
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12121-4
10.1016/j.heliyon.2024.e36090
e36090
Research Article
Memory type general class of estimators for population variance under simple random sampling
Kumar Anoop anoop.asy@gmail.com
a⁎
Anshika b
Emam Walid c
Tashkandy Yusra c
a Department of Statistics, Central University of Haryana, Mahendergarh, Haryana, 123031, India
b Department of Statistics, Amity University, Lucknow, 226028, India
c Department of Statistics and Operations Research, Faculty of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
⁎ Corresponding author. anoop.asy@gmail.com
14 8 2024
30 8 2024
14 8 2024
10 16 e360907 6 2024
19 7 2024
9 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
With an emphasis on memory-type approaches, this study presents a class of estimators specifically designed for estimating population variation in simple random sampling (SRS). The term ‘memory-type’ pertaining to the use of exponentially weighted moving averages (EWMA) statistic for the estimation, which utilizes the current and past information in temporal surveys. The study provides expressions for the bias and mean square error (MSE) of these estimators and establishes conditions under which their efficiency represses the conventional and other memory-type estimators. The theoretical findings are reinforced through a comprehensive simulation study conducted on hypothetically sampled populations. Additionally, the effectiveness of the proposed estimators is demonstrated utilizing real-life population data. The findings of simulation and real data application show the superiority of the proposed memory type estimator over the existing usual and memory type estimators.

MSC

62D05
Keywords

Population variance
Memory-based methods
Simple random sampling
Exponentially weighted moving averages (EWMA)
Mean square error (MSE)
Efficiency performance
==== Body
pmc1 Introduction

In survey sampling, the auxiliary information serves as an important tool for improving the accuracy and efficiency of estimates obtained from the sample. By elaborating the primary variables of interest with additional data, such as economic indicators, demographic characteristics, or geographic information, the survey practitioners may improve their sampling designs and improve the representativeness of the sample. Utilizing auxiliary information grants for strategic stratification of the population, weighting of sampled units, and adjustment of estimates to align with known population parameters. The incorporation of auxiliary information in survey sampling contributes to reducing sampling error, increasing precision, and minimizing biases, thereby enhancing the reliability of findings of the study and allowing more robust insights into the population. For detailed discussion on the use of auxiliary information, the reader can refer to the studies of [1], [2], [3], [4], [5], [6], [7], and [8].

Variation in data exists and has a significant impact on every field of study, namely, manufacturing industry, agriculture, pharmaceuticals, medical, and biological sciences. Therefore, the problem of variance estimation is of great importance. The sampling theory describes a wide set of estimation procedures for population variance utilizing different sampling techniques which utilizes auxiliary information to obtain more efficient estimators. In the context of survey sampling literature, the problem of variance estimation utilizing auxiliary information has received a considerable attention. In this context, ratio, product, difference, exponential, logarithmic estimators are some well-known estimation procedures which are suggested to use under different circumstances. The ratio estimator is fruitful if there is a strong positive correlation between the main and the auxiliary variables. The ratio estimator for population variance proposed by [9] is stated as:tr=sy2(Sx2sx2),

where Sx2 is the population variance of the auxiliary variable x and (sy2, sx2) are the sample variances of the main and auxiliary variables (y, x), respectively. The estimator tr's MSE is:(1.1) MSE(tr)=Sy4(δ40+δ04−2δ22),

where Sy2 is the population variance of the main variable. Also, δ40=γ(β2y−1), δ04=γ(β2x−1), δ22=γ(μ22−1), β2y=μ4y/μ2y2, β2x=μ4x/μ2x2, and γ=1/n. Furthermore, β2y and β2x are the coefficients of kurtosis of study and auxiliary variables, respectively.

Further, the product method of estimation may be effectively utilized if the main and the auxiliary variables are negatively correlated. It is given bytp=sy2(sx2Sx2).

The following is the MSE expression for the estimator tp as:(1.2) MSE(tp)=Sy4(δ40+δ04+2δ22).

Ref. [10] proposed a general class of estimator under SRS given bytg=sy2(Sx2sx2)α,

where α is a properly selected scalar. The minimum MSE of tg at α(opt)=δ22/δ04 is given by(1.3) min.MSE(tg)=Sy4(δ40−δ222δ04).

