Introduction

Usage of auxiliary information is made through the ratio & product techniques of estimation to enhance estimates of the population mean. Estimation of the population mean of the variable of interest with higher precision is an unremitting issue in a sample survey. So, precision could be increased by the use of an apposite estimation procedure that consumes auxiliary information which is meticulously associated with variables of interest. In the ratio method of estimation, the auxiliary information is available which is linearly related to the variable of the study. The population parameters such as populations’ median, coefficient of kurtosis, skewness, coefficient of variation, decile, quartile, correlation, etc are auxiliary variables. The efficiency of estimators of population parameters can be increased by suitable usage of auxiliary information in relationship with auxiliary variables. Cochran¹ came up with what is known as a ratio-type estimator for estimation of population mean which is more competent than the sample mean. Many authors have used different auxiliary information in order to enhance the precision of the estimates by using prior knowledge of population parameters. Researchers in sample surveys like Kadilar and Cingi^2,3 developed classes of ratio estimators using known auxiliary information on coefficients of variation, &  kurtosis. Abid et al.⁴ also suggested a set of ratio-type estimators for the population mean using non-conventional location parameters like mid-range, and tri-mean as auxiliary information. Other researchers are Upadhyaya and Singh⁵, Yan and Tian⁶, Subramani and Kumarapadiyan⁷, Subramani and Kumarapadiyan⁸, Subramani and Kumarapadiyan⁹, Subramani and Kumarapadiyan¹⁰, Jeelani et al¹¹, and Nasir et al¹².

The objective of this study is to develop an innovative set of ratio-type estimators to increase the precision of estimates of the population mean using known auxiliary information.

Let U = {U₁, U₂,........,U_N} be finite population having N units & each Ui = (Xi, Yi), i = 1, 2, 3,....., N has pair of values. Y is study variable & X is auxiliary variable which is associated (correlated) with Y, in which x = {x₁, x₂,........,x_N} & y = {y₁, y₂,........,y_N} are the n sample values. &  is the sample mean of the variable of interest &  is the sample mean of auxiliary variable. s²_y is the sample mean square of the study variable & s²_x is the sample mean square of an auxiliary variable based on a random sample of size drawn without replacement. and S²_y is the population means square of study variable and S²_x is the population means square of auxiliary variable. Following are other symbols used in this study.

Y:           Study variable       

N:          Population size

X:          Auxiliary variable

n:          Sample size

:         Sample means of study variable

:        Sample means of auxiliary variable

:         Population means of study variable

:        Population means of auxiliary variable

S_pw:      Probability weighted moments

p:         Coefficient of correlation

C_y        Coefficient of variation of study variable

Q₃:       The upper quartile

C_x:       Coefficient of variation of auxiliary variable

QD:      Population Quartile Deviation

ß2        Coefficient of skewness  

ß₁        Coefficient of kurtosis

G:        Gini’s Mean Difference

TM:      Tri-Mean

M_d:      Median

MR:     Population mid-range

HL:      Hodges-Lehman estimator

D:        Downton’s Method



The Existing Estimators in Literature

Cochran¹ developed the conventional ratio estimator for estimating population mean () of study variable (Y) given as:


where R = 



Nasir et al¹² modified class of ratio type estimators for finite population mean consuming known values of coefficient of variation (C_x) & decile mean (DM) of auxiliary information, biases, constants, and mean square errors are given as:



Subzar¹³ developed a class ratio type estimators using a linear combination of different known population parameters given as:



where,


 

Proposed Estimator

Motivated by the work of Subzar et al¹³, we proposed ratio-type estimators for estimating the population means using the value of Hodges-Lehmann as:



where,



In order to derive bias & mean square error,  and  such that, from the definition of e₀ and e₁, we obtain



where,



Percentage Relative Efficiency (PRE) is given as:



where _i are estimators in this study.

Efficiency Comparisons

Efficiencies of suggested estimators are compared with efficiencies of existing estimators in the study.

 _pi - family of proposed estimators of finite population mean is more efficient than _r if,



The _pi of proposed estimators of the population mean is more efficient than _j if,



When conditions (2.27), and (2.28) are contented, the conclusion will be made that anticipated estimators are better and relatively efficient than other estimators in the study.

Empirical Study

To evaluate the performance of anticipated estimators, the following real populations are used.

