Estimation of the common location parameter of several distributions when their common scale parameter is proportional to with known coefficient of variation by ranked set sampling

Introduction

Order statistics is concerned with the characteristics and uses of ordered random variables as well as functions of these variables. If the random variables 𝑋𝑋 1 , 𝑋𝑋 2 , … , 𝑋𝑋 𝑚𝑚 are arranged in ascending order of magnitude as 𝑋𝑋 1:𝑚𝑚 ≤ 𝑋𝑋 2:𝑚𝑚 ≤ ⋯ ≤ 𝑋𝑋 𝑚𝑚:𝑚𝑚 , then m j X: is called the th j order statistic, j=1,2,…,m. Additional information regarding order statistics and its uses may be found in Balakrishnan and Cohen (1991) [2] and David and Nagaraja (2003) [4] . The literature reports on certain biological and physical science problems where the scale parameter is proportional to the location parameter (see Glesser and Healy, 1976) [5] . The issue of estimating the mean has been thoroughly covered in the literature that is currently available if the parent distribution is univariate Normal, more details, see, Arnholt and Hebert (1995) [1] , Searls (1964) [19] , Khan (1968) [8] , Kunte (2000) [9] , and Guo and Pal (2003) [6] . The best linear unbiased estimator of the mean of the Normal distribution with known coefficient of variation using order statistics is discussed by Thomas and Sajeevkumar(2003) [21] . Estimating the mean of logistic distribution with known coefficient of variation using order statistics is discussed by Sajeevkumar and Thomas (2005) [13,[16][17] . Estimating a parameter of the exponential distribution with known coefficient of variation by order statistics are discussed in Sajeevkumar and Irshad (2011) [7] . In order to increase the accuracy of the sample mean as an estimator of the population mean, McIntyre (1952) [11] originally proposed the idea of ranked set sampling (RSS). When ranking a group of sampling units can be accomplished with ease using the judgment approach, the RSS outlined in McIntyre (1952) [11] is appropriate. The process of ranked set sampling involves selecting m sets of units of size m at random from the population of interest. The units in each set are then ranked using a judgment method or a low-cost technique without the units being measured. After measuring the unit ranked as one from the first set, the unit ranked as two from the second set is also measured. This method is repeated until the unit from the mth set that is ranked as m is measured. The observations that come from real measurements on units selected in the manner outlined above are then referred to as ranked set sampling. Ranked set sampling (RSS) offers several advantages over using order statistics (like median, quantiles etc.) from simple random sampling. The following are the advantages of RSS over order statistics.

i). Higher Efficiency

RSS provide more precise estimators (lower variance) for population parameters than simple random sampling (SRS), especially for the mean or median. This provide efficiency comes without increasing sample size.

ii). Cost-effectiveness

In RSS, ranking is done without actual measurement (eg. Visual inspection), and only a subset is measured. This reduces cost and effort when measurement is expensive or destructive. The theory and applications of ranked set sampling (RSS) and the estimation of parameters of certain distributions using RSS (efficiency better than order statistics) are covered in detail in Chen et al. (2004), Lam et al. (1994) [10] , Stokes (1995) [20] , and Zheng and Moderres (2006) [23] . Irshad and Sajeevkumar (2011) [7] discuss the use of ranked set sampling to estimate the location parameter µ of the exponential distribution for which the coefficient of variation is known.

The problem of estimating the common location and common scale parameter of several distributions using order statistics of independent but not identically distributed (inid) random variables are discussed by Sajeevkumar and Thomas (2005) [13,[16][17] . The ranked set sampling (RSS) of independent non-identically distributed (inid) random variables was suggested by Priya and Thomas (2016) [12] . RSS offers several advantages in the multiple population context -such as comparing two or more groups, populations, or treatments. Hence are the main benefits.

a). Increased Efficiency Across Population

RSS improve estimation precision (eg. Means, variances, quintiles) for each population individually, when comparing multiple populations, it lead to more powerful statistical tests (eg. ANOVA, t-tests).