Several studies on estimation of variance utilizing ratio, exponential, regression, and logarithmic type estimators based on various sorts of transformations have recently been published. For example, [11] suggested population variance estimation employing auxiliary data. Ref. [12] advocated variance estimation under SRS utilizing ratio type estimator. [13] introduced generalized class of estimators for population variance estimation. [14] investigated an improved population variance estimator. [15] estimated population variance under a class of ratio estimators based on auxiliary variable, while, [16] examined the ratio and product type exponential population variance estimators under transformed sample information of study and auxiliary variables. [17] offered the estimation of population variance utilizing regression type estimator in successive sampling. [18] suggested elevated population variance estimators under SRS. [19] proposed efficacious class of population variance estimator utilizing attributes, while [20] introduced some alternative classes of population variance estimators based on attributes. [21] introduced variance estimation with an efficient class of estimators under SRS. [22] offered an improved population variance estimation through the use of dual auxiliary information under stratified sampling. [23] estimated population variance through an enhanced estimation procedure under SRS.

A time series may be analysed or characterized utilizing EWMA statistics which is a quantitative or statistical approach. In finance sector, EWMA provides well-known applications in technical analysis and volatility modeling. The approach assigns less weights to past data points within the moving average construction, a characteristic referred to as “exponentially weighted”. The selection of smoothing parameter stands as the primary decision point for EWMA users because it affects the significance of current observations in the computation. A higher parametric value elevates the precision of EWMA in tracking the original time series.

The survey researchers have proved that employing auxiliary data enhances the accuracy of the estimator by reducing variance or MSE. However, the traditional practice often involves solely considering present observations (yt at time t), leveraging information from present and past samples (y¯t−1 at time t−1, y¯t−2 at time t−2, etc.) can further elevate the performance of the estimator. This approach proves specially valuable in surveys carried out over certain intervals like annually, monthly, or quarterly.

The population mean based memory-type ratio and product estimators employing EWMA and hybrid EWMA (HEWMA) in time-scaled surveys were introduced by [24] and [25], respectively. The memory-based population mean estimators in ranked-based sampling and stratified sampling were examined by [26] and [27], respectively. [28] proposed memory-based logarithmic estimators utilizing HEWMA statistics. [29] presented mean estimation for time-based surveys utilizing memory-type logarithmic estimators. [30] developed memory-based population variance estimators employing EWMA statistics for time-scaled surveys, while, [31] proposed memory-type variance estimators based on EWMA statistic in presence of measurement error for time-based surveys. In this study, we have some objectives related to memory-type general class of estimators for population variance under SRS:• Investigate the theoretical framework of memory-based estimators for population variance under SRS.

• Develop a thorough understanding of the principles underlying memory-based estimators and their applicability in the variance estimation.

• Assess the performance of memory-type estimators in comparison to the conventional estimators under various sampling scenarios.

• Explore the impact of different parametric settings on the efficiency and accuracy of memory-type estimators.

• Compare the computational complexity of memory-type estimators with alternative methods for population variance estimation using simulation and real data applications.

• Explore the sensitivity of memory-type estimators to sample size variations.

• Discuss the potential extensions or modifications of memory-type estimators to improve their performance or applicability in particular contexts.

• Assess the practical feasibility and implementation challenges associated with adopting memory-type estimators in real-world sampling scenarios.

• Provide recommendations and guidelines for the effective utilization of memory-type population variance estimators under SRS.

The remaining portion is organised as follows: In Section 2, the existing memory-type population variance estimators are discussed in detail with their corresponding characteristics. The memory-type general class of estimators is presented in Section 3 along with a comparison of its characteristics with the currently available memory-type estimators. An empirical comparison using actual data applications and simulations is presented in Section 4. Section 5 provides the conclusion at the end.

2 Review of estimators

The EWMA statistic is a technique employed in statistical analysis and time series forecasting. It was initially introduced by [32] to enhance the detection of changes in the mean of processes in statistical quality control charts.

The basic idea behind EWMA statistics is to assign more weight to the current observations, while, still considering older ones. To do this, a weighted mean of earlier data is computed with the weights decreasing exponentially with increasing retrogression. The advantage of this technique is that it allows for quick adaptation to changes in the data, while still retaining some memory of past observations. For t greater than unity, the EWMA statistics for the main and auxiliary variables are prescribed as:Vt=ηsˆyt2+(1−η)Vt−1andWt=ηsˆxt2+(1−η)Wt−1,

where t denotes the sample number, Vt denotes the current observation, and Vt−1 denotes the past observation. The initial value V0 is derived either from the mean value of the prior sample or set to one. The pair (sˆyt2,sˆxt2) denotes the sample variance of variables (y,x) and η denotes the smoothing constant (weight parameter) of the present sampled observations. As the parameter η varies between 0 to 1, greater weight is assigned to present observations, while simultaneously reducing the weight assigned to older observations. When η=1, all weight is assigned to present observation, and the EWMA statistic behaves identically to traditional sample variance.

