Inference on Downton s Bivariate Exponential Distribution Based on Moving Extreme Ranked Set Sampling

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1 AUSTRIAN JOURNAL OF STATISTICS Volue ), Nuber 3, 6 79 Inference on Downton s Bivariate Exponential Distribution Based on Moving Extree Ranked Set Sapling Ahed Ali Hanandeh and Mohaad Fraiwan Al-Saleh 2 Dept. Matheatical Sciences, University of Cincinnati, Ohio, USA 2 Dept. Statistics, Yarouk University, Irbid, Jordan Abstract: The purpose of this paper is to estiate the paraeters of Downton s bivariate exponential distribution using oving extree ranked set sapling MERSS). The estiators obtained are copared via their biases and ean square errors to their counterparts using siple rando sapling SRS). Monte Carlo siulations are used whenever analytical coparisons are difficult. It is shown that these estiators based on MERSS with a concoitant variable are ore efficient than the corresponding ones using SRS. Also, MERSS with a concoitant variable is easier to use in practice than RSS with a concoitant variable. Furtherore, the best unbiased estiators aong all unbiased linear cobinations of the MERSS eleents are derived for soe paraeters. Zusaenfassung: Das Ziel dieser Arbeit ist es, die Paraeter der Downton bivariaten Exponentialverteilung ittels oving extree ranked set sapling MERSS) zu schätzen. Die erhaltenen Schätzer werden bezüglich Bias und ittleren quadratischen Fehler it deren Gegenstücken basierend auf einfache Zufallsstichproben SRS) verglichen. Monte Carlo Siulationen werden verwendet, wenn analytische Vergleiche schwierig sind. Es wird gezeigt, dass die Schätzer basierend auf MERSS it einer begleitenden Variablen effizienter sind als die entsprechenden beruhend auf SRS. Auch ist MERSS it einer begleitenden Variablen einfacher in der Praxis zu verwenden als RSS it einer begleitenden Variablen. Außerde werden die besten Schätzer unter allen unverzerrten Linearkobinationen der MERSS Eleente für einige Paraeter hergeleitet. Keywords: Siple Rando Sapling, Moving Extree Ranked Set Sapling, Concoitant Variable. Introduction The ost popular sapling ethod that is usually used in statistical studies is Siple Rando Sapling SRS). A SRS of size n fro a population is a subset of the population consisting of n units selected in such a way that all subsets of size n are equally likely to be selected; but for an infinite population which we are interested in) each unit of the saple is selected independently and coes fro the sae population i.i.d.). Siple

2 62 Austrian Journal of Statistics, Vol ), No. 3, 6 79 rando sapling is the basic building block and point of reference for all other sapling ethods. McIntyre 952) suggested a sapling technique which was later called Ranked Set Sapling RSS). This technique of data collection was introduced for situations where taking the actual easureents on saple observations is difficult i.e., costly, tieconsuing) as copared to the judgent ranking of the. The ranked set sapling technique can be executed as follows: Step : Randoly draw siple rando saples each of size fro the population of interest. Step 2: Within each of the sets, the sapled ites are ranked fro lowest to largest according to the variable of interest based on the researcher s judgent or by any negligible cost ethod that does not require actual quantifications. Step 3: Fro the first set of units, the unit ranked lowest is easured. Fro the second set of units, the unit ranked second lowest is easured. The process is continued until the th ranked unit is easured fro the th set. Note that 2 units are sapled but only of the are easured with respect to the variable of interest. The above 3 steps describe one cycle of the RSS technique. Step 4: Repeat steps to 3, if necessary, k independent ties cycles) to obtain a total saple of size n = k units. In McIntyre s RSS procedure, it is assued that the researcher could order a set of size with respect to the characteristic of interest. Many authors recoended that should be 2, 3, or 4 to iniize the ranking error see Takahasi and Wakioto, 968). There are several variations of RSS. One of the is Moving Extree Ranked Set Sapling MERSS). In this procedure only the axiu or iniu) of sets of varied size is identified by judgent) for quantification. The MERSS, as described by Al-Odat and Al-Saleh 200) and Al-Saleh and Al-Hadrai 2003a, 2003b) can be executed as follows: Step : Select siple rando saples of sizes,...,, respectively. Step 2: For each of these saples, easure accurately the axiu ordered observation fro the first set identified by judgent, the axiu ordered observation fro the second set, etc.. The process continues in this way until the axiu ordered observation fro the last th saple is easured. Step 3: Repeated steps and 2, if necessary, r ties to obtain a saple of size n = r. 2 Main Results about RSS and its Variations RSS was introduced by McIntyre 952) in the context of estiating pasture yields. He claied, without providing a atheatical proof, that. ˆµ RSS = X k [i]r which is the ean of a saple of size k obtained using r= i= the RSS technique, is an unbiased estiator of the population ean µ. X [i]r is the ith judgent order statistic in the rth cycle. 2. With perfect ranking, the efficiency of RSS w.r.t. SRS in estiation the population ean is nearly + )/2 for typical uniodal distributions.

