 6(7)7 6(8)8 7() 7() 7(3)3 7(4)4 7(5)5 7(6)6 7(7)7 7(8)8 8() 8() 8(3)3 8(4)4 8(5)5 8(6)6 8(7)7 8(8)8. Witout doing any actual quantifications on tese sets rank te units witin eac set wit respect to te variable of interest and ten select te first quartile i() fro te it set ( i 34) and select te tird quartile i(7) fro te it set ( i 5678) as sown below: () () (3) (4) (5) (6) (7) (8) () () (3) (4) (5) (6) (7) (8) ()3 ()3 (3)3 (4)3 (5)3 (6)3 (7)3 (8)3 45

6 Aer Ibrai Al-Oari Loai M. Al-Zubi Aad Kazale ()4 ()4 (3)4 (4)4 (5)4 (6)4 (7)4 (8)4 ()5 ()5 (3)5 (4)5 (5)5 (6)5 (7)5 (8)5 ()6 ()6 (3)6 (4)6 (5)6 (6)6 (7)6 (8)6 ()7 ()7 (3)7 (4)7 (5)7 (6)7 (7)7 (8)7 ()8 ()8 (3)8 (4)8 (5)8 (6)8 (7)8 (8)8 v Tis process produces () () ()3 ()4 (7)5 (7)6 (7)7 (7)8 as a QDRSSE of size 8. Te edian of tese units can be considered as an estiator of te population edian. It is defined as ˆ Median. (8) QDRSSE () () ()3 ()4 (7)5 (7)6 (7)7 (7)8 Te ost interesting ting ere is tat te nuber of quantified units using QDRSS is 8 wic will be copared wit a SRS of size 8 is a sall relative to to te nuber of sapled units 5. Hence te inforation contained in te QDRSS saple is ore tan te inforation in te 8 units of te SRS. In te t cycle... n if te saple size is even let be te first quartile i( q ( )) of te it saple i... and i( q3 ( )) be te tird quartile of te it saple 4 i.... Hence te easured QDRSSE units are ( q ( )) Te suggested QDRSSE estiator of te ( q ( )) ( q3 ( )) ( q3 ( )) population edian is given by ˆ QDRSSE Median ( q( )) ( q3( )) (9) ( q( )) ( q3( )) If te saple size is odd let i... and i i( q ( )). be te first quartile of te it saple be te edian of te it saple of te rank i and 3 5 i( q3 ( )) be te tird quartile of te it saple i.... Terefore te QDRSSO easured units are ( q3 ( )) as ( q ( )) ( q ( )) 3 ( q3 ( )). Te suggested estiator of te population edian using QDRSSO is defined 45

7 On te Population Median Estiation using Quartile Double Ranked Set Sapling ˆ Median QDRSSO ( q ( )) ( 3 ( q3 ( )) q( )) ( q3( )) (0) 4. Siulation Study In tis section a siulation study is considered to copare te proposed estiators for te population edian using QDRSS QRSS RSS relative to SRS. Six probability distribution functions were considered for te populations: Unifor Noral Logistic Exponential Gaa and Weibull saples were generated and te averages of tese saples were copared. If te distribution is syetric te efficiency of RSS QRSS and QDRSS relative to SRS is defined as respectively and eff eff ˆ RSS ˆ ˆ QDRSS SRS ˆ SRS Var ˆ Var ˆ SRS eff ˆ ˆ QRSS SRS RSS Var ˆ SRS. Var ˆ QDRSS Var ˆ SRS Var ˆ QRSS If te distribution is asyetric te efficiency is defined by MSE ˆ SRS MSE ˆ SRS eff ˆ ˆ RSS SRS eff ˆ ˆ QRSS SRS MSE ˆ MSE ˆ and eff ˆ QDRSS ˆ SRS RSS QRSS MSE ˆ SRS were MSE ˆ QDRSS MSE Var E. Results of siulation in ters of te efficiency and bias values for RSS QRSS and DQRSS are suarized for 45 in Table for 67 in Table for 0 in Table 3 and for in Table 4. Table : Te efficiency and bias values of RSS QRSS and QDRSS wit respect to SRS in estiating te population ean wit 4 and 5. Distribution 4 5 RSS QRSS QDRSS RSS QRSS QDRSS Unifor (0) Eff Noral (0) Eff Logistic (-) Eff Exponential () Eff Bias Gaa () Eff Bias Weibull (3) Eff Bias