It is noted that the expectation and variance of Vt are, respectively, given as:E(Vt)=Sy2andVar(Vt)=Var(sˆyt2)[(η2−η){1−(1−η)2t}].

The limiting form of Var(Vt) is given as:Var(Vt)=Sy4δ40(η2−η).

To improve the efficiency of the traditional ratio and product estimators, [30] explored the concept of the EWMA and proposed memory-type ratio and product estimators in SRS as:trm=Vt(Sx2Wt),tpm=Vt(WtSx2),

where the superscript m in the above and other estimators stand for “memory.”

To find out the properties of the memory-type estimators, the following notations are used.Vt=Sy2(1+e0),Wt=Sx2(1+e1),such thatE(e0)=E(e1)=0,E(e02)=ςδ40,E(e12)=ςδ04,E(e0e1)=ςδ22,andς=η(2−η).

Utilizing above notations, the MSE of the memory-type ratio and product estimators trm and tpm is computed up to an approximation of order one and reported below:(2.1) MSE(trm)=Sy4ς(δ40+δ04−2δ22)

(2.2) andMSE(tpm)=Sy4ς(δ40+δ04+2δ22).

For more details see [30].

3 Proposed estimators

The desire to improve the effectiveness and accuracy of the statistical estimates in real-life applications is the motivation behind the proposal of the memory-type general class of estimators for population variance. The traditional estimators frequently generate skewed or inconsistent results because they depend on the assumptions which might not hold true in real-life data. Utilizing memory-type estimators, survey persons can reduce bias and variance/MSE in their estimators by utilizing additional data to improve their computations based on prior sample information. When data collection is costly or challenging, or when sample numbers are very few, this approach is very helpful. Memory-type estimators also provide more robust and accurate estimates of population variance which is important for decision-making in domains like engineering, medicine, and economics. Furthermore, these estimators can adjust to shifting data patterns over time. Following [30], we suggest the memory-type general class of estimators (MTGCEs) for population variance under SRS astgm=Vt(Sx2Wt)λ,

where λ is a properly selected scalar. Remark 3.1 For λ=1, the proposed estimator tgm converts into memory-type ratio estimator trm.

Remark 3.2 For λ=−1, the proposed estimator tgm converts into memory-type product estimator tpm.

The estimator tgm may be rewritten by leveraging the notations expressed as error terms in the preceding section.tgm=Sy2(1+e0){Sx2Sx2(1+e1)}λ,=Sy2(1+e0)(1+e1)−λ.

Employing Taylor series expansion to the right hand side in the above expression, multiplying and ignoring the error terms having power more than two, we gettgm=Sy2(1+e0){1−λe1+λ(λ+1)2!e12},=Sy2{1+e0−λe1+λ(λ+1)2e12−λe0e1}.

Subtracting Sy2 both sides to the above expression, we get(3.1) tgm−Sy2=Sy2{e0−λe1+λ(λ+1)2e12−λe0e1}.

Considering the expectations on both side of (3.1), we establish the bias of the suggested estimator asBias(tgm)=Sy2ς{λ(λ+1)2δ04−λδ22}.

By squaring and taking into account expectation on both sides of the (3.1), it gives the estimator tgm's MSE to the first order approximation in the following way:E(tgm−Sy2)2=Sy4E(e02+λ2e12−2λe0e1),MSE(tgm)=Sy4ς(δ40+λ2δ04−2λδ22).

The following is the optimum value of λ that minimises the MSE(tgm):λ(opt)=δ22δ04.

Putting the optimum value of λ in MSE(tgm) provides the optimum MSE given as:(3.2) MSE(tgm)min=Sy4ς(δ40−δ222δ04).

Moreover, we compute the performance of the proposed memory-type general class of estimator in comparison to both conventional and memory-type population variance estimators by comparing their respective MSE provided below.(i). From (1.1) and (3.2):IfMSE(tr)>MSE(tgm),⇒Sy4(δ40+δ04−2δ22)>Sy4ς(δ40−δ222δ04),⇒ς<(δ40+δ04−2δ22)(δ40−δ222δ04).

(ii). From (1.2) and (3.2):IfMSE(tp)>MSE(tgm)⇒ς<(δ40+δ04+2δ22)(δ40−δ222δ04).

(iii). From (1.3) and (3.2):IfMSE(tg)>MSE(tgm)⇒ς<1.