3 A.A. Hanandeh, M.F. Al-Saleh The efficiency of the estiators of higher population oents based on RSS are only slightly better than those based on SRS. Takahasi and Wakioto 968) obtained the following ain theoretical results. Under perfect ranking, the ean of a RSS is an unbiased estiator of the population ean and its variance is always saller than the variance of the ean of a SRS of equal size. Also, they showed that fx) = f i x), i= µ = i= µ i and σ 2 = σi 2 + i= µ i µ) 2, where fx) denotes the probability density function pdf) of a rando variable X with EX) = µ and varx) = σ 2. Furtherore, f i x) is the pdf of the ith order statistic X i) with EX i) ) = µ i and varx i) ) = σ 2 i. The perforance of the estiators is assessed using either the relative precision RP) or the relative saving RS), which are defined as i= RP = effˆµ RSS, ˆµ SRS ) = varˆµ SRS) varˆµ RSS ) RS = varˆµ SRS) varˆµ RSS ) varˆµ SRS ) = RP. They also showed that varˆµ RSS ) = σ 2 ) µ i µ) 2 i= and RS = ) 2 µi µ, σ i= and thus where 0 RS + effˆµ RSS, ˆµ SRS ) + 2 ˆµ SRS = is the ean of a saple of size obtained using the SRS technique, and ˆµ RSS = k is the ean of a saple of size k obtained using the RSS technique. Stokes 977) studied RSS with concoitant variables. In this case, the variable of interest is difficult to rank but it is related to soe other variable that is easy to rank concoitant variable). We use X, Y ) in the sequel, where X is the variable of interest and r= i= X i i= X [i]r,

4 64 Austrian Journal of Statistics, Vol ), No. 3, 6 79 Y is the concoitant one. Saawi, Ahed, and Abu-Dayyeh 996) introduced the Extree Ranked Set Sapling ERSS) procedure. It was shown that the ERSS estiator of the ean is ore efficient than the usual SRS ean and unbiased if the underlying distribution is syetric. In this ERSS, only the two extrees iniu, axiu) are identified by judgent for different sets. Al-Saleh and Al-Kadri 2000) considered double RSS DRSS) as a procedure that increases the efficiency of RSS estiator without increasing the set size. Al-Saleh and Al-Oari 2002) generalized DRSS to ultistage ranked set sapling. They showed that the efficiency is always between and 2 for all distributions and equals 2 for the unifor distribution when the nuber of stages goes to infinity. Al-Saleh and Zheng 2002) proposed a new RSS for two characteristics and called it a bivariate ranked set sapling. MERSS is a useful odification of RSS. Unlike RSS, MERSS allows for an increase of the set size without introducing too uch ranking error. MERSS was introduced by Al-Odat and Al-Saleh 200) who also introduced the concept of varied set size RSS. Al-Saleh and Al-Hadrai 2003a, 2003b) used varied set size of RSS coined by the MERSS) for estiating the ean of the noral and exponential distribution, and they showed that this procedure could be ore useful than SRS for estiating the ean of any syetric distribution. Al-Saleh and Saawi 200) estiated the odds based on MERSS. The suggested estiator based on MERSS is otivated by soe of the theoretical properties of the su of geoetric series. RSS and soe of its variation were used by any authors in paraetric estiation like bivariate noral, exponential, Downton s bivariate exponential see Downton, 970), etc.. Moran 967) introduced a bivariate exponential distribution, which is one of the ost iportant bivariate distributions in reliability. There are various bivariate exponential distributions. In this research, we are interested in Downton s bivariate exponential distribution with pdf [ )] [ ] fx, y)= λ λ 2 ρ) exp x λ ρ) + y 2ρxy) 2 I 0, ) λ 2 ρ) λ λ 2 ) 2 ρ) where x, y, λ, λ 2 > 0, 0 ρ < and I 0 z) = z/2) 2k k=0 k! 2 is the odified Bessel function of the first kind of order zero. If X, Y ) is a bivariate rando variable that has Downton s bivariate exponential distribution, then fro ) the arginal distributions of X and Y are exponential with paraeters λ and λ 2, respectively. Thus, in particular, EX) = λ and EY ) = λ 2. Downton 970) further showed that EY X = x) = ρ)λ 2 + ρ λ 2 λ x vary X = x) = ρ) 2 λ ρ ρ) λ2 2 λ x.