Table 1: Characteristics of Populations [Subzar et al.¹³]

Parameter	Population I	Population II	Population III
N	34	34	80
n	20	20	20
	856.4117	856.4117	5182.637
	199.4412	208.8823	1126.463
p	0.4453	0.4491	0.941
Sy	733.1407	733.1407	1835.659
Cy	0.8561	0.8561	0.354193
Sx	150.2150	150.5059	845.610
Cx	0.7531	0.7205	0.7506772
ß2	1.0445	0.0978	-0.063386
ß1	1.1823	0.9782	1.050002
Md	142.5	150	757.5
MR	320	284.5	1795.5
TM	165.562	162.25	931.562
HL	320	190	1040.5
QD	184	80,25	588.125
G	162.996	155.446	901.081
D	144.481	140.891	801.381
S pw	206.944	199.961	791.364
DM	206.944	234.82	1150.7

Table 2: Constant and Bias of Some Selected Existing and Proposed Estimators.

Estimator	Constant			Bias
	Pop-I	Pop-II	Pop-III	Pop-I	Pop-II	Pop-III
r	4.294	4.100	4.601	4.940	4.270	60.877
1	2.107	1.9301	2.276	2.137	2.2087	26.800
2	1.806	1.6013	1.949	1.483	1.3964	19.650
3	1.289	1.1703	2.206	0.800	0.7459	25.188
4	1.6960	1.6651	1.9813	1.5604	1.5098	20.312
5	1.7606	1.7136	2.0598	1.6815	1.599	21.953
6	1.5602	1.5324	2.0681	1.3205	1.2788	22.129
7	1.8955	1.9263	1.8608	1.9490	2.0207	17.916
8	1.9764	1.9915	1.9299	2.1191	1.9915	2.1598
9	1.7274	1.751	1.9371	1.6187	1.6696	19.416
10	0.9671	0.9633	1.7938	0.5074	0.5053	16.650
12	1.0148	0.9997	1.8622	0.5586	0.5443	17.942
13	0.8701	0.8666	1.8693	0.4107	0.409	18.079
14	1.1177	1.1672	1.9130	0.6777	0.7419	18.936
15	1.1818	1.2211	1.9909	0.7577	0.8121	20.508
16	0.9902	1.0283	1.9991	0.5318	0.5758	20.677
17	1.4153	1.3533	1.5540	1.0866	0.9973	12.495
18	1.4752	1.3979	1.6184	1.1806	1.0642	13.552
19	1.2908	1.2329	1.6252	0.9038	0.8278	13.666
20	1.6021	1.5977	1.6667	1.3923	1.3901	14.373
21	1.6793	1.6603	1.7410	1.5298	1.5011	15.684
p1	1.4444	1.4326	1.7489	1.1317	1.1176	15.825
p2	0.0005	0.0335	0.0061	0.0313	0.0006	0.0002
p3	0.0007	0.0369	0.0069	0.0353	0.0007	0.0002
p4	0.0003	0.0261	0.0070	0.0247	0.0004	0.0003
p5	0.0005	0.0334	0.0062	0.0313	0.0006	0.0002
p6	0.0007	0.0368	0.0069	0.0352	0.0007	0.0002
p7	0.0003	0.0261	0.0070	0.0247	0.0004	0.0003
p8	0.0005	0.0336	0.0062	0.0315	0.0006	0.0002
p9	0.0007	0.0371	0.0069	0.0355	0.0007	0.0002
p10	0.0003	0.0262	0.0070	0.0248	0.0004	0.0003
p11	0.0005	0.0336	0.0062	0.0314	0.0006	0.0002
p12	0.0007	0.0370	0.0069	0.0354	0.0007	0.0002
p13	0.0003	0.0262	0.0070	0.0248	0.0004	0.0003
p14	0.0005	0.0336	0.0062	0.0314	0.0006	0.0002
p15	0.0007	0.0370	0.0069	0.0354	0.0007	0.0002
p16	0.0003	0.0262	0.00701	0.0248	0.0004	0.0003
p17	0.0005	0.0335	0.0062	0.0313	0.0006	0.0002
p18	0.0007	0.0369	0.0069	0.0353	0.0007	0.0002
p19	0.0003	0.0261	0.0070	0.0248	0.0004	0.0003

 

Table 2 shows the constant and bias of estimators

Table 3: MSE and PRE of Estimators.