b). Better Resource Allocation

In multiple population studies, RSS allows you to minimize measurement costs while still obtaining reliable comparisons. Especially useful when measurements are expensive but ranking is cheap. Sajeevkumar and Thomas (2006) [18] describe the use of order statistics of inid random variables to estimate the common location parameter µ of multiple distributions where their common scale parameter is proportional to µ . Sajeevkumar (2025) [14] discusses the use of ranked set sampling to estimate the common mean of logistic and normal distributions with known coefficients of variation. With the use of ranked set sampling of inid random variables, we hope to estimate the common location parameter µ of multiple distributions where their common scale parameter is proportional to µ in this article by applying the notion of Sajeevkumar (2025) [14] . The estimation of a common mean across multiple population is statistically significant in many applied fields because it allows for a pooled understanding of a shared parameter. Here's why it matters.

Estimation of the Common Location

i) i). Efficient Data Utilization

When populations are believed to have similar or identical means, estimating a common mean allows for combining data from all groups, increasing statistical power and precision.

ii) ii). Simplified Modeling

A common mean reduces model complexity. Instead of modeling each group separately, one parameter represents the central tendency of all, making interpretation simpler.

iii) iii). Cost Reduction

Infields like agriculture, medicine or industry, assuming a common mean means fewer measurements may be needed from each population, reducing costs and resources.

The distribution theory of order Statistics of independent but not identically distributed (inid) random variables are discussed in Vaughan and Venables (1972) [22] . Sajeevkumar and Thomas (2005) [13,[16][17] used order statistics of inid random variables to determine the best linear unbiased estimator of common location and common scale parameters of multiple distributions.

Let t jm j j X X X ,..., , 2 1 be random sample of size i m drawn from a distribution with pdf of the form ( )

.For convenience we assume that all the t distributions defined by the pdf defined in (1) have same support.

For the pooled sample of all m observations, let ' ) ,..., , (

be the vector of order statistics. Thus, the vector of order statistics above can be seen as a vector of order statistics of m independent random variables that are not identically distributed (inid). Vaughan and Venables (1972) [22] review the distribution theory of order statistics of inid random variables. Let Then the BLUE of µ namely µ ˆ and its variance is given by ) (

More details, see, Sajeevkumar and Thomas (2006) [16][17] . The ranked set sampling of independent non-identically distributed (inid) random variables was suggested by Priya and Thomas (2016) [12] and it is described below.

Estimation of Common Location

Suppose there are t populations and the

, taken together is said to form the ranked set sampling drawn from t populations with j m observations from the th j population, j=1, 2,…,t. Using this RSS of inid random variables Priya and Thomas (2016) [12] estimate the common location and common scale parameter of several distributions using RSS of inid random variables. In this article we have used the RSS of inid random variable suggested by Priya and Thomas (2016) [12] to estimate the common location Parameter µ of several distributions,when the Scale Parameter of each of the distributions is equals to c µ for known c using RSS of inid random variables. ,( ) ( ) Clearly both the populations defined in (12) and ( 13) have common mean µ and common standard deviation c µ .

Let

Simulation Study.

We take sample of size 1 m from (12) and take sample of size 2 m taken from (13), such that

. Using the means, variances and covariance's of order statistics of inid random variables of standard normal and standard double exponential distributions available in Sajeevkumar (2005) [13,[16][17] , we have calculated the BLUE µ ~ of µ using RSS of inid random variables and calculate the relative efficiency of our estimator µ ~ using RSS of inid random variables with that of the estimator µ ˆ using order statistics of inid random variables(defined in (2) and Sajeevkumar and Thomas( 2006)) [16][17] were calculated (for

and for c=0.1(0.05)0.3), "Using R software" and are presented in Table 4.1 and Table 4.2 respectively.

Concluding Remarks

In all the cases, efficiency of the estimator µ ~ defined in equation ( 5), calculated using RSS of inid random variables is much better than that of the estimator µ ˆ defined in equation ( 2) calculated using Order Statistics of inid random variables given in Sajeevkumar and Thomas (2006) [16][17] . Also it is very difficult to calculate the covariance between any two order statistics in the inid case, but in the case of RSS covariance between any two RSS in the inid case is zero.