5 A.A. Hanandeh, M.F. Al-Saleh 65 The paraeter ρ is the correlation coefficient between X and Y with independence corresponding to ρ = 0 since I 0 0) =. This distribution is a candidate distribution for positively correlated bivariate exponential data, in which the conditional ean and variance of one variable is increasing function of the other variable. This distribution has real applications in several fields. Lefebvre 2004) considered forecasting the flow values of the Mistassini river in Quebec Canada). The relation between oil pollution of sea water and tar deposit near sea shore is another application considered by Bain 978). In this application, the oil pollution is hard and expensive to easure while the tar deposit can be ranked visually. For ore applications, see also Balakrishna and Lai 2009, pp ). Iliopoulos 2003) showed that the joint sufficient statistic for the three paraeters based on the SRS X, Y ),..., X, Y ) fro this distribution is ) X Y,..., X Y, X i, Y i. Nagao and Kadoya 97) showed that the axiu likelihood estiator of λ and λ 2 is X and Ȳ, respectively. Two classes of estiators of the paraeter ρ based on the coplete bivariate saple were derived in Al-Saadi and Young 980). The ethod of oents estiator is based on the statistic i= ˆρ = X iy i XȲ, where X = X i and Ȳ = Y i. Using the condition 0 ρ <, a odified estiator for ρ is i= i= i= i= 0 if ˆρ < 0 ˆρ = ˆρ if 0 ˆρ if ˆρ >. Another estiator is based on the saple correlation coefficient r = X i X)Y i Ȳ ) i=. X i X) 2 Y i Ȳ )2 Because 0 ρ <, they finally suggested to use { 0 if r < 0 ˆρ 2 = r if r 0. i= Note that the first estiator is a function of the sufficient statistic, whereas the second estiator is not, so we will use the first estiator in what follows. Al-Saleh and Diab 2009) estiated the paraeters of Downton s bivariate exponential distribution based on a RSS. He and Nagaraja 20) estiated the correlation coefficient of Downton s bivariate exponential distribution when all other paraeters are i=

6 66 Austrian Journal of Statistics, Vol ), No. 3, 6 79 unknown using incoplete saples. For ore details about RSS and its variations see Chen, Bai, and Sinha 2004). For ore details about this work see Hanadeh 20). In this paper, we consider the estiation of the paraeters of a DBEDλ, λ 2, ρ) distribution using MERSS. The suggested estiators are copared with the corresponding ones using SRS. Estiation of λ and λ 2 for the two cases of known and unknown correlation coefficient ρ is considered. Furtherore, estiation of the correlation coefficient ρ for the two cases of known and unknown λ and λ 2 is also discussed. 3 Mean and Variance of X k:k) and Y [k:k] Let X, Y ),..., X, Y ) be a rando saple fro Downton s bivariate exponential distribution, DBEDλ, λ 2, ρ), with coon density given in ). Let f X x) and f Y y) be the arginal densities of X and Y, respectively, and let F X x) and F Y y) be the corresponding cuulative distribution function cdf) of X and Y, respectively. Now, suppose that {X :), Y [:] ),..., X k:k), Y [k:k] )}, k =,...,, is a MERSS saple fro the DBED, X i:) is the ith order statistic of X,..., X, and Y [i:] is the Y - variate paired with X i:k), then Y [i:k] is called a concoitant order statistic. If the judgent ranking on the X variate is perfect, then X k:k), Y [k:k] ) has pdf f Xk:k),Y [k:k] x, y) = f Xk:k) x)f Y X y x), 2) for k =,...,, where f Xk:k) x) is the pdf of the kth order statistic of a SRS of size k fro an exponential distribution, f Y[k:k] x) is the pdf of the corresponding variable Y concoitant order statistic) and f Y X y x) is the conditional pdf of Y X = x) cp. Yang, 977). But, f Xk:k) x) = kf X x)) k f X x) = k exp x )) k f X x). 3) λ Also, f Xk:k),Y [k:k] x, y) = f X,Y x, y) f X x) = k exp k exp x )) k f X x) λ x λ )) k f X,Y x, y). 4) It can be verified that if Z,..., Z k are i.i.d. fro an Exponentialλ ) distribution then X k:k) has the sae distribution as k Z j see Yang, 977). Thus, it follows that k j+ E X k:k) ) = var X k:k) ) = j= j= j= EZ j ) k j + = λ j= varz j ) k j + ) 2 = λ2 k j + j= 5) k j + ) 2. 6)

