Estimation of the Population Mean Based on Extremes Ranked Set Sampling
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1 Aerican Journal of Matheatics Statistics 05, 5(: 3-3 DOI: 0.593/j.ajs Estiation of the Population Mean Based on Extrees Ranked Set Sapling B. S. Biradar,*, Santosha C. D. Departent of Studies in Statistics, University of Mysore, Mysore, India All India Institute of Speech Hearing, Mysore, India Abstract This paper is concerned with ranked set sapling theory which is useful to estiate the population ean when the order of a saple of sall size can be found without easureents or with other ethods. In practice ranking a saple of oderate size observing the i-th ranked unit (ranking of iddle ordered units is a difficult task. Therefore, in this paper we propose two estiators of the population ean based on extrees ranked set sapling ethods. The proposed estiators are unbiased for the population ean when the underlying distribution is syetric. It is shown that the proposed estiators are ore efficient than their counter part siple ro sapling ethod for distributions considered in this study. Keywords Ranked set sapling, Extrees ranked set sapling, Population ean, Relative efficiency, Errors in Ranking. Introduction Ranked set sapling (RSS was introduced by McIntyre [] (reprinted in [] for estiating the pasture forage yields. McIntyre indicates that RSS is a ore efficient sapling ethod than siple ro sapling (SRS ethod for estiating the population ean. It is appropriate for situations where quantification of sapling units is either costly or difficult, but ranking the units in a sall set is easy inexpensive. Ranking can be perfored based on expert judgent, visual inspection or any eans that does not involve actually quantifying the observations. In RSS one first draws units at ro fro the population partitions the into sets of units. The units in each set are ranked without aking actual easureents. Fro the first set of units the unit ranked lowest is chosen for actual quantification. Fro the second set of units the unit ranked second lowest is easured. The process is continued until the unit ranked largest is easured fro the -th set of units. If a larger saple size is required then the procedure can be repeated r ties to obtain a saple of size n=r. These chosen eleents are called a ranked set saple. Takashi Wakioto [3] established a very iportant statistical foundation for the theory of RSS. They proved that the saple ean of the RSS is an unbiased estiator of the population ean with saller variance than the saple ean of a SRS with the sae saple size. Dell Clutter [] showed that the ean of the RSS is an unbiased estiator of * Corresponding author: biradarbs@statistics.uni-ysore.ac.in (B. S. Biradar Published online at Copyright 05 Scientific & Acadeic Publishing. All Rights Reserved the population ean, whether or not there are errors in ranking ore efficient than the ean of SRS. Stokes [5] showed that the estiator of the variance based on RSS is an asyptotically unbiased estiator of the population variance, it is ore efficient than the usual estiator based on SRS data with sae saple size, when the saple size is large. Stokes Sager [] studied the epirical distribution function based on RSS showed that it is an unbiased estiate of the underlying distribution function. A recent suaries of RSS literature appear in two survey articles by Wolfe [7] [] a onograph by Chen et al. [9]. Saawi et al. [0] used extree ranked set saple(erss in case of even saple size which is easier to use than the usual RSS procedure to estiate the population ean. Muttlak [] proposed the use of edian ranked set sapling (MRSS ethod for estiating the population ean. Muttlak [] investigated quartile ranked set sapling (QRSS for estiating the population ean. Jeain et al. [3] suggested balanced groups ranked set sapling (BGRSS for estiating population ean. These procedure are based on ranking the i-th unit (other than the extrees of the saple which is a difficult task. Recently, Balci et al. [] introduced another RSS based on extrees for both even odd saple sizes. They have studied odified axiu likelihood estiator (MMLE Best linear unbiased estiator (BLUE when the underlying distribution is noral. In this paper another odification RSS is introduced naely independent extree ranked set sapling (RSS in the case of both even odd saple sizes. The ain objective of this paper is to propose nonparaetric estiators of the population ean using these two extrees RSS are copared with estiator based on SRS ethod.