(iv). From (2.1) and (3.2):IfMSE(trm)>MSE(tgm)⇒(δ40−δ222δ04)<(δ40+δ04−2δ22).

(v). From (2.2) and (3.2):IfMSE(tpm)>MSE(tgm)⇒(δ40−δ222δ04)<(δ40+δ04+2δ22).

Under these circumstances, the proposed memory type estimator demonstrates superior performance compared to existing estimators.

4 Empirical study

This section considers an empirical study of the proposed MTGCEs within the conditions reported in previous section. Through simulation study and real-life data applications, the effectiveness of the suggested MTGCEs is exhibited.

4.1 Simulation study

This section presents the comparison of the developed MTGCE with the conventional and memory-based estimators through a simulation experiment. The process of the simulation experiment is narrated in the following points.(i). Utilize R software to draw a N=1000 units population sampled from a bivariate normal distribution with the following constants: Y¯=15, X¯=12, σy2=36, σx2=38, and varying values of the correlation coefficient ρxy such as ±0.1, ±0.3, ±0.5, ±0.7, and ±0.9. Additionally, choose different values for the smoothing constant η including 0.15, 0.35, 0.55, 0.75, 0.95.

(ii). The 10,000 samples are collected having sample of sizes n=10, 50, 100, and 500 drawn from each sample group.

(iii). The 10,000 values for each variance estimator taken into consideration in this study are computed using the samples produced in the previous stage.

(iv). All variance estimators have their MSE calculated using (4.1), and the results are combined in Table 1 and Table 2. Using (4.2), the relative efficiency (RE) is also calculated, and the results are compiled in Table 3 and Table 4.Table 1 MSE of basic and memory-based estimators for chosen positive amounts of ρxy, n, and η.

Table 1					η=0.15		η=0.35		η=0.55		η=0.75		η=0.95	
ρxy	n	tm	tr	tg	trm	tgm	trm	tgm	trm	tgm	trm	tgm	trm	tgm	
0.1	10	238927.9	509901.1	204956.4	41343.3	16618.0		108160.8	43475.6		193410.8	77742.1		305940.7	122973.9		461339.1	185436.8	
50	62576.9	123876.0	60970.0	10044.0	4943.5		26276.7	12933.0		46987.4	23126.5		74325.6	36582.0		112078.3	55163.3	
100	32239.4	63322.0	31820.1	5134.2	2580.0		13431.9	6749.7		24018.6	12069.7		37993.2	19092.0		57291.3	28789.6	
500	6571.0	12780.1	6560.2	1036.2	531.9		2710.9	1391.5		4847.6	2488.3		7668.0	3936.1		11562.9	5935.4	


	
0.3	10	237891.8	460802.9	199287.7	37362.4	16158.4		97746.0	42273.1		174787.3	75591.8		276481.8	119572.6		416916.9	180307.9	
50	62292.0	112569.5	59828.2	9127.2	4850.9		23878.3	12690.8		42698.7	22693.4		67541.6	35896.9		101848.6	54130.3	
100	32146.2	57615.2	31394.9	4671.5	2545.5		12221.4	6659.5		21854.0	11908.4		34569.1	18836.9		52128.1	28404.9	
500	6569.0	11655.8	6517.1	945.0	528.4		2472.4	1382.4		4421.1	2472.0		6993.4	3910.3		10545.7	5896.4	


	
0.5	10	234695.4	367038.6	181546.9	29759.8	14720.0		77856.6	38509.9		139221.5	68862.6		220223.1	108928.1		332082.5	164256.7	
50	61175.5	90970.4	55147.6	7375.9	4471.4		19296.7	11697.9		34506.0	20918.0		54582.2	33088.6		82306.6	49895.5	
100	31600.8	46669.8	29098.0	3784.0	2359.2		9899.6	6172.3		17702.3	11037.1		28001.9	17458.8		42225.1	26326.7	
500	6476.7	9480.3	6083.7	768.6	493.2		2010.9	1290.4		3596.0	2307.6		5688.2	3650.2		8577.4	5504.3	


	
0.7	10	231249.9	236710.7	140794.0	19192.7	11415.7		50211.3	29865.4		89786.8	53404.6		142026.4	84476.4		214166.8	127385.1	
50	59824.1	60169.2	43271.4	4878.5	3508.4		12763.1	9178.7		22822.8	16413.3		36101.5	25962.8		54438.8	39150.3	
100	30891.8	30976.1	22929.4	2511.5	1859.1		6570.7	4863.8		11749.5	8697.3		18585.7	13757.6		28026.0	20745.6	
500	6342.3	6329.5	4822.0	513.2	390.9		1342.6	1022.8		2400.8	1829.0		3797.7	2893.2		5726.7	4362.8	