7 A.A. Hanandeh, M.F. Al-Saleh 67 Moreover, we have Using 5) we get Also, Hence, vary [k:k] ) = λ 2 2 EY [k:k] ) = EEY [k:k] X k:k) )) = ρ)λ 2 + ρ λ 2 λ EX k:k) ). EY [k:k] ) = ρ)λ 2 + ρλ 2 j= k j +. 7) vary [k:k] ) = EvarY [k:k] X k:k) )) + varey [k:k] X k:k) )). { ρ) 2 + 2ρ ρ) j= k j + + ρ2 4 Unbiased Estiation of λ, λ 2 with Known ρ j= }. 8) k j + ) 2 We now consider the estiation of λ and λ 2 when ρ is known using both MERSS and SRS. Then the perforance of the suggested estiators is investigated. Denote the unbiased estiator of λ and λ 2 based on SRS by ˆλ SRS and ˆλ 2SRS, respectively, and those based on MERSS by ˆλ MERSS and ˆλ 2MERSS, respectively, with where Now where Hence, ˆλ SRS = i= X i and ˆλMERSS = S S = + k k. X k:k) 9) varˆλ MERSS ) = C λ 2 S 2 0) C = + ) k k 2. effˆλ MERSS, ˆλ SRS ) = varˆλ SRS ) varˆλ MERSS ) = S2 C = + ) k k +) k k, ) +) k

8 68 Austrian Journal of Statistics, Vol ), No. 3, 6 79 which is clearly larger than and increasing in. Also, ˆλ 2SRS = Y i and ˆλ2MERSS = i= ρ) + ρs Y [k:k]. 2) It follows that varˆλ 2MERSS ) = which can be siplified to ρ) + ρs ) 2 { ρ)λ 2 2 ρ) + 2ρ +ρ 2 λ 2 2 j= ) k j + j= }, k j + ) 2 varˆλ 2MERSS ) = Hence, using 2) and 3) we get λ 2 2 ρ) + ρs ) 2 ρ) 2 + 2ρ ρ)s + ρ 2 C ). 3) effˆλ 2MERSS, ˆλ 2SRS ) = varˆλ 2SRS ) varˆλ 2MERSS ) which is also clearly larger than, since = ρ) + ρs ) 2 ρ) 2 + 2ρ ρ)s + ρ 2 C ), 4) effˆλ 2MERSS, ˆλ 2SRS ) = 2 ρ) 2 + 2ρ ρ)s + ρ 2 S 2 2 ρ) 2 + 2ρ ρ)s + ρ 2 C S2 C. Table contains efficiencies of ˆλ MERSS and ˆλ 2MERSS with respect to the respective estiators using SRS. Based on Table we conclude that effˆλ MERSS, ˆλ SRS ) and effˆλ 2MERSS, ˆλ 2SRS ) are always greater than. effˆλ MERSS, ˆλ SRS ) is fixed in ρ and increasing in. effˆλ 2MERSS, ˆλ 2SRS ) increases in for fixed ρ and increases in ρ for fixed. effˆλ 2MERSS, ˆλ 2SRS ) effˆλ MERSS, ˆλ SRS ) for any as ρ.

