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

Abstract: The purpose of this paper is to estimate the parameters of Downton’s bivariate exponential distribution using moving extreme ranked set sampling (MERSS). The estimators obtained are compared via their biases and mean square errors to their counterparts using simple random sampling (SRS). Monte Carlo simulations are used whenever analytical comparisons are difficult. It is shown that these estimators based on MERSS with a concomitant variable are more efficient than the corresponding ones using SRS. Also, MERSS with a concomitant variable is easier to use in practice than RSS with a concomitant variable. Furthermore, the best unbiased estimators among all unbiased linear combinations of the MERSS elements are derived for some parameters.


Introduction
The most popular sampling method that is usually used in statistical studies is Simple Random Sampling (SRS).A SRS of size n from 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 sample is selected independently and comes from the same population (i.i.d.).Simple random sampling is the basic building block and point of reference for all other sampling methods.
McIntyre (1952) suggested a sampling technique which was later called Ranked Set Sampling (RSS).This technique of data collection was introduced for situations where taking the actual measurements on sample observations is difficult (i.e., costly, timeconsuming) as compared to the judgment ranking of them.The ranked set sampling technique can be executed as follows: • Step 1: Randomly draw m simple random samples each of size m from the population of interest.
• Step 2: Within each of the m sets, the sampled items are ranked from lowest to largest according to the variable of interest based on the researcher's judgment or by any negligible cost method that does not require actual quantifications.
• Step 3: From the first set of m units, the unit ranked lowest is measured.From the second set of m units, the unit ranked second lowest is measured.The process is continued until the mth ranked unit is measured from the mth set.Note that m2 units are sampled but only m of them are measured with respect to the variable of interest.
The above 3 steps describe one cycle of the RSS technique.• Step 4: Repeat steps 1 to 3, if necessary, k independent times (cycles) to obtain a total sample of size n = mk units.
In McIntyre's RSS procedure, it is assumed that the researcher could order a set of size m with respect to the characteristic of interest.Many authors recommended that m should be 2, 3, or 4 to minimize the ranking error (see Takahasi and Wakimoto, 1968).
There are several variations of RSS.One of them is Moving Extreme Ranked Set Sampling (MERSS).In this procedure only the maximum (or minimum) of sets of varied size is identified (by judgment) for quantification.The MERSS, as described by Al-Odat and Al-Saleh (2001) and Al-Saleh andAl-Hadrami (2003a, 2003b) can be executed as follows: • Step 1: Select m simple random samples of sizes 1, . . ., m, respectively.• Step 2: For each of these samples, measure accurately the maximum ordered observation from the first set identified by judgment, the maximum ordered observation from the second set, etc..The process continues in this way until the maximum ordered observation from the last mth sample is measured.
• Step 3: Repeated steps 1 and 2, if necessary, r times to obtain a sample of size n = rm.

Main Results about RSS and its Variations
RSS was introduced by McIntyre (1952) in the context of estimating pasture yields.He claimed, without providing a mathematical proof, that r which is the mean of a sample of size mk obtained using the RSS technique, is an unbiased estimator of the population mean µ.X [i]r is the ith judgment order statistic in the rth cycle.
