A class of ratio-cum-product estimators of population mean under extreme ranked set sampling (ERSS) and simple random sampling (SRS)

Introduction

Under sampling techniques, the estimation of population variables is of utmost interest. This estimation most times seek to make use of better methods of estimation that would give an improved result. More often, interest is on the mean of a definite feature of a finite population on the ground of the portion taken from the population following a specific sampling scheme. This is so because the mean has a wider use in sampling and statistical analysis. Many sampling techniques depend on the possession of advance information about an auxiliary variate. Researches which adopt supplementary information in statistical survey (sampling) are broad and traceable to the pioneer work of Bowley (1926) [27] who carried out the groundwork of contemporary sampling theory involving stratified random sampling and Neyman (1934, 1938) [37,38] . Nevertheless, application of supplementary knowledge in estimation technique to enhance the performances of estimators was introduced by Watson (1937) [51] and Cochran (1940, 1942) [28,29] . McIntyre (1952) [35] initiated and put in the procedure of ranked-set-sampling (RSS) in approximating the average pasture output as a better/reliable approaches and inexpensive scheme than the procedure of Simple Random Sampling (SRS). This technique is functional in circumstances where units of interest are very simple and economical to order than to observe in relation to a variable under consideration.

Takahasi and Wakimoto (1968) [50] separately set out the procedure of RSS and revealed an impressive mathematical reasoning, which is in tandem with McIntyre's instinctive postulation. Dell and Clutter (1972) [30] proved that deviations in ordering lowers the accuracy of the RSS average comparative to SRS average. Nevertheless, RSS average is always superior over the SRS average till ordering is too substandard as to produce a probabilistic sample when its performances are akin to that of SRS average. The techniques of Extreme-Ranked Set Sampling (ERSS) as first introduced by Samawi et al. (1996) [41] to estimate the population mean and showed that the mean based on ERSS though unbiased but is more efficient that the sample me due to SRS. Furthermore, Samawi (1996) [41] introduced the principle of Stratified Ranked Set Sampling (SRSS); to improve the precision of estimating the population means in case of SSRS. Ali and Iqbal (2021) [3] proposed an efficient generalized family of estimators to estimate finite population mean of study variable under Ranked Set Sampling utilizing information on an auxiliary variable and concluded that when correlation between the study and auxiliary variables increases, the proposed generalized family of estimators proved to be efficient estimator of population mean of the study variable. Further researches on RSS method include but not limited to Al-Omari el tal.(2009) [2] , Al-Omari (2019) [1] , Haq and Shabbir (2010) [31] , Kaur et al. (1995) [33] Al Saleh and Al-Kadiri (2000) [18] , Al-Saleh and Al-Omari (2002) [22] , Abu Dayyeh et al. (2002) [4] , Al-Saleh and-Zheng (2002) [25] , Al-Saleh and Samawi (2000) [24] , Ozturk and Wolfe (2000) [39] , Ozturk (2002) [40] , Al-Saleh and Ababneh (2015) [20] , Zheng and Al-Saleh (2002) [25] , Al-Saleh and Darabseh (2017)) [23] . The expeditious development in the area of RSS over the past twenty (20) years provided a boost for the uprising of other key connected methods to inferential statistics. In this paper, a class of ratio-cum-product estimators of population mean of the study variable is proposed by employing ERSS method following information on a single accompanying variable. The expressions of the biases and Mean Square Errors (MSEs) of the proposed estimators were calculated. Analytical and simulation study of performances and efficiencies of the estimators over the usual SRS method using their (MSEs) were carried out in an attempt to support the theoretical results with numerical illustration. Application to real life data was successfully done to illustrate the method, from where conclusion was drawn following the results obtained from the work.

Sampling methods

Here, we present the sampling scheme which is employed in the course of this work i.e Ranked Set Sampling (RSS), (ERSS), as well as the frequently used (SRS).

RSS Description

Step 1: Select 𝑚 random samples each of size 𝑚 bivariate units from the population under consideration.

Step 2: Rank the units within each set-in relation to the variable of interest by eyeball approach or any cost-free method.