	
0.9	10	229985.6	81849.6	60922.4	6636.4	4939.6		17362.0	12922.9		31046.4	23108.5		49109.7	36553.4		74054.4	55120.2	
50	59022.9	21648.8	18899.2	1755.3	1532.3		4592.1	4008.9		8211.6	7168.6		12989.3	11339.5		19587.0	17099.3	
100	30396.4	11197.4	10034.2	907.9	813.5		2375.2	2128.4		4247.3	3806.1		6718.4	6020.5		10131.0	9078.6	
500	6224.7	2305.3	2116.9	186.9	171.6		489.0	449.0		874.4	802.9		1383.2	1270.1		2085.8	1915.2	

Table 2 MSE of basic and memory-based estimators for chosen negative amounts of ρxy, n, and η.

Table 2					η=0.15		η=0.35		η=0.55		η=0.75		η=0.95	
ρxy	n	tm	tp	tg	tpm	tgm	tpm	tgm	tpm	tgm	tpm	tgm	tpm	tgm	
-0.1	10	236896.2	426780.0	204194.2	34603.7	16556.2		90529.1	43313.9		161882.1	77452.9		256068.0	122516.5		386134.3	184747.2	
50	61772.0	116911.3	60505.5	9479.2	4905.8		24799.3	12834.5		44345.6	22950.3		70146.7	36303.3		105776.9	54743.1	
100	31762.9	60407.3	31455.4	4897.8	2550.4		12813.6	6672.3		22913.1	11931.3		36244.4	18873.2		54654.3	28459.7	
500	6464.4	12328.0	6455.4	999.5	523.4		2615.0	1369.3		4676.1	2448.6		7396.8	3873.2		11153.9	5840.6	


	
-0.3	10	232013.3	463868.2	197272.5	37610.9	15995.0		98396.2	41845.6		175950.0	74827.5		278320.9	118363.5		419690.3	178484.7	
50	59989.1	124547.7	58600.3	10098.4	4751.3		26419.2	12430.3		47242.2	22227.7		74728.6	35160.1		112686.0	53019.3	
100	30786.4	64163.8	30430.8	5202.4	2467.3		13610.5	6455.0		24338.0	11542.7		38498.2	18258.5		58052.9	27532.7	
500	6264.5	13072.6	6240.1	1059.9	505.9		2772.9	1323.6		4958.5	2366.9		7843.5	3744.1		11827.5	5645.8	


	
-0.5	10	225695.1	542418.4	179033.9	43979.8	14516.2		115058.4	37976.8		205744.9	67909.4		325451.0	107420.3		490759.5	161983.1	
50	57727.8	141703.7	53616.3	11489.4	4347.2		30058.3	11373.1		53749.6	20337.2		85022.2	32169.8		128208.1	48510.0	
100	29576.3	72766.2	27895.8	5899.9	2261.8		15435.2	5917.3		27600.9	10581.1		43659.7	16737.5		65836.1	25239.1	
500	6022.4	14797.6	5738.3	1199.8	465.2		3138.8	1217.2		5612.8	2176.6		8878.5	3442.9		13388.3	5191.8	


	
-0.7	10	220676.3	661286.1	138821.2	53617.7	11255.7		140272.8	29446.9		250832.6	52656.3		396771.6	83292.7		598306.4	125600.2	
50	55907.2	169188.0	42069.2	13717.9	3411.0		35888.3	8923.7		64174.7	15957.3		101512.8	25241.5		153074.9	38062.6	
100	28611.6	86684.1	21985.6	7028.4	1782.6		18387.5	4663.6		32880.1	8339.3		52010.4	13191.4		78428.4	19891.8	
500	5828.9	17605.0	4550.8	1427.4	368.9		3734.4	965.3		6677.7	1726.2		10563.0	2730.5		15928.3	4117.4	


	
-0.9	10	221496.4	818654.6	60345.6	66377.4	4892.8		173654.0	12800.5		310524.2	22889.7		491192.8	36207.3		740687.5	54598.4	
50	56015.5	208117.8	18547.7	16874.4	1503.8		44146.2	3934.3		78941.2	7035.3		124870.7	11128.6		188297.1	16781.3	
100	28666.7	106586.4	9758.3	8642.1	791.2		22609.2	2069.9		40429.3	3701.4		63951.8	5855.0		96435.3	8828.9	
500	5833.4	21650.2	2037.6	1755.4	165.2		4592.4	432.2		8212.1	772.8		12990.1	1222.5		19588.3	1843.5	

Table 3 RE of basic and memory-based estimators for chosen positive values of ρxy, n, and η.

