On Concomitants of Ordered Ranked Set Sampling from Morgenstern Family

Email: msakotb1712@yahoo.com Abstract: This paper develops the distribution theory of concomitants in the context of ordered ranked set sample from the Farlie-GumbelMorgenstern bivariate family. Here, we derive the joint distribution of concomitants of two ordered ranked set sample and obtain their single and product moments alongside the correlation coefficient. Finally, an application of the results in establishing the moments and the variancecovariance for the Gumbel’s bivariate exponential distribution.


Introduction
The concept of RSS, was first suggested by McIntyre (1952), has been introduced as a method to increase the precision of estimated yield without the bias of researcherchosen samples and reduce cost by using simple judgment or qualitative information. RSS is a cost-efficient alternative to SRS, particularly in situations where the measurements on the selected subject are expensive or difficult to obtain, see Yu (2001). Many authors have tried to evaluate the performance of RSS, see for example Halls and Dell (1966), Martin et al. (1980) and Alodat and Al-Sagheer (2007). Takahasi and Wakimoto (1968) have presented a more detailed mathematical development for the RSS. During the last three decades, there has been much work on RSS to improve and study the estimation of population parameters, see for example Sinha et al. (1996), Samawi et al. (1996), Adatia (2000), Al-Saleh and Al-Omari (2002), Shaibu and Muttlak (2004), Balakrishnan and Li (2008), Kadilar et al. (2009), Al-Saleh and Samawi (2010), Ozturk (2011), Mohie El-Din et al. (2015) and Sadek et al. (2015). The prediction problems based on RSS are studied by several authors, for instance Salehi et al. (2015), Kotb (2016), Mohie El-Din et al. (2017) and Kotb and Raqab (2017).
Let X 1 , X 2 ,···, X n be a sequence of IID r.v.'s of size n with a continuous distribution F. McIntyre's concept depends on measuring, the smallest observation from the first sample (denoted as X 11 ), the second smallest observation X 22 from the second sample and so on until the maximum observation X nn from the last sample. We call this process as a one-cycle RSS of size n and the measured RSS units are denoted by X 11 , X 22 ,···, X nn . By repeating this process independently m times, we obtain secondary sample of size nm form RSS.
The RSS estimator RSS µ % and variance of the population mean are given by: In addition, Stokes (1980)

RSS
σ% are usual unbiased estimators of σ 2 from the SRS and the RSS schemes of size n, respectively.
The idea of RSS based on order statistics from INID r.v.'s has been introduced by Balakrishnan and Li (2008). They showed that the BLUE based on ORSS are more efficient than the BLUE based on RSS for the normal, logistic and two-parameter exponential distributions. Now, let the n pairs (X i , Y i ) be IID r.v.'s drawn from a bivariate distribution with df F(x, y). If the (X i , Y i ) are ordered by their X-variates (in increasing order of magnitude), then the Y values associated with the X r:n , 1≤r ≤ n of X (denoted as Y [r:n] ) is called the concomitant of the r-th order statistics.
The paired (X, Y) is said to be a bivariate FGM distributed if the bivariate df F(x, y) is an absolutely continuous such that: where, −1 ≤ α ≤ 1. The corresponding pdf is given by: Therefore, the conditional df and pdf of Y given X = x are given by: and: respectively. This family, was first suggested by Morgentern (1956), is characterized by the specified marginal dfs F X (x) and F Y (y) of r.v.'s X and Y, respectively. Gumbel (1960) investigated and studied this model for exponential marginals. The FGM is a flexible family useful in applications provided the correlation between the r.v.'s is not too large. References on this family may be found, among many others in Farlie (1960), Johnson and Kott (1975;Johnson and Kotz, 1977), Bairamov and Bekci (1999) and Beg and Ahsanullah (2008). A wide variety of applications for concomitants been used in many fields such as inference problems, ocean engineering, selection procedure, double sampling plans and prediction analysis. These applications for concomitants were introduced by Gross (1973), O'Connel and David (1976), Yang (1977), Yeo and David (1984), Castillo (1988), Do and Hall (1991), Balasubramanian and Beg (1997) and Domma and Giordano (2015). For comprehensive and superb review of work on concomitants of order statistics, we refer the reader to the Handbook of Statistics by David and Nagaraja (1998). The rest of the present paper is organized as follows: The next Section contains some preliminaries. In section 3, we use the idea of order statistics from INID r.v.'s to derive the pdf, joint pdf and distribution of the concomitants from the FGM bivariate family (1) based on ORSS. In section 4, we obtain the single moments, product moments and the variance-covariances of the concomitants of ORSS. In section 5, the Gumbel's (1960) bivariate exponential distribution are used as special case of our results. We end the paper with a short conclusion in section 6.

Preliminaries
Let the pairs (X i , Y i ), i = 1, 2,···, n be a sequence of IID random variables with df F(x, y). This sequence will be referred to here as a SRS. Let X j1 , X j2 ,···, X jn , j = 1, 2,···, n be the visual (judgment)-ordered observations of n sets within a one-cycle. The procedure of a one-cycle RSS can be described in the following Scheme 1: Scheme 1: A ranked set sample design with sample size n Also, we will need the following definition in the next section to obtain the joint pdf of the concomitants of ORSS.

Definition 1
Let B = (b i,j ) be a real square matrix of size n×n. The permanent of B is defined to be the scalar: where, Σ P denotes the summation over all n! permutations (i 1 , i 2 ,···, i n ) of (1, 2,···, n). If we arrange the X's in ascending X ORSS = 1: and are called concomitants of ORSS. By using the results for order statistics from INID r.v.'s (see Balakrishnan (2008) and Balakrishnan and Li (2008)), the one dimensional marginal density function of ORSS is given by: where, Σ P was defined in Definition 1, Arnold et al. (1992) and David and Nagaraja (2003).
The following function is the two dimensional marginal density function of ORSS: see Balakrishnan and Li (2008), where:

Distribution of Concomitants of ORSS
For the FGM bivariate distributions with df and pdf given by (1) and (2) where,

Moments of Concomitants Based on ORSS
Using the results of the previous section, we obtain the single and product moments as well as the variance-covariances of the concomitants of ORSS as follows.

Conclusion
In this study we investigated the distribution theory of concomitants of ORSS from the FGM bivariate distribution family. Also, we derived an analytical expression of the joint distribution of concomitants of two ORSS and find their single and product moments alongside the correlation coefficient. The family considered in this study is widely used in practice and gives more flexibility to model various types of data as well as it includes several important bivariate distributions. The Gumbel's bivariate exponential distribution is used as an example of the FGM bivariate family. Finally, we simulated the values of the means, variances and covariances of the concomitants of ORSS based on Monte Carlo simulations.