Induced Ranked Set Sampling when Units are Inducted from Several Populations

STATISTICA, anno LXXVIII, n. 1, 2018 INDUCED RANKED SET SAMPLING WHEN UNITS ARE INDUCTED FROM SEVERAL POPULATIONS P. Yageen Thomas Department of Statistics, University of Kerala, Trivandrum-695 581, India Anne Philip 1 Department of Statistics, University of Kerala, Trivandrum-695 581, India 1. INTRODUCTION Random sampling is well known as an unrestricted method of collecting the units from a population. However, if collecting the units and measuring the characteristic of interest on them is expensive (or risky or painful), then one is compelled to look for alternative sampling methods which are capable of accommodating observational economy considerations. McIntyre (1952) first used a judgement method to rank the randomly chosen units within each of independent groups of units and devised a method of final selection of units from the groups of units based on their ranks and termed this method as ranked set sampling (RSS). Chen et al. (2004) have discussed extensively about the observational economy considerations accommodated in RSS as defined by McIntyre (1952). For more details, see Shaibu and Muttlak (2004), Al-Rawwash et al. (2010) and Samuh (2017). Ranking by judgement method is not suitable if there is a fear that ranking error which is otherwise known as imperfect ranking creeps in the ranking process of the units. In such situations Stokes (1977) introduced a scheme of sampling the units based on the measurements made on an easily measurable auxiliary variable X which is jointly distributed with the variable Y of primary interest, whose measurement is expensive and thereby defined the RSS in the following manner. Choose n 2 units randomly from the population and arrange them in the random order in n sets of n units each. Rank units within each set based on the measurement made on the auxiliary variable X of the units. Then from the i th set choose the unit ranked i and measure the variable Y of primary interest on this unit for i = 1, 2, . . . , n. Clearly, the observation obtained from the ith set is the concomitant of ith order statistic of X -observations in that set and we write it as Y[i :n]i for i = 1, 2, . . . , n. Then the observations Y[1:n]1 , Y[2:n]2 , . . . , Y[n:n]n are said to 1 Corresponding Author. E-mail: 58 P. Y. Thomas and A. Philip constitute the ranked set sample (r s s) as proposed by Stokes. The sampling procedure as described above to obtain the r s s is known as the Stokes method of RSS. Since there is potential difference in the approaches and mathematical preciseness in the RSS methods proposed by McIntyre (1952) and Stokes (1977), rather than calling both procedures as RSS, we prefer to call the RSS method proposed by Stokes as induced ranked set sampling (IRSS) all through this paper. We may further call the sample arising due to IRSS as induced ranked set sample (i r s s). It is to be noted that contrary to general perception, concomitants of order statistics generated from IRSS are not causing any disadvantage but create advantages by the mathematical rigour we observe on their distributions for devising inference procedures on the parameters involved in the distribution of Y . Distribution theory of concomitants of order statistics in applying IRSS method in inference problems is available in David (1973), Bhattacharya (1984) and David and Nagaraja (1998, 2003). For some recent survey on IRSS, one may refer to Chen et al. (2004), Chacko and Thomas (2007, 2008, 2009), Ahmad et al. (2010), Lesitha et al. (2010), Lesitha and Thomas (2013), Singh and Mehta (2013) and Philip and Thomas (2015). Taking the advantage of recent developments on the theory of order statistics of independent non-identically distributed (INID) random variables as portrayed in Vaughan and Venables (1972), Balakrishnan (1988), Balakrishnan et al. (1992), Bapat and Beg (1989a, 1989b), Beg (1991) and Samuel and Thomas (1998), some applications of these theories in parameter estimation have been illustrated in Sajeevkumar and Thomas (2005) and Thomas and Sajeevkumar (2005). One difficulty experienced in the large scale applications of the results of Thomas and Sajeevkumar (2005) and Sajeevkumar and Thomas (2005) in further inference problems is about the tediousness involved in the evaluation of the covariance between different pairs of order statistics using the permanent expression (for details, see Vaughan and Venables, 1972) for the joint probability density function (pdf) of two order statistics of INID random variables. However, if we make use of McIntyre (1952) method of RSS involving selection of units belonging to different populations for inference problems, then we are redeemed from the burden of obtaining the values of the covariances as those covariances are zero in a r s s due to the reason that the observations in the r s s occur from independent samples. With this theoretical background Priya and Thomas (2013, 2016) have defined RSS when units from different univariate populations are to be considered and used the resulting r s s observations to estimate the common parameters of several univariate distributions. Eryilmaz (2005) first derived the expression for the cumulative distribution function (cdf) of concomitants of order statistics of INID random variables. It may be noted that for using concomitants of order statistics of INID random variables, we require the pdf of those concomitants. Thus Veena and Thomas (2015) have derived the marginal pdfs and joint pdfs of concomitants of order statistics of INID random variables using permanents. For some of the applications of concomitants of order statistics of INID random variables in estimation, see Veena and Thomas (2015). But there is a limitation in large scale applications of the results of Veena and Thomas (2015) as the expansion of the permanent expression for the joint pdf of different pairs of concomitants of order Induced Ranked Set Sampling from Several Populations 59 statistics and computation of those covariances turns out to cause unbearable burden. However, like the extension made by Priya and Thomas (2013, 2016) to McIntyre’s RSS to the case when units from different univariate populations are inducted in the sampling scheme, if one extend IRSS scheme and the pdf representation of concomitants of order statistics of INID random variables as given in Veena and Thomas (2015), then the resulting i r s s relieves the user from the burden of computation of covariances of different pairs of observations while applying IRSS to inference problems. This has motivated the authors to define IRSS to the case when units from several populations are to be inducted in the sample. It is to be noted that unlike the possibility of occurrence of imperfect ranking in the extended method of RSS proposed by Priya and Thomas (2013, 2016), such imperfect ranking never creeps in the proposed extended method of IRSS in this paper. In Section 2, the procedure for the induced ranked set sampli (...truncated)