Table 3				η=0.15	η=0.35	η=0.55	η=0.75	η=0.95	
ρxy	n	tr	tg	trm	tgm	trm	tgm	trm	tgm	trm	tgm	trm	tgm	
0.1	10	0.47	1.17	5.78	14.38	2.21	5.50	1.24	3.07	0.78	1.94	0.52	1.29	
50	0.51	1.03	6.23	12.66	2.38	4.84	1.33	2.71	0.84	1.71	0.56	1.13	
100	0.51	1.01	6.28	12.50	2.40	4.78	1.34	2.67	0.85	1.69	0.56	1.12	
500	0.51	1.00	6.34	12.35	2.42	4.72	1.36	2.64	0.86	1.67	0.57	1.11	


	
0.3	10	0.52	1.19	6.37	14.72	2.43	5.63	1.36	3.15	0.86	1.99	0.57	1.32	
50	0.55	1.04	6.82	12.84	2.61	4.91	1.46	2.74	0.92	1.74	0.61	1.15	
100	0.56	1.02	6.88	12.63	2.63	4.83	1.47	2.70	0.93	1.71	0.62	1.13	
500	0.56	1.01	6.95	12.43	2.66	4.75	1.49	2.66	0.94	1.68	0.62	1.11	


	
0.5	10	0.64	1.29	7.89	15.94	3.01	6.09	1.69	3.41	1.07	2.15	0.71	1.43	
50	0.67	1.11	8.29	13.68	3.17	5.23	1.77	2.92	1.12	1.85	0.74	1.23	
100	0.68	1.09	8.35	13.39	3.19	5.12	1.79	2.86	1.13	1.81	0.75	1.20	
500	0.68	1.06	8.43	13.13	3.22	5.02	1.80	2.81	1.14	1.77	0.76	1.18	


	
0.7	10	0.98	1.64	12.05	20.26	4.61	7.74	2.58	4.33	1.63	2.74	1.08	1.82	
50	0.99	1.38	12.26	17.05	4.69	6.52	2.62	3.64	1.66	2.30	1.10	1.53	
100	1.00	1.35	12.30	16.62	4.70	6.35	2.63	3.55	1.66	2.25	1.10	1.49	
500	1.00	1.32	12.36	16.22	4.72	6.20	2.64	3.47	1.67	2.19	1.11	1.45	


	
0.9	10	2.81	3.78	34.65	46.56	13.25	17.80	7.41	9.95	4.68	6.29	3.11	4.17	
50	2.73	3.12	33.63	38.52	12.85	14.72	7.19	8.23	4.54	5.21	3.01	3.45	
100	2.71	3.03	33.48	37.36	12.80	14.28	7.16	7.99	4.52	5.05	3.00	3.35	
500	2.70	2.94	33.30	36.27	12.73	13.86	7.12	7.75	4.50	4.90	2.98	3.25	

Table 4 RE of basic and memory-based estimators for chosen negative values of ρxy, n, and η.

Table 4				η=0.15	η=0.35	η=0.55	η=0.75	η=0.95	
ρxy	n	tp	tg	tpm	tgm	tpm	tgm	tpm	tgm	tpm	tgm	tpm	tgm	
-0.1	10	0.46	1.16	5.73	14.31	2.19	5.47	1.22	3.06	0.77	1.93	0.51	1.28	
50	0.50	1.02	6.13	12.59	2.34	4.81	1.31	2.69	0.83	1.70	0.55	1.13	
100	0.50	1.01	6.17	12.45	2.36	4.76	1.32	2.66	0.83	1.68	0.55	1.12	
500	0.50	1.00	6.22	12.35	2.38	4.72	1.33	2.64	0.84	1.67	0.56	1.11	


	
-0.3	10	0.50	1.18	6.20	14.51	2.37	5.54	1.33	3.10	0.84	1.96	0.56	1.30	
50	0.53	1.02	6.51	12.63	2.49	4.83	1.39	2.70	0.88	1.71	0.58	1.13	
100	0.53	1.01	6.54	12.48	2.50	4.77	1.40	2.67	0.88	1.69	0.59	1.12	
500	0.53	1.00	6.59	12.38	2.52	4.73	1.41	2.65	0.89	1.67	0.59	1.11	


