Estimation of parameters μ₂ and σ₂ of extended morgenstern type bivariate bilal distribution by concomitants of order statistics and ranked set sampling

Introduction

The concept of ranked set sampling was first introduced by McIntyre(1952) as a process of improving the precision of the sample mean as an estimator of the population mean. Ranked set sampling as described in McIntyre (1952) is applicable whenever ranking of a set of sampling units can be easily by judgment method, for details, see, Chen et al., (2004) [4] . The process of ranked set sampling involves selecting n sets of units, each of size n, and ordering the units of each of the set by judgment method or by using values obtained on the units by a relatively inexpensive method, without making actual measurement on the units. Then the unit ranked as one from the first set is actually measured, the unit ranked as two from the second set is measured. The process continues in this way until the unit ranked as n from the set is measured. Then the observations resulting out of actual measurements made on the units chosen as described in the above Process is known as ranked set sample. Ranking by a judgment method is not recommendable in case there is ambiguity in discriminating the rank of one unit with another (for example, ranking based on intensity of yellowish color of newborn babies, as mentioned in Chen et al., (2004, p.7)) [4] , become difficult, as we known that probability of ties in this situation is not zero as the characteristic under consideration is not continuous).In certain situation, one may prefer exact measurement of some easily measurable variable associated with the study variable rather than ranking the units by a crude judgment method. Suppose the variable of interest say, Y, is difficult or much expensive to measure, but an auxiliary variable X correlated with Y is readily measurable and can be ordered exactly . In this case as an alternative to McIntyre's (1952) method of ranked set sampling, Stokes (1977) [17] used an auxiliary variable for the ranking of the sampling units. Stokes (1977) [17] using auxiliary variate is as follows: Choose independent units, arrange them randomly into n sets each with n units and observe the value of the auxiliary variable X on each of these units. In the first set, that units for which the measurement on the auxiliary variable is the smallest is chosen. In the second set, that units for which the measurement on the auxiliary variable is the second smallest is chosen. The procedure is repeated until in the last set, that unit for which the measurement on the auxiliary variable is the largest is chosen. The measurements on the Y variate of the resulting new set of n units chosen by the above method gives a ranked set sample as suggested by Stokes(1977) [17] . If r r X ) ( is the observation measured on auxiliary variable X from the unit chosen from the th r set then we write  r r Y to denote the corresponding measurement made on the study variable Y on this unit so that  r r Y , r = 1,2,…,n form the ranked set sampling. Clearly  r r Y is the concomitants of th r order statistic arising from the th r sample. By using the expression given by Yang(1977) for the mean and variance of concomitants of order statistics, we have the following . For

Stokes (1977) [17] has suggested the ranked set sample mean as an estimator for the mean of the study variable Y, when an auxiliary variable X is used for ranking the sample units, under the assumption that (X,Y) follows a bivariate normal distribution. Barnett and Moore (1997) [2] have improved the estimator of Stokes (1977) [17] by deriving the Best linear unbiased estimator (BLUE) of the mean of the study variate Y, based on ranked set sample obtained on the study variate Y.

Morgenstern familly of distributions.

The most important aspect to the considered in a modeling problem is regarding the flexibility of the chosen family, in the sense that it should contain a wide variety of models capable of representing any data situation.Another consideration is the sort of prior information available in the choice of the model. In problems involving several random variables, the analyst may make reasonable assumption about the marginal distributions. Then the question is to construct a joint distribution function with a set of given marginal.The Morgenstern family of distributions assumes importance in such contexts as a highly flexible system.Let X and Y be two random variables with cdf's given by

and jointly distributed with cdf F(x,y) given by (see, Kotz et al.,(2000)

Then the family of distributions having the above form of cdf is called the Morgenstern family of distributions (MFD) . The system provides a very general expression of a bivariate distribution from which member can derived by substituting expression of any desired set of marginal distributions. The joint pdf corresponding to the cdf defined in (1.4) is given by,

The parameter  is considered as the association parameter, the two random variables are independent when the value of  is zero and the maximum correlation attained between X and Y is 3

, the MFD is primarily useful as an alternative to the standard bivariate normal distribution (See, Conway (1983) [5] ,p.28-31).The distribution theory of concomitance of order statistics from MFD are discussed in Scaria and Nair(1999) [16] . Chako and Thomas (2008) derived the BLUE of a parameter of a three parameter Morgenstern type bivariate exponential distribution using concomitance of order statistics. Sajeevkumar and Sumi(2023) [15] derived the BLUE's of the parameters 2  and 2  of a five parameter Morgenstern type bivariate exponential distribution (MTBED) using concomitants of ranked set sampling. Irshad et al. (2019) [7] estimate a parameter of a three parameter Morgenstern type bivariate Lindly distribution (MTBLD) using ranked set sampling. 1 and Priya(2026a) [11] derived the BLUE's of the parameters 2  and 2  of a five parameter Morgenstern type bivariate Lindly distribution (MTBLD) using ranked set sampling. Arun et.al (2023) [1] derived the BLUE of a parameter of a three parameter Morgenstern type bivariate moment exponential distribution(MTBMED) using concomitance of order statistics. Sajeevkumar and Priya (2026b) [11,13,13,14] estimate the BLUE's of the parameters 2  and 2  of a five parameter Morgenstern type bivariate moment exponential distribution (MTBMED) using Ranked set sampling . Irshad et.al.(2021) [6] estimated a parameter of a four parameter extended morgenstern type bivariate Lindley distribution(EMTBLD) using concomitants of order statistics .Sajeevkumar and Priya (2026c) [11,12,13,14] [11,12,13,14] [8] estimated a parameter of a four parameter extended Morgenstern type bilal distribution by concomitants of order statistics and ranked set sampling.In this work our aim is to estimate the parameters 2

 and 2  involved in a six parameter extended Morgenstern type bilal distribution by concomitants of order statistics and ranked set sampling.This work is an extention of the work done by Irshad at.al(2022) [8] .

Estimation of the parameters 2

 and 2  involved in emtbbd using RSS.

An important member of Morgenstern family of distribution is (MFD) is Extended Morgenstern type bivariate Bilal distribution (EMTBBD) .The bivariate random variable (X,Y) is said to have six parameter EMTBBD, if its pdf is given by

The pdf defined in (2.1) is an extension of the four parameter EMTBLD defined in Irshad et.al (2021) [6] .The joint pdf of standard EMTBLD is obtained by making the substitution

Suppose X and Y are the two characteristics of interest on the units of a population and (X,Y) has a distribution with pdf given in (2.1) . Suppose there are n independent random samples each of size n drawn from (2.1) and in each sample, units are ranked according to the values of the observations measured on the easily measurable variable X on the units. Though the variable Y is of primary interest, we assume that it is difficult to measure . Then to deal with the inference problem on the distribution of the variable Y we obtain a ranked set sample by measuring the value of Y on the th r ordered unit of the th r sample, r=1,2,…,n.

1) .Clearly we have

, for I=1,2,…,n. Then using (2.4) to (2.6), we have for

Where r  , r r e , and s r e , are defined by (2.4), (2.5) and (2.6) respectively. Clearly from (2.4) to (2.6) it follows that r  , r r e , and s r e , are known constants provided  is known.

And corresponding variances are given by

 ) (

Using (3.1) and (3.2), we have , (3.4) Also for

sample of size n from a population with pdf defined in (2.1). Clearly

, where 1 is a column vector of n ones and

and corresponding variances are given by

Computation section

We have evaluated the coefficients of the BLUE * 2  of 2  given in (2.12) and evaluated the coefficients of the BLUE * 2  of 2  given in (2.13) using ranked set sampling, for different values of n,  and 50 . 0   ,are given in Table 1. Also evaluated the coefficient of the BLUE 2  of 2  given in (3.9) and evaluated the coefficient of the BLUE 2  given in (3.10) using concomitants of order statistics for different values of n,  and 50 . 0   ,are given in Table 2 . Also we have evaluated 3.

. Conclusion

From Table 3 given in section 4, it may be noted that for sample sizes n=2  calculated using concomitants of order statistics is much better than the estimators calculated using ranked set sampling.

References

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