8 Aer Ibrai Al-Oari Loai M. Al-Zubi Aad Kazale Table : Te efficiency and bias values of RSS QRSS and QDRSS wit respect to SRS in estiating te population ean wit 6 and 7. Distribution 6 7 RSS QRSS QDRSS RSS QRSS QDRSS Unifor (0) Eff Noral (0) Eff Logistic (-) Eff Exponential () Eff Bias Gaa () Eff Bias Weibull (3) Eff Bias Table 3: Te efficiency and bias values of RSS QRSS and QDRSS wit respect to SRS in estiating te population ean wit 0 and. Distribution 0 RSS QRSS QDRSS RSS QRSS QDRSS Unifor (0) Eff Noral (0) Eff Logistic (-) Eff Exponential () Eff Bias Gaa () Eff Bias Weibull (3) Eff Bias Table 4: Te efficiency and bias values of RSS QRSS and QDRSS wit respect to SRS in estiating te population ean wit respect to SRS wit. Distribution RSS QRSS QDRSSE Unifor (0) Eff Noral (0) Eff Logistic (-) Eff Exponential () Eff Bias Gaa () Eff Bias Weibull (3) Eff Bias According to tese results we conclude: ) If te underlying distribution is syetric about its ean ten are unbiased estiators of te population edian wit saller a) ˆQDRSSE and ˆQDRSSO variance tan te ˆSRS estiator based on te sae saple size. As an exaple for 45

9 On te Population Median Estiation using Quartile Double Ranked Set Sapling 7 te efficiency of ˆQDRSSO is for estiating te population edian of te standard noral distribution. and ˆQDRSSO are ore efficient tan ˆRSS. For exaple wen te b) ˆQDRSSE efficiency of ˆQDRSSO and ˆRSS are 3.65 and.838 respectively for estiating te population edian of te standard unifor distribution. and ˆQDRSSO are ore efficient tan ˆQRSS. For 0 te efficiency c) ˆQDRSSE values of ˆQDRSSE and ˆQRSS are.46 and 4.3 respectively for estiating te edian of te Logistic distribution wit paraeters - and. ) If te underlying distribution is asyetric we noted tat ave a sall bias. As an exaple for te efficiency of a) ˆQDRSSE ˆQDRSSE and ˆQDRSSO is 5.0 wit bias for estiating te edian of te exponential distribution wit paraeter. and ˆQDRSSO are ore efficient tan ˆRSS if 4 and tey are ore b) ˆQDRSSE efficient tan ˆQRSS for all cases considered in tis study based on te sae nuber of easured units. For exaple wit 0 te efficiency values of ˆRSS ˆQRSS and ˆQDRSSE are respectively and 8.70 for estiating te edian of Weibull distribution wit paraeters and 3. 3) Coparing ˆQDRSSE to ˆQDRSSO it is found tat ˆQDRSSE is ore efficient. For exaple for 6 and 7 te efficiency of ˆQDRSSE and ˆQDRSSO are respectively and for estiating te edian of standard noral distribution. Tis ay be due to tat: in te case of odd saple size we select only te edian of te set of te rank i wile wit even saple size we identify te first or te tird quartile of te it saple. 5. Real Data Application In tis section to evaluate te perforance of QDRSS in estiating te population edian of a real data a study is conducted to estiate te edian weigt of 34 students. Balanced ranked set sapling is considered and all saples were done witout replaceent. Let i for i...34 be te weigt of te i t student in te population. Te ean edian and te variance of te population are respectively and Z7 Z7 Zi kg Median Z i i i 34 ( Zi ) 58.93kg 34 i. Te skewness of te 34 observations is.44 wic eans tat tese data are asyetrically distributed and so te QDRSS estiators will be biased. Hence te bias 4