	
-0.5	10	0.61	1.26	7.56	15.55	2.89	5.94	1.62	3.32	1.02	2.10	0.68	1.39	
50	0.63	1.08	7.72	13.28	2.95	5.08	1.65	2.84	1.04	1.79	0.69	1.19	
100	0.63	1.06	7.72	13.08	2.95	5.00	1.65	2.80	1.04	1.77	0.69	1.17	
500	0.63	1.05	7.76	12.94	2.97	4.95	1.66	2.77	1.05	1.75	0.70	1.16	


	
-0.7	10	0.93	1.59	11.46	19.61	4.38	7.49	2.45	4.19	1.55	2.65	1.03	1.76	
50	0.91	1.33	11.27	16.39	4.31	6.26	2.41	3.50	1.52	2.21	1.01	1.47	
100	0.91	1.30	11.22	16.05	4.29	6.14	2.40	3.43	1.52	2.17	1.01	1.44	
500	0.91	1.28	11.23	15.80	4.29	6.04	2.40	3.38	1.52	2.13	1.01	1.42	


	
-0.9	10	2.70	3.67	33.28	45.27	12.72	17.30	7.11	9.68	4.50	6.12	2.98	4.06	
50	2.55	3.02	31.49	37.25	12.04	14.24	6.73	7.96	4.26	5.03	2.82	3.34	
100	2.53	2.94	31.19	36.23	11.92	13.85	6.67	7.74	4.22	4.90	2.80	3.25	
500	2.51	2.86	30.92	35.31	11.82	13.50	6.61	7.55	4.18	4.77	2.77	3.16	

Based on the simulation with 10,000 iterations, the MSE and RE are computed, respectively by utilizing the following formulae:(4.1) MSE(t)=110,000∑i=110,000(t−Sy2)2wheret=sy2,tr,tp,tg,trm,tpm,tgm,

(4.2) andRE(t)=MSE(sy2)MSE(t).

The significant simulation results are interpreted as follows:

• The MSE and RE of the developed MTGCE tgm are least and highest, respectively, in comparison to the conventional variance estimator sy2, conventional ratio estimator tr, conventional product estimator tp, general class of estimator tg, memory-type ratio estimator trm, and memory-type product estimator tpm that are presented, respectively, in Table 1 to Table 4. For example, at ρxy=0.9, when n=10 and η=0.15, the MSEs of tgm and tg are 4939.6 and 60922.4, respectively.

• For increasing correlation coefficient ρxy from 0.1 to 0.9, it is observed that the developed MTGCE's MSE boils down whereas its RE boils up, as exhibited in Table 1 to Table 4, respectively. This exhibits that leveraging auxiliary information meliorates the efficacy of the estimators. For example, at η=0.95 and n=500, the MSE values of tgm are 5935.4 and 1915.2, while the RE values are 1.11 and 3.25 for ρxy=0.1 and 0.9, respectively.

• As the size of the sample n increases from 10 to 500, for any constant value of the correlation coefficient ρxy, the MSE of the MTGCE reduces. It shows that the MTGCEs are more efficient compared to the available counterparts. For example, at ρxy=0.1 and η=0.15, the MSE values of the MTGCE tgm are 16618.0 and 531.9 for n=10 and 500, respectively.

• The results obtained from the memory-based estimators presented in Table 1, Table 2, Table 3, Table 4 are further influenced by the smoothing constant η, which assigns weight to both new and old observations.

• Moreover, as the smoothing constant η varies, greater emphasis is assigned on current information over past information. Consequently, this leads to an increase in the MSE and a decrease in the RE of the memory-based estimators.

4.2 Real data applications

To validate the theoretical and simulation findings, we conduct a numerical study utilizing two distinct real populations outlined below:

Population 1: (Sourced from [33].)

y=Per capita consumption of chickens (in pounds) in United States in 1960-1982,

x=Real disposable income per capita (in dollars) in United States in 1960–1982,

N=23, n=5, μy=39.699, μx=1035.065, Sx2=381735.000, Sy2=54.360, ρxy=0.8277, β2(y)=2.030, β2(x)=2.696, λ22=2.094, δ40=0.1612, δ04=0.2655 and δ22=0.1713.

Population 2: (Sourced from [34].)

y=Leaf area for the new developed strain of wheat,

x=Weight of leaves,

N=39, n=14, μy=26.8433, μx=106.2, Sy2=38.9900, Sx2=124.1286, ρxy=0.8899, β2(y)=2.4032, β2(x)=2.9930, λ22=2.4882, δ40=0.0643, δ04=0.0913 and δ22=0.0682.

Employing the data specified in these two populations, we have calculated MSE and RE of the estimators. The RE is calculated with the help of following formula:RE(t)=MSE(sy2)MSE(t)

and the results are reported in Table 5 by MSE and RE for both the populations. The results are showing the superiority of the proposed MTGCE tgm over the conventional ratio, product, and power ratio estimators and memory-type ratio and product estimators for different values of smoothing parameter η. The impact of η can easily be seen over the MSE and RE values of the memory-based estimators. As we increase the value of η, the MSE of the memory type estimators increases, while the RE decreases.Table 5 MSE and RE of estimators for real populations.

Table 5		Population 1	Population 2	
η	Estimator	MSE	RE	MSE	RE	
0.1	tr	248.52	1.92	29.19	3.35	
tp	2273.29	0.21	443.90	0.22	
tg	149.75	3.18	20.30	4.81	
trm	13.08	36.42	1.54	63.63	
tpm	119.65	3.98	23.36	4.18	
tgm	7.88	60.44	1.07	91.48	


	
0.3	tr	248.52	1.92	29.19	3.35	
tp	2273.29	0.21	443.90	0.22	
tg	149.75	3.18	20.30	4.81	
trm	43.86	10.86	5.15	18.98	
tpm	401.17	1.19	78.34	1.25	
tgm	26.43	18.03	3.58	27.28	


	
0.5	tr	248.52	1.92	29.19	3.35	
tp	2273.29	0.21	443.90	0.22	
tg	149.75	3.18	20.30	4.81	
trm	82.84	5.75	9.73	10.05	
tpm	757.76	0.63	147.97	0.66	
tgm	49.92	9.54	6.77	14.44	


	
0.7	tr	248.52	1.92	29.19	3.35	
tp	2273.29	0.21	443.90	0.22	
tg	149.75	3.18	20.30	4.81	
trm	133.82	3.56	15.72	6.22	
tpm	1224.08	0.39	239.03	0.41	
tgm	80.64	5.91	10.93	8.94	


	
0.9	tr	248.52	1.92	29.19	3.35	
tp	2273.29	0.21	443.90	0.22	
tg	149.75	3.18	20.30	4.81	
trm	203.33	2.34	23.88	4.09	
tpm	1859.96	0.26	363.19	0.27	
tgm	122.53	3.89	16.61	5.88	

5 Conclusion

The proposition of the MTGCEs for population variance under SRS has provided valuable insights into enhancing estimation techniques. Through rigorous experimentation and analysis, we have observed the potential of these estimators in enhancing the accuracy of variance estimation, specially in situations involving auxiliary information. The findings underscore the importance of leveraging memory-type methods in sampling methodologies to achieve more precise estimations of population variance. Moreover, the simulation and real data application results highlight the potential for future enhancements in sampling theory, emphasizing the need for continued research and refinement of estimation techniques to address real-world challenges effectively.

In conclusion, our investigation into memory-type general class estimators for population variance under SRS has yielded insightful findings with significant implications for statistical estimation methodologies. Through comprehensive experimentation and analysis based on simulation and real data applications, we have demonstrated the effectiveness of the MTGCEs over the conventional and memory-based estimators. Thus, our study underscores the importance of incorporating memory-type approaches into sampling techniques to achieve more precise estimations of population variance.

Furthermore, our research highlights the potential of memory-type estimators to address challenges inherent in traditional variance estimation methods, such as the reliance on past observations and the need for efficient utilization of available data. By leveraging memory-type techniques, we can enhance the efficiency and reliability of variance estimation in real-world applications.

In forthcoming studies, we would like to examine the proposed estimator for the population variance under different sampling schemes, including, stratified sampling, ranked set sampling, two-phase sampling, stratified ranked set sampling, etc.

CRediT authorship contribution statement

Anoop Kumar: Writing – review & editing, Supervision, Software, Methodology, Investigation, Data curation, Conceptualization. Anshika: Writing – original draft, Software, Formal analysis, Data curation. Walid Emam: Resources, Project administration, Funding acquisition. Yusra Tashkandy: Resources, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Walid Emam reports a relationship with King Saud University that includes: funding grants. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability statement

Data included in article/supplementary material/referenced in article.

Acknowledgements

The study was funded by Researchers Supporting Project number (RSP2024R488 ), 10.13039/501100002383 King Saud University , Riyadh, Saudi Arabia.
==== Refs
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