2 Aerican Journal of Matheatics Statistics 05, 5(: Since it is not difficult to identify axiu or iniu units, extrees RSS is a very useful odification of RSS. It allows for an increase in set size without introducing too any ranking errors.. Estiation of Population Mean Based on Two New RSS In this study two new estiators of population ean based on odified ranked set sapling ethods (i choosing both extrees of each saple (ii choosing extree for two independent saples have been developed... Ranked Set Sapling by Choosing Extrees of Saples (RSS(E Balci et al. [] introduced ranked set saple by choosing extrees of the saples they have called this sapling schee as RSS(E. The procedure of RSS(E is described as follows:. Select ro saples each of size.. Each saple is ranked in itself as in ranked set sapling design. 3. Then sallest largest order statistics fro each saple are observed.. Repeat above steps r ties until the desired saple size n= r is obtained. We assue that the lowest largest units of this set can be detected visually, or by any other eans easily. Let X, X,..., X be a ro saple of size with probability density function f(x with ean µ varianceσ. Let { Xi, Xi,..., X i}, i=,, be sets of independent ro saples each of size fro a population with distribution function F(x pdf f(x with ean µ variance σ. Denote Xi( = in{ Xi, Xi,..., Xi} Xi( = ax{ Xi, Xi,..., Xi} i=,,,. Then { X, X, X X,..., X, X } is a ( ( ( ( ( ( RSS(E of size. Note that the order statistics within the saple are dependent between the saples are independent. For all i=,,, let µ = EX (, σ = Var( X i, µ ( = EX ( i (, µ ( = EX ( i (, ( Var X i ( σ = (, σ ( = Var( Xi ( an σ = Cov( X, X. (, i( i ( Let X be the ean of the SRS of size. The ean variance of X are EX ( = µ Var( X = σ /, respectively. The estiator of the population ean based on RSS(E can be defined as i E= ( i( + i ( ( i = X X X X E can be written as X E = ( X( + X(, where X( = Xi( X( = Xi (. i = i = The ean variance of X E can be shown to be EX ( E = ( µ ( + µ ( ( Var( X E = ( ( ( (, σ + σ + σ (3 If the underlying distribution is syetric about zero, then X( d -X(. Arnold et al. [5] have shown that µ ( = -µ( σ( = σ(. Using these results EX ( E = 0, Var( X E = ( ( (, σ + σ. ( Thus, if the underlying distribution is syetric about its ean then X E is an unbiased estiator of the population ean... Ranked Set Sapling by Choosing Extrees of Tow Independent Saples (RSS Here we introduce another odified RSS called independent Extrees ranked set sapling (RSS based on two independent saples of size : first we select ro saples of size each then identify the axiu within each set of first saples by visual inspection or by soe other cheap ethod, without actual easureent of the variable of interest. Repeat this for other siple ro saples but for the inia. Repeat above steps for r ties until the desired saple size n=r is obtained. Let { Xi, Xi,..., X i} { Y, Y,..., Y } i=,, be sets of ro saples i i i each of size fro a population with distribution function F(x pdf f(x with ean µ variance σ. Denote Xi( = in{ Xi, Xi,..., Xi} Yi ( = ax{ Yi, Yi,..., Yi}, i=,,,. Then { X, X,..., X, Y, Y,..., Y } be ( ( ( ( ( (
3 3 B. S. Biradar et al.: Estiation of the Population Mean Based on Extrees Ranked Set Sapling RSS of size. Note that eleents of saple are independent of each other. The estiator of the population ean based on RSS with one cycle can be defined as: X = ( Xi( Yi ( + i (5 The ean variance of X can be shown to be EX ( = ( µ ( + µ ( ( Var( X ( ( ( (7 Where σ ( is as defined above σ ( = Var( Yi (. If the underlying distribution is syetric about zero then using the above results of Arnold et al.(99, we have EX ( E = 0 Var( X ( ( We can easily see that if the underlying distribution is syetric about its ean then X is an unbiased estiator of the population ean. 3. Efficiency The efficiency of X E with respect to X for estiating the population ean is defined as: X E, X = Var( X MSE( X E. (9 If the distribution is syetric then MSE( X E = Var( X E. Siilarly, X, X = Var( X MSE( X. (0 If the distribution is syetric then MSE( X = Var( X. And finally X, X E = MSE( X E MSE( X. ( For unifor distribution over (0, the efficiency of X E with respect to X is given by X E, X = ( + ( + >, for >. ( Siilarly the efficiency of X with respect to X is given by Table. The Relative Efficiency of estiators of population ean using RSS(E RSS Distribution N Biase( X E Biase( X Eff ( X E, X Eff ( X, X Eff ( X, X E Unifor Exponential Noral Logistic