Step 3: From the first set of 𝑚 units, the smallest ranked unit X is selected together with the corresponding Y, and from the second set of 𝑚 units the second smallest ranked unit X is selected together with the corresponding Y. The process is continued until from the 𝑚 𝑡ℎ set of 𝑚 units the largest ranked unit X is selected with the associated Y. This process can be repeated 𝑟 times to increase the sample size to 𝑟𝑚 RSS bivariate units. In this work we assume that the ranking is done on the variable X for estimating the population mean of the study variable Y. Nevertheless, the entire process can be carried out again while the ranking can be done on the variable Y.

ERSS Description

Let (𝑋 𝑖 (1) , 𝑌 𝑖 [1] ), (𝑋 𝑖 (2) , 𝑌 𝑖 [2] ), … (𝑋 𝑖(𝑚) , 𝑌 𝑖[𝑚] ) be the order Statistics of 𝑋 𝑖1 , 𝑋 𝑖2 , 𝑋 𝑖3 , … 𝑋 𝑖𝑚 and the judgment order of 𝑌 𝑖1 , 𝑌 𝑖2 , 𝑌 𝑖3 , … 𝑌 𝑖𝑚 , (𝑖 = 1, 2, … 𝑚). Then the RSS units are: (𝑋 1(1) , 𝑌 1 [1] ), (𝑋 2(2) , 𝑌 2 [2] ), … (𝑋 𝑚(𝑚) , 𝑌 𝑚[𝑚] ), here, (•) and [•] implies that the ordering of X is faultless or without error and the ordering of Y has error, be 𝑚 independent random samplesof size 𝑚 and assume that each member (𝑋 𝑖(𝑗) , 𝑌 𝑖 [𝑗] ) in the sample has the same bivariate distribution-function 𝐹(𝑥, 𝑦) with mean 𝜇 𝑋 , 𝜇 𝑌 variance 𝜎 𝑋 , 𝜎 𝑌 , and 𝜌 𝑋𝑌 . The ERSS method, as suggested by Samawi et al. (1996) [41] , can be described as given below: a) Select 𝑚 random samples, each of size 𝑚 units, from an infinite population and order the units within each sample with respect to a variable under consideration by impressionistic method or any other cost-free procedure. For exact quantification, if the sample size 𝑚 is even, from the first sets, we take the mean of the measures of the largest ranked unit. Such a sample shall be represented by 𝐸𝑅𝑆𝑆 0(𝑎) . (ii) From the remaining measure of the 𝑚 𝑡ℎ unit we take the median. Such a sample will be represented by 𝐸𝑅𝑆𝑆 0(𝑚) .

𝑒: is even 0(𝑎): is odd average 0(𝑚): is odd median The procedure can be continued 𝑟 times, if need be, to get a sample of size 𝑟𝑚 units. The choices of (𝑎) and 𝑏(𝑖𝑖) is usually less difficult in application than the choice of 𝑏(𝑖). In this work, we considered the choices of (𝑎) and 𝑏(𝑖𝑖), (i.e the even case and the case of taking the median from the 𝑚 𝑡ℎ sample if 𝑚 is odd).

If 𝑚 is even, then the observed ERSSe units are:

), then under ERSSe the sample means and variances of the study and accompanying variables are defined as:

If 𝑚 odd, then measured 𝐸𝑅𝑆𝑆 0(𝑚) units are:

] ), (𝑋𝑚+3 ), then under 𝐸𝑅𝑆𝑆 0(𝑚) the sample means and variances of the study and accompanying variables are defined as:

SRS Description

In SRS, 𝑚 unit out of 𝑀 units of a population are chosen in such a way that every individual unit has equiprobable chance of being selected. According to our description, 𝑦 ̅ ( 𝑥 ̅ 𝑋 ̅ ) 𝛼 , Srivastava (1970) [46] (

], Bahl and Tuteja (1991) [26] 𝑌 ̅

Bahl and Tuteja (1991) [26] 𝑌 ̅ ) ],

Singh & Choudhury (2012) [42] 𝑌 ̅ 2 ( [45] ( Kadilar and Cingi (2004) [32] (

Singh and Espejo (2003) [44] (

Notations and some useful equations

Let 𝑌 be the study variable and 𝑋 be the auxiliary variable which is correlated with 𝑌. Then following notations and expressions shall be useful in the course of this work. For all 𝑖 = 1, 2, … , 𝑚.

Then proposed class of estimator based on ERSS Motivated by Singh and Espejo (2003) [44] ratio-cum-product estimator under SRS, we advocated a class of ratio-cum-product estimators of the population mean 𝑌 ̅ of the study variable Y with a single accompanying variable using SRS and ERSS schemes as follows:

Where (𝑎 ≠ 0, 𝜌 ≠ 0) are real numbers and may take the values of parameters associated with either the study variable 𝑦 or the concomitant variable 𝑥 or both(𝑥, 𝑦); in this case, the coefficient of variation and the correlation coefficient respectively. (𝛼 1 , 𝛼 2 ,) are scalars or real constants which helps in designing the estimator and can be determined suitably. We may fix 𝛼 1 , 𝛼 2 , and 𝑤 will be selected in an optimum manner by minimizing the (MSEs) of 𝑇 1𝐸𝑖 , 𝑖 = 1 𝑡𝑜 𝑚 with respect to 𝑤.

The proposed class of estimator based on SRS

for all the 𝐶 𝑚 𝑀 samples.

(2) 𝐵(𝑇 1(0(𝑚))𝑗 ) = [𝐸(𝑇 1(0(𝑚))𝑗 ) -𝑌 ̅ 𝐸𝑅𝑆𝑆 0(𝑚) ], 𝑗 = 1,2, … 𝑚 (14)

[𝐸(𝑇 1(0(𝑚))𝑗 )-𝑌 ̅ 𝐸𝑅𝑆𝑆 0(𝑚) ] 𝑌 ̅ 𝐸𝑅𝑆𝑆 0(𝑚)

𝑇 1𝑆𝑗 , 𝑗 = 1, 2, . . 𝑚 is a class of estimator of the Population mean 𝑌 ̅ 𝑆𝑅𝑆 under SRS

𝑇 1𝑆𝑗 , 𝑗 = 1, 2, . . 𝑚 is a class of estimator of the Population mean 𝑌 ̅ 𝑆𝑅𝑆 under SRS, for 𝐸𝑅𝑆𝑆𝑒 case, 𝑒: is even and 𝐸𝑅𝑆𝑆 0(𝑚) case, 0(𝑚): is median odd respectively, 𝑚 the total number of members of the proposed class of estimator.

Biases 𝑻 𝟏𝑬𝒋 , 𝑻 𝟏(𝟎(𝒎))𝒋 , 𝑻 𝟏𝑺𝒋 To obtain the bias and Mean Square Error of the class of estimators 𝑇 1𝐸𝑗 we write

𝑇 1𝐸𝑗 , 𝑇 1(0(𝑚))𝑗 , 𝑇 1𝑆𝑗 in equations 10, 11 and 12 can be expressed in terms of 𝑒 ′ 𝑠 as:

Expanding the right and side of ( 23), neglecting terms of 𝑒 ′ 𝑠 having power greater than two, we have and then taking the mathematical expectations of the emergent expression, yields the bias of 𝑇 1𝐸𝑗 as:

In like manners, we obtained the biases of 𝑇 1(0(𝑚))𝑗 , 𝑇 1𝑆𝑗 𝐵( 𝑇 1(0(𝑚))𝑗 ) = [𝐸(𝑇 1(0(𝑚))𝑗 ) -𝑌 ̅ 𝐸𝑅𝑆𝑆 0(𝑚) ] = 𝑌 ̅ 𝐸𝑅𝑆𝑆 0(𝑚)

MSEs of 𝑻 𝟏𝑬𝒋 , 𝑻 𝟏(𝟎(𝒎))𝒋 , 𝑻 𝟏𝑺𝒋

Recall that from equation ( 23)

By Squaring both sides of (29) and neglecting terms of 𝑒 ′ 𝑠 having powers greater than two we have:

By taking the mathematical expectation of (30), gives the MSE of 𝑇 1𝐸𝑗 as:

Similarly, the MSEs of 𝑇 1(0(𝑚))𝑗 , 𝑇 1𝑆𝑗 were obtained to be:

Optimal MSEs of 𝑻 𝟏𝑬𝒋 , 𝑻 𝟏(𝟎(𝒎))𝒋 , 𝑻 𝟏𝑺𝒋 To obtain the optimum MSE of 𝑇 1𝐸𝑗 , 𝑇 1(0(𝑚))𝑗 , 𝑇 1𝑆𝑗 we differentiate ( 31), (37), and (43) partially and separately with respect to 𝑤 and (1 -𝑤) and equate the resulting expression to zero respectively. This procedure yields the optimal MSEs of 𝑇 1𝐸𝑗 , 𝑇 1(0(𝑚))𝑗 , 𝑇 1𝑆𝑗 as:

Efficiency comparison

Let 𝑀𝑆𝐸(𝑇 1𝐸𝑗 ) 𝑜𝑝𝑡 , 𝑀𝑆𝐸(𝑇

(𝑖𝑖𝑖) 𝑀𝑆𝐸(𝑇 1[0(𝑚)]𝑗 ) 𝑜𝑝𝑡 , is most efficient than 𝑀𝑆𝐸(𝑇 1𝐸𝑗 ) 𝑜𝑝𝑡 , and MS𝐸(𝑇 1𝑆𝑗 ) 𝑜𝑝𝑡 , under optimal condition if

(𝑖𝑣) 𝑀𝑆𝐸 (𝑇 1𝐸𝑗 ) 𝑜𝑝𝑡 is more efficient than MS𝐸(𝑇 1𝑆𝑗 ) 𝑜𝑝𝑡 in terms of 𝑃𝑅𝐸, if

× 100 > 100 for 𝑗 = 1 𝑡𝑜 14 } (56)

Empirical study

In order to investigate the efficiency of the proposed class of estimators and its members under ERSS over its corresponding counterpart estimator based on SRS, and some existing ratio type estimators, we have considered three natural populations data sets. The real-life data sets were obtained from various sources and the description of the population and the values of the required parameters are specified below:

Population I: [Source: Murthy (1967)] [36] . The population consists of 80 factories in a region, the character 𝑋 and Y being fixed capital and output respectively.

Simulation study

A computer simulation study was conducted using the R-software to examine the theoretical underpinnings of the work and the performances of the proposed class of estimator of population mean based on ERSS (even and odd median) and on SRS, when ranking is done on a single accompanying variable X. Bivariate random observations were generated from a bivariate normal distribution having parameters 𝜇 𝑋 = 25, 𝜇 𝑌 = 15, 𝜎 𝑋 2 = 𝜎 𝑦 2 = 1, and 𝜌 𝑋𝑌 = ±0.99, ±0.90, ±0.70 and ±0.50. Using 5000 simulations, estimates of MSE's for the estimators in question were computed. We considered sample sizes m = 3, 4, 5, 6, 7, 8, 9, 10 and r = 1 respectively to study the performances of the proposed ratio-cum-product estimators under ERSSe, ERSSo(m) and on SRS, the results is as shown in table 12 and table 13.

Conclusion

A class of ratio-cum-product estimators of population mean of the study variable Y have been successfully proposed following information on a single accompanying variable under ERSS as shown in equations ( 10) and ( 11) while keeping track record of the SRS version of the proposed estimators as shown in (12) for the purpose of efficiency comparison. Members of the proposed class of the estimators were obtained by varying the scalars that helps in designing the estimator and were presented in table 2, table 3, and table 4 respectively. Their properties such as biases, and MSEs were all derived as can be envisage in equations ( 26), ( 27), (28) for biases and ( 31), ( 37), (43) for MSEs. The Optimal Mean Square Errors were also calculated to the quadratic polynomial form of Taylor's series approximation and presented in ( 49), ( 50) and ( 51) respectively. Theoretical underpinnings and the condition for which the proposed class of estimator would provide an appreciable gain in efficiency over its counterpart estimator were established and shown in (52). ( 53), ( 54), (55), and (56). Empirical and simulation studies were conducted to ascertain the veracity of the theoretical underpinnings of the work. From where it was discovered from the results that the proposed class of estimators based on ERSS provided smaller MSEs, R.E, P.R.E, for all values of the correlation coefficients and sample sizes considered in the work and are therefore adjudged to be more efficient than the corresponding counterpart under SRS. This evidence is presented in table 7 The estimator 𝑇 1(𝑜(𝑚))𝑗 performs better than that of 𝑇 1𝐸𝑗 and 𝑇 2𝑆𝑗 for all the values of the correlated coefficient and the samples sizes considered in this work. Therefore, the estimator𝑠 𝑇 1(𝑜(𝑚))𝑗 was adjudged to be the most efficient estimators among their brethren 𝑇 1𝐸𝑗 , 𝑇 1𝑆𝑗 since it produces the smallest MSEs in all the population, correlation coefficients, and sample sizes considered in this work. The estimators in question were therefore adjudged to be efficient and provide a better alternative whenever efficiency is required.

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