Hence it is very easy to calculate BLUE of the estimator µ ~ using RSS of inid random variables than the estimator µ ˆ using order statistics of inid random variables. All the computational work in this article are done using "R " software.

References

1. Arnholt AT, Hebert JL. Estimating the normal mean with known coefficient of variation. American Statistician. 1995;49(4):367-369.
2. Balakrishnan N, Cohen AC. Order Statistics and Inference: Estimation Methods. Academic Press; 1991.
3. Chen Z, Bai Z, Sinha BK. Lecture Notes in Ranked Set Sampling: Theory and Applications. Springer; 2004.
4. David HA, Nagaraja HN. Order Statistics. 3rd ed. John Wiley & Sons; 2003.
5. Gleser LJ, Healy JD. Estimation of a normal distribution with known coefficient of variation. Journal of the American Statistical Association. 1976;71:977-981.
6. Guo H, Pal N. On a normal mean with known coefficient of variation. Calcutta Statistical Association Bulletin. 2003;54(12):17-30.
7. Irshad MR, Sajeevkumar NK. Estimating a parameter of the exponential distribution with known coefficient of variation by ranked set sampling. Journal of the Kerala Statistical Association. 2011;22:41-52.
8. Khan RA. A note on estimating the mean of a normal distribution with known coefficient of variation. Journal of the American Statistical Association. 1968;63:1039-1041.
9. Kunte S. A note on consistent maximum likelihood estimation for family. Calcutta Statistical Association Bulletin. 2000;50:325-328.
10. Lam K, Sinha BK, Wu W. Estimation of parameters in a two-parameter exponential distribution using ranked set sampling. Annals of the Institute of Statistical Mathematics. 1994;46(4):723-736.
11. McIntyre GA. A method for unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research. 1952;3:385-390.
12. Priya RS, Thomas P. An application of ranked set sampling when observations from several distributions are to be included in the sample. Communications in Statistics -Theory and Methods. 2016;46:7040-7052.
13. Sajeevkumar NK. On Estimation Techniques Using Order Statistics. [PhD thesis]. University of Kerala; 2005.
14. Sajeevkumar NK. Estimation of the common mean of normal and logistic distributions with known coefficient of variation by ranked set sampling. The Aligarh Journal of Statistics. 2025; (Accepted for publication).
15. Sajeevkumar NK, Irshad MR. Estimating a parameter of the exponential distribution with known coefficient of variation by order statistics. Aligarh Journal of Statistics. 2013;33:22-32.
16. Sajeevkumar NK, Thomas PY. Estimating the mean of logistic distribution with known coefficient of variation by order statistics. Recent Advances in Statistical Theory and Applications. ISPS Proceedings. 2005;1:170-176.
17. Sajeevkumar NK, Thomas PY. Applications of order statistics of independent non-identically distributed random variables in estimation. Communications in Statistics -Theory and Methods. 2005;34:775-783.
18. Sajeevkumar NK, Thomas PY. Estimation of common location parameter of several distributions when their common scale parameter is proportional to it. Journal of the Kerala Statistical Association. 2006;17:37-52.
19. Searls DT. The utilization of known coefficient of variation in the estimation procedure. Journal of the American Statistical Association. 1964;59:1225-1226.
20. Stokes SL. Parametric ranked set sampling. Annals of the Institute of Statistical Mathematics. 1995;47:465-482.
21. Thomas PY, Sajeevkumar NK. Estimating the mean of normal distribution with known coefficient of variation by order statistics. Journal of the Kerala Statistical Association. 2003;14:26-32.
22. Vaughan RJ, Venables WN. Permanent expression for order statistics densities. Journal of the Royal Statistical Society. Series B. 1972;34(2):308-310.
23. Zheng G, Modarres R. A robust estimate of correlation coefficient for bivariate normal distribution using ranked set sampling. Journal of Statistical Planning and Inference. 2006;136:298-309.