9 A.A. Hanandeh, M.F. Al-Saleh 69 Table : effˆλ MERSS, ˆλ SRS ) effˆλ 2MERSS, ˆλ 2SRS )) ρ ) ) ) ) ) ) ) ) ) ) ) ) ).733.3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 5 Optial Estiator of λ, λ 2 Based on MERSS Eleents If τ = {ˆθ,..., ˆθ N } is a set of N unbiased estiators of the paraeter θ with varˆθ i ) = δi 2 and covˆθ i, ˆθ j ) = 0 for all i j, then aong all unbiased linear cobinations N b iˆθi, where b i is a constant, has the sallest variance ˆθ = N ˆθ i /δi 2 i= N /δi 2 i= varˆθ ) = N /δi 2 see e.g. Casella and Berger, 2002, p. 363). Using this result, we can find the best unbiased estiators for λ and λ 2 aong all unbiased linear cobinations of the MERSS eleents. For siplicity, let a k = j= We know fro 5) and 6) that k j + i= and b k = j= k j + ) 2. EX k:k) ) = a k λ and varx k:k) ) = b k λ 2. Thus, the best unbiased estiator for λ aong all unbiased linear cobinations of X k:k) is X k:k) a 2 k a k X k:k) a k λ 2 T b k b k = =. 5) a 2 k a 2 k λ 2 b k b k i=

10 70 Austrian Journal of Statistics, Vol ), No. 3, 6 79 It follows that Hence, using 6) we get vart ) = λ2 a 2 k. 6) b k efft, ˆλ SRS ) = varˆλ SRS ) vart ) = λ2 λ 2 a 2 k b k = j= j= k j+) k j+) 2 ) 2 ) 7) Table 2 gives values of the efficiencies efft, ˆλ SRS ) and effˆλ MERSS, ˆλ SRS ). Based on these results, we conclude that efft, ˆλ SRS ) is always larger than. It is larger than effˆλ MERSS, ˆλ SRS ) for all values of, i.e. T is ore efficient than ˆλ MERSS and hence than ˆλ RSS and ˆλ SRS. Note that the best linear unbiased estiator and the traditional one, though different, are relatively close to each other. Table 2: Efficiencies efft, ˆλ SRS ) and effˆλ MERSS, ˆλ SRS ) efft, ˆλ SRS ) effˆλ MERSS, ˆλ SRS ) Siilarly, the best unbiased estiator of λ 2 aong all unbiased linear cobinations of Y [k:k] is ρ + ρa k )Y [k:k] λ 2 ω [ ρ) 2 + 2ρ ρ)a k + ρ 2 b k ] =. 8) ρ + ρa k ) 2 λ 2 [ ρ) 2 + 2ρ ρ)a k + ρ 2 b k ] It follows that varω ) = λ 2 2 ρ + ρa k ) 2 ρ) 2 + 2ρ ρ)a k + ρ 2 b k ) 2. 9) ρ + ρa k ) 2 ρ) 2 + 2ρ ρ)a k + ρ 2 b k

11 A.A. Hanandeh, M.F. Al-Saleh 7 Hence, using 9) we get effω, ˆλ 2SRS ) = varˆλ 2SRS ) varω ) ρ + ρa k ) 2 ) 2 = ρ) 2 + 2ρ ρ)a k + ρ 2 b k ρ + ρa k ) 2 ρ) 2 + 2ρ ρ)a k + ρ 2 b k 20) Table 3 includes the efficiencies of ω copared to ˆλ 2SRS ˆλ 2MERSS copared to ˆλ 2SRS ). Based on Table 3 we conclude that effω, ˆλ 2SRS ) is always larger than. It is increasing in for fixed ρ and also increasing in ρ for fixed. effω, ˆλ 2SRS ) is larger than effˆλ 2MERSS, ˆλ 2SRS ) for any size and ρ, i.e., ω is ore efficient than ˆλ 2MERSS and hence than ˆλ 2RSS and ˆλ 2SRS. Table 3: effω, ˆλ 2SRS ) effˆλ 2MERSS, ˆλ 2SRS )) ρ ) ).03.02).06.05) ) ) ) ) ) ) ).9.6) ).5.3).60.58).99.97) ).74.73).24.24) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 6 Estiation of ρ with Known λ, λ 2 Now we consider the estiation of ρ when λ and λ 2 are known using MERSS and SRS. Diab 2006) see also Al-Saleh and Diab, 2009) derived the following forula for the variance of ˆρ based on SRS: varˆρ SRS ) = 3ρ2 + 4ρ + 3, where ˆρ SRS = λ λ 2 X i Y i. 2) i=