4 Aerican Journal of Matheatics Statistics 05, 5(: X, X E = >, for > tends to as goes to infinity. This indicates that as the set size increases σ (, = Cov( Xi (, Xi ( tends to zero both estiators X X E are equally efficient. The results of efficiency bias of the estiators are presented in Table for unifor, exponential, noral logistic distributions using SRS, RSSE RSS sapling schees. Fro Table we observed that in the case of unifor, noral logistic distributions the estiators based on X X E are both ore efficient than X X is ore efficient than X E. For exponential distribution Eff ( X E, X Eff ( X, X decrease as increases. This is because X X E are bias estiators bias diverges as goes to. We can easily see that the estiator of population ean based on X is equivalent to the one of the estiators proposed by Saawi et al. [0] based on extree ranked set saple with nuber of cycles equal two when set size is even.. Extree RSS with Errors in Ranking Dell Clutter [] considered the case in which there are errors in ranking; that is, the quantified observation fro the i-th saple in the j-th cycle ay no be the i-th order statistic but rather the i-th judgeent order statistic. They showed that the saple ean of RSS with errors in ranking is unbiased estiator of the population ean regardless of the errors in ranking has saller variance than the usual estiator based on SRS with the sae saple size. Let X i[] X i [ ] denote the sallest largest judgent order statistic of the saple respectively (i=,,,. Then { X[], X[ ], X[] X[ ],..., X[], X [ ]} { X, X,..., X, Y, Y,..., Y } denote [] [] [] [ ] [ ] [ ] RSS(E RSS saples with errors in ranking. The estiators of the population ean using RSS(E RSS with errors in ranking can be defined as X E= ( Xi[] Xi [ ] +, (3 i = X = ( Xi[] Yi[ ] + ( i = The variance of X E X can be defined as Var ( X E = ( [] [ ] [, ] σ + σ + σ (5 Var( X = ( [] [ ] σ + σ ( where σ [] = Var( X i [], σ [ ] = Var( Xi [ ] σ [, ] = Cov( Xi[], Xi [ ] for i=,,. It can easily seen that X E X are unbiased estiators of the population ean if the underlying distribution is syetric about its ean. 5. Conclusions We proposed two new estiators of the population ean using two odified ranked set sapling ethods. The proposed estiators are unbiased of the population ean when the underlying distribution is syetric about its ean. We showed that both estiators have saller variances than the estiator using SRS provide ore efficient estiators. The estiators using extrees RSS (RSS (E RSS will reduce errors in ranking copared to RSS, MRSS, QRSS BGRSS, since we have to identify easure the sallest largest of the ith saple. ACKNOWLEDGEMENTS The Authors are very grateful to the referee editor for their thoughtful coents suggestions. REFERENCES [] McIntyre, G.A. (95. A ethod for unbiased selective sapling using ranked sets.aust.j Agri.Res.3: [] McIntyre, G.A., (005. A ethod for unbiased selective sapling, using ranked sets. The Aerican Statistician 59, Originally appeared in Australian Journal of Agricultural Research 3, [3] Takahasi, K. Wakioto, K. (9. On unbiased estiates of the population ean based on the saple stratified by eans of ordering. Ann. Inst. Stat. Math., 0, -3. [] Dell, T.R. Clutter, J.L. (97. Ranked Set Sapling Theory with Order Statistics Background. Bioetrics,, [5] Stokes, S.L. (90 Estiation of variance using judgent ordered ranked set saples, Bioetrics,3,35-. [] Stokes, S.L Sager, T. (9. Characterization of a ranked set saple with application to estiating distribution
5 3 B. S. Biradar et al.: Estiation of the Population Mean Based on Extrees Ranked Set Sapling functions, J.Aer.Statist.Assoc., 3, [7] Wolfe, D.A. (00. Ranked set sapling: an approach to ore efficient data collection. Statistical Science 9, 3 3. [] Wolfe, D.A., (00. Ranked set sapling. Wiley Interdisciplinary Reviews: Coputational Statistics, 0. [9] Chen, Z., Bai, Z., Sinha, B.K., (00. Ranked Set Sapling: Theory Applications. Springer, New York. [0] Sawi, H., Ahad, M. Abu-Dayyeh, W. (99. Estiating the population ean using extree ranked set sapling. Bio. Journal, 3, [] Muttlak, H.A (997 Median Ranked set sapling, Journal of applied statistical Sciences,,5-55. [] Muttlak H.A. (003. Investigating the use of quartile ranked set saples for estiating the population ean. Journal of Applied Matheatics Coputation.,37-3. [3] Jeain A.A., Al-oari A, Ibrahi, K (00. Soe variations of Ranked set saling. Electronic J.App. Stat. Anal.,, -5. [] Balci, S, Akkaya A.D., Ulgen, B.E (03 Modified Maxiu Likelihood estiators using ranked set sapling. Journal of Coputational Applied Matheatics 3,7-79. [5] Arnold, B.C. Balkrishna, N Nagaraj, H.N. (99. A First course in oreder statistics. John Wiley Sons, New York.
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