10 Aer Ibrai Al-Oari Loai M. Al-Zubi Aad Kazale and ean squared error (MSE) of te estiators were coputed. Te efficiency of RSS QRSS and QDRSS wit respect to SRS are obtained using Equations (8) (9) and (0). Te siulated edian bias MSE and te efficiency values are suarized in Table 5. Table 5: Te efficiency and bias values of RSS QRSS and QDRSS relative to SRS wit saple size for estiating te edian weigt of 34 students. Metod Saple size SRS Median Bias MSE RSS Median Bias MSE Efficiency QRSS Median Bias MSE Efficiency QDRSS Median Bias MSE Efficiency Table 5 sows tat tere is a sall difference between te true and te estiated edian. Tis difference is due to skewness of te data used in tis exaple. For 4 RSS is ore efficient tan QDRSS. Wile QDRSS is ore efficient tan RSS. In addition it can be noted tat QDRSS is ore efficient tan QRSS for all saple sizes considered in Table 5. Furterore te results of real data exaple are agreed wit te results of te siulation study conducted in Section Conclusion In estiating te population edian a good acieveent is gained in efficiency using QDRSS QRSS RSS regardless te underlying distribution weter it is syetric or asyetric. QDRSS estiators are unbiased estiators of te population edian wen distributions are syetric. In addition it is found tat QDRSS is ore efficient tan RSS if 4 and ore efficient tan QRSS in all cases considered in tis study. However te QDRSS is recoended for estiating te population edian of syetric distributions. Acknowledgeent Te autors are tankful to te reviewer and te associate editor for valuable coents tat significantly iproved te current version of te article. 4

11 On te Population Median Estiation using Quartile Double Ranked Set Sapling References. Al-Oari A.I. (00). Estiation of te population edian of syetric and asyetric distributions using double robust extree ranked set sapling. Revista Investigación Operacional 3(3): Al-Oari A.I. and Haq A. (05). Entropy estiation and goodness-of-fit tests for te inverse Gaussian and Laplace distributions using paired ranked set sapling. Journal of Statistical Coputation and Siulation accepted DOI:0.080/ Al-Oari A.I. and Jaber K. (008). Percentile double ranked set sapling. Journal of Mateatics and Statistics 4(): Al-Sale M.F. and Al-Kadiri M.A. (000). Double ranked set sapling. Statistics and Probability Letters 48(): Al-Sale M.F. and Al-Oari A.I. (00). Multistage ranked set sapling. Journal of Statistical Planning and Inference 0(): Al-Oari A.I. and Al-Sale M.F. (00.). Quartile double rankled set sapling for estiating te population ean. Econoic Quality Control 4(): Balakrisnan N. and Li T. (006). Confidence intervals for quantiles and tolerance intervals based on ordered ranked set saples. Annals of te Institute of Statistical Mateatics 58: Biradar B.S. and Santosa C.D. (05). Estiation of te population ean using paired ranked set sapling. Open Journal of Statistics Bouza C.N. (00). Ranked set subsapling te non-response strata for estiating te difference of eans. Bioetrical Journal 44: David H.A. and Nagaraja H.N. (003). Order Statistics 3rd ed. Jon Wiley & Sons Inc. Hoboken NJ.. Hatefi A. Jozani M.J. and Ziou D. (04). Estiation and classification for finite ixture odels under ranked set sapling. Statistica Sinica 4: McIntyre G.A. (95). A etod for unbiased selective sapling using ranked sets. Australian Journal of Agricultural Researc 3: Muttlak H.A. (997). Median ranked set sapling. Journal of Applied Statistical Sciences 6(4): Muttlak H.A. (003). Investigating te use of quartile ranked set saples for estiating te population ean. Journal of Applied Mateatics and Coputation Ozurk O. and Jozani M.J. (04). Inclusion probabilities in partially rank ordered set sapling. Coputational Statistics and Data Analysis 69:

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