12 72 Austrian Journal of Statistics, Vol ), No. 3, 6 79 But, Thus, Now, suppose that λ and λ 2 are known, then EX k:k) Y [k:k] ) = E[EX k:k) Y [k:k] X k:k) )] = E[X k:k) EY [k:k] X k:k) )] = E [X k:k) ρ)λ 2 + ρ λ )] 2 X k:k) λ = ρ)λ 2 λ j= E ) Xk:k) 2 = varxk:k) ) + [ EX k:k) ) ] 2 = λ 2 EX k:k) Y [k:k] ) = ρ)λ 2 λ +ρ λ 2 λ j= k j + + ρλ 2 E ) Xk:k) 2. λ [ k j + ) + 2 λ2 j= j= λ 2 j= = λ λ 2 j= +ρ j= k j + k j + ) 2 + λ2 k j + k j + ) 2 Hence, ) E X k:k) Y [k:k] = ρλ λ 2 C S + Thus, Hence, ˆρ MERSS = ρ = λ λ 2 j= [ j= ] 2. k j + ] 2 k j + [ k j + + j= j= ) E X k:k) Y [k:k] λ λ 2 S λ λ 2 C S + X k:k) Y [k:k] S C S + j= k j + ] 2 [ ] 2 + λ λ 2 S. k j + [ ] 2 k j + j= [ ] 2 = k j + λ λ 2 X k:k) Y [k:k] S C S + a 2 k 22)

13 A.A. Hanandeh, M.F. Al-Saleh 73 is an unbiased estiator of ρ. It follows that varˆρ MERSS ) = varx k:k) Y [k:k] ) λ 2 λ 2 2 C S + a 2 k ) 2. 23) Thus, varˆρ MERSS )= Hence, we finally get C S + a 2 k ) 2 { 2 ρ) 2 b k +a 2 k) k ) ) k ) j +24 ρ)ρ k 24) j +j) 4 j=0 k ) ) k ) +ρ 24 2 j [ k ρ)ak +ρb j +j) 5 k +a 2 k) ] } 2. j=0 effˆρ MERSS, ˆρ SRS ) = 3ρ2 + 4ρ + 3)/. 25) varˆρ MERSS ) Table 4 contains the efficiencies of ˆρ MERSS w.r.t. ˆρ SRS. We conclude that the efficiency of ˆρMERSS w.r.t. ˆρ SRS is always larger than and it is increasing in for fixed ρ. Table 4: effˆρ MERSS, ˆρ SRS ) ρ

14 74 Austrian Journal of Statistics, Vol ), No. 3, Estiation of ρ with Unknown λ, λ 2 Suppose that λ and λ 2 are unknown, then we can replace the in ˆρ SRS by ˆλ SRS and ˆλ 2SRS to get ˆρ 2SRS = X i Y i. 26) ˆλ SRSˆλ2SRS Since 0 ρ <, we odify this estiator by 0 if ˆρ 2SRS < 0 ˆρ 2SRS = ˆρ 2SRS if 0 ˆρ 2SRS if ˆρ 2SRS >. Siilarly, where ˆλ 2MERSS = this estiator by ˆρ 2MERSS = i= X k:k) Y [k:k] S ˆλ MERSSˆλ 2MERSS, 27) C S + Y [k:k] since ˆλ 2MERSS depends on ρ. Since 0 ρ < we odify 0 if ˆρ 2MERSS < 0 ˆρ 2MERSS = ˆρ 2MERSS if 0 ˆρ 2MERSS if ˆρ 2MERSS >. a 2 k We copare the bias and the ean squared error MSE) of the two estiators ˆρ 2MERSS and ˆρ 2SRS by siulation using 0000 repetitions. To siulate a pair X, Y ) fro the DBEDλ, λ 2, ρ), Iliopoulos 2003) rewrote the pdf of the DBED ) as an infinite ixture of independent gaa distributions with geoetric ixing weights see also Diab, 2006; Al-Saleh and Diab, 2009) fx, y) = πk; ρ)γ k+) x; λ ρ))γ k+) y; λ 2 ρ)), k=0 where Γ α ; β) is the pdf of the Gaaα, β) distribution and πk; ρ) = ρ)ρ k, k = 0,,... is the geoetric probability function. Let K be a rando variable with this geoetric pdf. Then, X and Y are conditionally given K = k) independent gaa rando variables with α = k + and β = λ ρ), β 2 = λ 2 ρ), respectively. Also, the unconditional distribution of X, Y ) is a DBEDλ, λ 2, ρ). The algorith of siulating observations for the DBED consists of the following steps:. Given λ, λ 2 and ρ, siulate rando nubers K i fro the geoetric distribution with p = ρ).