Estimator	MSE			PRE
	Pop-I	Pop-II	Pop-III	Pop-I	Pop-II	Pop-III
r	10960.76	10539.27	189775.1	100	100	100
1	10934.74	10571.58	153292.6	100.238	99.69437	123.7993
2	10386.83	10030.11	116239.3	105.5256	105.0763	163.2624
3	9644.04	9472.95	144936.7	113.6532	111.2565	130.9365
4	10208.16	10333.07	119741.5	107.3725	101.9955	158.4873
5	10311.83	10203.59	128249.9	106.2931	103.2898	147.9729
6	10002.72	9929.39	129161.3	109.5778	106.1422	146.9288
7	10540.91	10564.74	107326.5	103.9831	99.75892	176.8204
8	10686.6	10683.87	114349.5	102.5655	98.64656	165.9606
9	10258.09	10264.06	115098.9	106.8499	102.6813	164.88
10	9306.32	9266.94	100762.1	117.7776	113.7298	188.3398
11	9350.19	9300.31	107457.3	117.225	113.3217	176.6051
12	9223.53	9184.47	108172.8	118.8348	114.751	175.437
13	9452.18	9469.56	112609.8	115.9601	111.2963	168.5245
14	9250.7	9529.64	120761.3	118.4857	110.5946	157.1489
15	9327.27	9327.31	121636.1	117.5131	112.9937	156.0187
16	9802.37	9688.30	79228.58	111.8174	108.7835	239.5286
17	9882.86	9745.56	84709.31	110.9068	108.1443	224.031
18	9645.86	9543.10	85298.12	113.6318	110.4386	222.4845
19	10064.19	10024.69	88963.27	108.9085	105.1331	213.3185
20	10181.93	10119.77	95756.01	107.6491	104.1454	198.1861
21	9841.01	9791.32	96489.29	111.3784	107.6389	196.68
p1	8872.22	8834.676	14471.83	123.5402	119.2944	1311.341
p2	8872.342	8834.787	14472.1	123.5385	119.2929	1311.317
p3	8872.048	8834.471	14472.13	123.5426	119.2971	1311.314
p4	8872.218	8834.674	14471.83	123.5402	119.2944	1311.341
p5	8872.34	8834.786	14472.1	123.5385	119.2929	1311.317
p6	8872.047	8834.47	14472.13	123.5426	119.2971	1311.314
p7	8872.223	8834.68	14471.83	123.5402	119.2943	1311.341
p8	8872.348	8834.794	14472.1	123.5384	119.2928	1311.317
p9	8872.05	8834.473	14472.13	123.5426	119.2971	1311.314
p10	8872.222	8834.679	14471.83	123.5402	119.2943	1311.341
p11	8872.346	8834.792	14472.1	123.5385	119.2928	1311.317
p12	8872.049	8834.472	14472.13	123.5426	119.2971	1311.314
p13	872.221	8834.678	14471.83	123.5402	119.2943	1311.341
p14	8872.345	8834.791	14472.1	123.5385	119.2928	1311.317
p15	8872.049	8834.472	14472.13	123.5426	119.2971	1311.314
p16	8872.22	8834.677	14471.83	123.5402	119.2943	1311.341
p17	8872.343	8834.789	14472.1	123.5385	119.2928	1311.317
p18	8872.048	8834.471	14472.13	123.5426	119.2971	1311.314

 

Table 3 shows the mean square error (MSE) & percentage relative efficiency (PRE) for the three populations.

Results and Discussion

Class of ratio estimators of finite population means is proposed and performance of anticipated estimators over existing estimators was established. The scope of the study is to analyze and estimate the biases, mean square errors of anticipated estimators, and efficiency comparison with some existing estimators. Tables 2 and 3 show the results of the Constant, Bias, Mean Square Error (MSE) & Percentage Relative Efficiency (PRE) of anticipated & existing estimators considered in the study for all populations used. Outcomes also discovered that anticipated estimators have the least MSE and advanced PRE than other estimators. The outcomes also show that the average dispersion of anticipated estimators gives better estimates on the average compared to other estimators considered.

Future Scope

The future scope of the study is to transform the sampling technique from simple random sampling to other sampling techniques like stratified sampling, two-stage sampling, cluster sampling, or successive sampling.

Conclusion

In Table 3, anticipated estimators performed better than prevailing estimators considered in the study. So, it is clear that anticipated estimators performed superior to other estimators having minimum Mean Square Error (MSE) & highest Percentage Relative Error (PRE). We, therefore, conclude that anticipated estimators are relatively efficient and better than other estimators for the estimation of the population mean.

Funding Sources

The authors received no financial support for the research, authorship, and/or publication of this article.

Conflict of Interest

The authors declare no conflict of interest.

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