3. The efficiency of the estimators of higher population moments based on RSS are only slightly better than those based on SRS.Takahasi and Wakimoto (1968) obtained the following main theoretical results.Under perfect ranking, the mean of a RSS is an unbiased estimator of the population mean and its variance is always smaller than the variance of the mean of a SRS of equal size.Also, they showed that where f (x) denotes the probability density function (pdf) of a random variable X with E(X) = µ and var(X) = σ 2 .Furthermore, f i (x) is the pdf of the ith order statistic X (i) with E(X (i) ) = µ i and var(X (i) ) = σ 2 i .The performance of the estimators is assessed using either the relative precision (RP) or the relative saving (RS), which are defined as They also showed that and thus is the mean of a sample of size m obtained using the SRS technique, and is the mean of a sample of size mk obtained using the RSS technique.Stokes (1977) studied RSS with concomitant variables.In this case, the variable of interest is difficult to rank but it is related to some other variable that is easy to rank (concomitant variable).We use (X, Y ) in the sequel, where X is the variable of interest and Austrian Journal of Statistics, Vol. 42 (2013), No. 3, 161-179 Y is the concomitant one.Samawi, Ahmed, and Abu-Dayyeh (1996) introduced the Extreme Ranked Set Sampling (ERSS) procedure.It was shown that the ERSS estimator of the mean is more efficient than the usual SRS mean and unbiased if the underlying distribution is symmetric.In this ERSS, only the two extremes (minimum, maximum) are identified by judgment for different sets.Al-Saleh and Al-Kadri (2000) considered double RSS (DRSS) as a procedure that increases the efficiency of RSS estimator without increasing the set size m.Al-Saleh and Al-Omari (2002) generalized DRSS to multistage ranked set sampling.They showed that the efficiency is always between 1 and m 2 for all distributions and equals m 2 for the uniform distribution when the number of stages goes to infinity.Al-Saleh and Zheng (2002) proposed a new RSS for two characteristics and called it a bivariate ranked set sampling.
MERSS is a useful modification of RSS.Unlike RSS, MERSS allows for an increase of the set size without introducing too much ranking error.MERSS was introduced by Al-Odat and Al-Saleh (2001) who also introduced the concept of varied set size RSS.Al-Saleh andAl-Hadrami (2003a, 2003b) used varied set size of RSS (coined by them MERSS) for estimating the mean of the normal and exponential distribution, and they showed that this procedure could be more useful than SRS for estimating the mean of any symmetric distribution.Al-Saleh and Samawi (2010) estimated the odds based on MERSS.The suggested estimator based on MERSS is motivated by some of the theoretical properties of the sum of geometric series.
RSS and some of its variation were used by many authors in parametric estimation like bivariate normal, exponential, Downton's bivariate exponential (see Downton, 1970), etc.. Moran (1967) introduced a bivariate exponential distribution, which is one of the most important bivariate distributions in reliability.There are various bivariate exponential distributions.In this research, we are interested in Downton's bivariate exponential distribution with pdf where x, y, λ 1 , λ 2 > 0, 0 ≤ ρ < 1 and is the modified Bessel function of the first kind of order zero.If (X, Y ) is a bivariate random variable that has Downton's bivariate exponential distribution, then from (1) the marginal distributions of X and Y are exponential with parameters λ 1 and λ 2 , respectively.Thus, in particular, E(X) = λ 1 and E(Y ) = λ 2 .Downton (1970) further showed that The parameter ρ is the correlation coefficient between X and Y with independence corresponding to ρ = 0 since I 0 (0) = 1.This distribution is a candidate distribution for positively correlated bivariate exponential data, in which the conditional mean 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 (1978).In this application, the oil pollution is hard and expensive to measure while the tar deposit can be ranked visually.For more applications, see also Balakrishna and Lai (2009, pp. 401-466).Iliopoulos (2003) showed that the joint sufficient statistic for the three parameters based on the SRS (X 1 , Y 1 ), . . ., (X m , Y m ) from this distribution is ( Nagao and Kadoya (1971) showed that the maximum likelihood estimator of λ 1 and λ 2 is X and Ȳ , respectively.Two classes of estimators of the parameter ρ based on the complete bivariate sample were derived in Al-Saadi and Young (1980).
The method of moments estimator is based on the statistic Another estimator is based on the sample correlation coefficient Note that the first estimator is a function of the sufficient statistic, whereas the second estimator is not, so we will use the first estimator in what follows.
Al-Saleh and Diab (2009) estimated the parameters of Downton's bivariate exponential distribution based on a RSS.He and Nagaraja (2011) estimated the correlation coefficient of Downton's bivariate exponential distribution when all other parameters are unknown using incomplete samples.For more details about RSS and its variations see Chen, Bai, and Sinha (2004).For more details about this work see Hanadeh (2011).
In this paper, we consider the estimation of the parameters of a DBED(λ 1 , λ 2 , ρ) distribution using MERSS.The suggested estimators are compared with the corresponding ones using SRS.Estimation of λ 1 and λ 2 for the two cases of known and unknown correlation coefficient ρ is considered.Furthermore, estimation of the correlation coefficient ρ for the two cases of known and unknown λ 1 and λ 2 is also discussed.

Mean and Variance of X (k:k) and Y [k:k]
Let (X 1 , Y 1 ), . . ., (X m , Y m ) be a random sample from Downton's bivariate exponential distribution, DBED(λ 1 , λ 2 , ρ), with common density given in (1).Let f X (x) and f Y (y) be the marginal densities of X and Y , respectively, and let F X (x) and F Y (y) be the corresponding cumulative distribution function (cdf) of X and Y , respectively.Now, suppose that {(X Yang, 1977).But, It can be verified that if Z 1 , . . ., Z k are i.i.d.from an Exponential(λ 1 ) distribution then Yang, 1977).Thus, it follows that (6) Moreover, we have Using (5) we get Hence, 4 Unbiased Estimation of λ 1 , λ 2 with Known ρ We now consider the estimation of λ 1 and λ 2 when ρ is known using both MERSS and SRS.Then the performance of the suggested estimators is investigated.
Denote the unbiased estimator of λ 1 and λ 2 based on SRS by λ1SRS and λ2SRS , respectively, and those based on MERSS by λ1MERSS and λ2MERSS , respectively, with where where Hence, eff( λ1MERSS , λ1SRS ) = var( λ1SRS ) which is clearly larger than 1 and increasing in m.Also, It follows that Hence, using ( 12) and ( 13) we get eff( λ2MERSS , λ2SRS ) = var( λ2SRS ) which is also clearly larger than 1, since Table 1 contains efficiencies of λ1MERSS and λ2MERSS with respect to the respective estimators using SRS.Based on Table 1 we conclude that • eff( λ1MERSS , λ1SRS ) and eff( λ2MERSS , λ2SRS ) are always greater than 1.
• eff( λ2MERSS , λ2SRS ) increases in m for fixed ρ and increases in ρ for fixed m.
Based on Table 3 we conclude that eff(ω * , λ2SRS ) is always larger than 1.It is increasing in m for fixed ρ and also increasing in ρ for fixed m. eff(ω * , λ2SRS ) is larger than eff( λ2MERSS , λ2SRS ) for any size m and ρ, ω * is more efficient than λ2MERSS and hence than λ2RSS and λ2SRS .6 Estimation of ρ with Known λ 1 , λ 2 Now we consider the estimation of ρ when λ 1 and λ 2 are known using MERSS and SRS.Diab (2006) (see also Al-Saleh and Diab, 2009) derived the following formula for the variance of ρ based on SRS: Now, suppose that λ 1 and λ 2 are known, then ) . But, Thus, .
Hence, we finally get Table 4 contains the efficiencies of ρ1MERSS w.r.t.ρ1SRS .We conclude that the efficiency of ρ1M ERSS w.r.t.ρ1SRS is always larger than 1 and it is increasing in m for fixed ρ.
8 Estimation of λ 1 , λ 2 with Unknown ρ We noticed that the formula of λ1MERSS = Bias and MSE of the estimators λ2SRS and λ * 2M ERSS are given in Tables 8 and 9, respectively, and the efficiency values of λ * 2M ERSS w.r.t.λ2SRS are contained in Table 10.Based on Tables 8, 9, and 10 we conclude that 1.The bias of λ * 2M ERSS is larger than that of λ2SRS as ρ is getting large.
2. The MSE of λ * 2M ERSS behaves similarly; it is larger than that of λ2SRS for most of the cases that we considered.
3. The Estimators based on SRS are more efficient than those based on MERSS for ρ ≥ 0.20.The suggested estimator is not recommended if some prior information suggests that the unknown ρ is not very small.The reason could be that the estimator of ρ is not a suitable one.