15 A.A. Hanandeh, M.F. Al-Saleh Given K i = k i, siulate two independent rando variable X i and Y i, where X i is fro Gaak i + ; λ ρ)) and Y i is fro Gaak i + ; λ 2 ρ)). 3. Use the pairs X i, Y i ) just generated to copute the values of ˆρ SRS and ˆρ 2MERSS. 4. Repeat steps to 3 a 0000 ties to get 0000 values of ˆρ 2SRS and ˆρ 2MERSS. Bias and MSE of each of the estiators ˆρ 2SRS and ˆρ 2MERSS are given in Tables 5 and 6, respectively, and the efficiency of ˆρ 2MERSS w.r.t. ˆρ 2SRS is given in Table 7. Based on Tables 5, 6, and 7 we conclude that the efficiency of ˆρ 2MERSS w.r.t. ˆρ 2SRS is larger than for ρ 0.4. If ρ > 0.4 then ˆρ 2MERSS is not recoended. 8 Estiation of λ, λ 2 with Unknown ρ We noticed that the forula of ˆλ MERSS = S X k:k) does not depend on ρ, while ˆλ 2MERSS does. Suppose that ρ is unknown, then we can replace it by ˆρ 2MERSS in the forula of ˆλ 2MERSS to get ˆλ 2MERSS = ˆρ 2MERSS ) + ˆρ 2MERSS S Y [k:k]. 28) Bias and MSE of the estiators ˆλ 2SRS and ˆλ 2MERSS are given in Tables 8 and 9, respectively, and the efficiency values of ˆλ 2MERSS w.r.t. ˆλ 2SRS are contained in Table 0. Based on Tables 8, 9, and 0 we conclude that. The bias of ˆλ 2MERSS is larger than that of ˆλ 2SRS as ρ is getting large. 2. The MSE of ˆλ 2MERSS behaves siilarly; it is larger than that of ˆλ 2SRS for ost of the cases that we considered. 3. The Estiators based on SRS are ore efficient than those based on MERSS for ρ The suggested estiator is not recoended if soe prior inforation suggests that the unknown ρ is not very sall. The reason could be that the estiator of ρ is not a suitable one. 9 Conclusions Moving extree ranked set sapling is a useful variation of ranked set sapling and ore applicable since it allows for an increase of set size without introducing extra ranking errors. In this procedure, only the two extree values axiu or iniu) of sets of varied sizes were identified by judgent) for quantification. In this paper, the ain goal was to estiate the paraeters of Downton s bivariate exponential distribution using MERSS with concoitant variable. It was assued that X can be ranked visually while Y was highly correlated with X. It was shown that the use of MERSS with concoitant variable gives ore efficient estiators for the paraeters of Downton s bivariate exponential distribution than the corresponding ones using Siple Rando Saple.

16 76 Austrian Journal of Statistics, Vol ), No. 3, 6 79 Table 5: Bias of ˆρ 2SRS ˆρ 2MERSS ) ρ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Table 6: MSE of ˆρ 2SRS ˆρ 2MERSS ) ρ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Table 7: Efficiency of ˆρ 2SRS w.r.t. ˆρ 2MERSS ρ Acknowledgeents The authors would like to thank the editor and the referee for their useful coents. Their coents were very helpful in revising the previous drafts of the paper.

17 A.A. Hanandeh, M.F. Al-Saleh 77 Table 8: Bias of ˆλ 2SRS ˆλ 2MERSS ) ρ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Table 9: MSE of ˆλ 2SRS ˆλ 2MERSS ) ρ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Table 0: Efficiency of ˆλ 2MERSS w.r.t. ˆλ 2SRS ρ References Al-Odat, M. T., and Al-Saleh, M. F. 200). A variation of ranked set sapling. Journal of Applied Statistical Science, 0, Al-Saadi, S. D., and Young, D. H. 980). Estiation for the correlation coefficient in a

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19 A.A. Hanandeh, M.F. Al-Saleh 79 in Statistics Theory and Methods, 6, Takahasi, K., and Wakioto, K. 968). On unbiased estiates of the population ean based on the saple stratified by eans of ordering. Annals of the Institute of Statistical Matheatics, 20, -3. Yang, S. S. 977). General distribution theory of the concoitants of order statistics. The Annals of Statistics, 5, Authors addresses: Ahed Hanandeh Departent of Matheatical Sciences University of Cincinnati Cincinnati, OH USA E-ail: Mohaad Fraiwan Al-Saleh Departent of Statistics College of Sciences Yarouk University Irbid- 263 Jordan E-ail: