Estimation of parameters μ2 and σ2 of Morgenstern type bivariate Lindley distribution by ranked set sampling

Introduction

The concept of ranked set sampling was first introduced by McIntyre (1952) [7] 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) [7] is applicable whenever ranking of a set of sampling units can be easily by judgment method, for details, see, Chen et al., (2004) [3] . 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 𝑛 𝑡ℎ 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) [3] , 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) [7] method of ranked set sampling, Stokes (1977) [10] used an auxiliary variable for the ranking of the sampling units. Stokes (1977) [10] using auxiliary variate is as follows: Choose 𝑛 2 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) [10] .

If 𝑋 (𝑟)𝑟 is the observation measured on auxiliary variable X from the unit chosen from the 𝑟 𝑡ℎ set then we write 𝑌 [𝑟]𝑟 to denote the corresponding measurement made on the study variable Y on this unit so that 𝑌 [𝑟]𝑟 , r = 1,2,…,n form the ranked set sampling.

Clearly 𝑌 [𝑟]𝑟 is the concomitants of 𝑟 𝑡ℎ order statistic arising from the 𝑟 𝑡ℎ 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) [10] 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) [1] have improved the estimator of Stokes (1977) [10] 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 𝐹 𝑋 (𝑥) and 𝐹 𝑌 (𝑦) respectively with pdf's 𝑓 𝑋 (𝑥) and 𝑓 𝑌 (𝑦) and jointly distributed with cdf F(x, y) given by (see, Kotz et al., (2000), p.52) [6] .

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

. For suitable choice of 𝐹 𝑋 (𝑥) and 𝐹 𝑌 (𝑦), the MFD is primarily useful as an alternative to the standard bivariate normal distribution (See, Conway (1983), p.28-31) [4] . The distribution theory of concomitance of order statistics from MFD are discussed in Scaria and Nair (1999) [9] . Chako and Thomas (2008) [2] derived the BLUE of a three parameter Morgenstern type bivariate exponential distribution using concomitance of order statistics. Sajeevkumar and Sumi (2022) derived the BLUE of the parameters 𝜇 2 and 𝜎 2 of a five parameter Morgenstern type bivariate exponential distribution (MTBED) using concomitants of ranked set sampling. Irshad et al. (2019) [5] estimate a parameter of Morgenstern type bivariate Lindly distribution (MTBLD) using ranked set sampling. In this work we derived the BLUE of the parameters 𝜇 2 and 𝜎 2 of Morgenstern type bivariate Lindly distribution (MTBLD) using ranked set sampling. This work is an extension of the work done by Irshad et al. (2019) [5] .

Estimation of the parameters 𝝁 𝟐 and 𝝈 𝟐 involved in MTBLD using RSS An important member of Morgenstern family of distribution is (MFD) is Morgenstern type bivariate Lindley distribution (MTBLD).The bivariate random variable (X, Y) is said to have five parameter MTBLD, if its pdf is given by

The pdf defined in (2.1) is an extension of the three parameter MTBLD defined in Irshad et.al (2019) [5] . The joint pdf of standard MTBLD is obtained by making the substitution 𝑈 = 𝑋-𝜇 1 𝜎 1

and 𝑉 = 𝑌-𝜇 2 𝜎 2 in (2. 1) and it is given by, g(u,v)=

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.

And corresponding variances are given by

Where 𝜉 𝑟 , 𝑏 𝑟,𝑟 and 𝑏 𝑟,𝑠 are respectively defined in (3.3), (3.4) and (3.5).

Computation Section

We have evaluated the coefficients of the BLUE 𝜇 2 * of 𝜇 2 given in (2.12) and evaluated the coefficients of the BLUE 𝜎 2 * of 𝜎 2 given in (2.13) for different values of n and 𝛼and are given in Table 1. Also we have evaluated 𝑉𝑎𝑟(𝜇 2 * ) given in (2.14), 𝑉𝑎𝑟(𝜎 2 * ) given in (2.15), 𝑉𝑎𝑟(𝜇̂2) given in (3.11), 𝑉𝑎𝑟(𝜎 ̂2) given in (3.12) and evaluated the relative efficiency 𝑅𝐸 1 of 𝜇 2 * relative to 𝜇̂2, the relative efficiency 𝑅𝐸 2 of 𝜎 2 * relative to 𝜎 ̂2for different values of n and 𝛼 and are given Table 2.

Conclusion

From Table 2 given in section 4, it may be noted that for sample sizes (n) greater than five, all the cases calculated the performance of the estimators 𝜇 2 * and 𝜎 2 * calculated using Ranked set sampling is much better than the estimators calculated using concomitants of order statistics.

References

1. Barnett V, Moore K. Best linear unbiased estimates in ranked set sampling with particular reference to imperfect ordering. Journal of Applied Statistics. 1997;24(6):697-710.
2. Chacko M, Thomas PY. Estimation of parameters of a Morgenstern type bivariate exponential distribution by ranked set sampling. Annals of the Institute of Statistical Mathematics. 2008;60(2):301-318.
3. Chen Z, Bai Z, Sinha BK. Ranked set sampling: Theory and Applications. New York: Springer-Verlag; 2004. Lecture notes in Statistics (vol. 176).
4. Conway DA. Farlie-Gumbel-Morgenstern distributions. In: Kotz S, Johnson NL, editors. Encyclopedia of statistical sciences. Vol. 3. New York: Wiley; 1983.
5. Irshad MR, Maya R, Arun SP. Estimation of a parameter of morgenstern type bivariate Lindley distribution by Ranked set sampling. ISTATISTIK: Journal of the Turkish statistical Association. 2019;12:25-34.
6. Kotz S, Balakrishnan N, Johnson NL. Continuous multivariate distributions. Vol. 1. 2nd ed. New York: Wiley; 2000.
7. MacIntyre GA. A method for unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research. 1952;3(4):385-390.
8. Sajeevkumar NK, Sumi AR. Use of ranked set sampling in the estimation of parameters and of Morgenstern type bivariate exponential distribution. IAPQR Transactions. 2023;47(1-2):16-42.
9. Scaria J, Nair NU. On concomitants of order statistics from Morgenstern family. Biometrical Journal. 1999;41(4):483-489.
10. Stokes SL. Ranked set sampling with concomitant variables. Communications in Statistics -Theory and Methods. 1977;6:1207-1211.
11. Yang SS. General distribution theory of the concomitants of order statistics. The Annals of Statistics. 1977;5:996-1002.