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Article

Efficient Difference and Ratio-Type Imputation Methods under Ranked Set Sampling

1
Department of Statistics, University of Lucknow, Lucknow 226007, India
2
Department of Statistics, Amity University, Lucknow 226028, India
3
Department of Statistics, Faculty of Science, Çankiri Karatekin University, Çankiri 18100, Turkey
4
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Axioms 2023, 12(6), 558; https://doi.org/10.3390/axioms12060558
Submission received: 10 May 2023 / Revised: 1 June 2023 / Accepted: 2 June 2023 / Published: 5 June 2023

Abstract

:
It is well known that ranked set sampling (RSS) is more efficient than simple random sampling (SRS). Furthermore, the presence of missing data vitiates the conventional results. Only a minuscule amount of work has been conducted under RSS with missing data. This paper makes a modest attempt to provide some efficient difference- and ratio-type imputation methods in the presence of missing values under RSS. The envisaged imputation methods are demonstrated to provide better results than the existing imputation methods. The theoretical results are enhanced by a computational analysis using real and hypothetically generated symmetric (Normal) and asymmetric (Gamma and Weibull) populations. The computational results show that the proposed imputation method outperforms the existing imputation methods in terms of its higher percent relative efficiency. Additionally, the impact of skewness and kurtosis on the efficiency of the suggested imputation methods has also been calculated.

1. Introduction

The most common problem reported by a survey statistician in their daily life is making inferences from data containing missing values. Such problems of missing values in survey sampling may be tackled through the technique of imputation. A wide range of imputation methods have been suggested by various authors. The authors of [1] discussed three noteworthy concepts on missing values as missing at random ( M A R ), observed at random ( O A R ), and parameter distribution ( P D ). Subsequently, [2,3,4,5,6] suggested different types of imputation methods. The authors of [7] showed that missing at random and missing completely at random ( M C A R ) are totally different approaches. Many renowned authors [8,9,10,11,12] assumed the M C A R approach in their studies for the imputation of missing values. The authors of [13] introduced some imputation methods which outperformed the imputation methods suggested by [14]. The authors of [15] developed logarithmic-type imputation methods under S R S . The authors of [16] utilized robust measures and suggested compromised imputation-based mean estimators.
In real-life applications, situations may arise where the measurement of the study variable is not easy or expensive to do so but can be ranked visually or by a cost-free measure. In this situation, ref. [17] envisaged the concept of ranked set sampling (RSS) but did not provide any rigorous mathematical support. The authors of [18] explored the idea of [17] and furnished the essential mathematical foundation to the theory of RSS. In sample surveys, when each group has very few observations, each observation then becomes essential to make an effective prediction. Furthermore, the utilization of these types of datasets based on missing values may alter the final conclusion and decrease the efficiency of the estimation procedure. To deal with such problems, refs. [19,20,21,22] introduced an analytical comparison of imputation methods under RSS.
This paper conducted a search for efficient imputation procedures. We adapted some efficient difference- and ratio-type imputation methods under RSS based on [11,12], which are more efficient compared to the mean imputation method and the imputation methods suggested by [21,22] under RSS.
The paper is organised as follows: Section 2 describes the detailed methodology of RSS as well as the notations utilized throughout the paper. In Section 3, we consider a concise recap of some imputation methods under RSS, whereas in Section 4, we consider the proposed methods of imputation. In Section 5, we provide the efficiency conditions. Section 6 is devoted to the computational analysis and finally, Section 7 considers the conclusions of this study.

2. Sampling Methodology and Notations

The methodology of RSS was initiated by [17], based on drawing m simple random samples of size m from the parent population. These m units are now ranked inside each set regarding the auxiliary variable. The r a n k 1 unit is chosen from the first set for the measurement of the auxiliary variables along with the associated study variable. The r a n k 2 unit is chosen from the second ranked set for the measurement of auxiliary variable X along with the associated study variable Y, and the process is proceeded until the r a n k m unit is chosen from the last set. The above process is referred to as a cycle. This whole procedure is repeated k times, providing n = m k ranked set samples.
In the presence of missing values in a dataset, an alteration in the aforesaid methodology is proposed for the estimation of the population mean of the study variables under the consideration of usable auxiliary information. To facilitate ranking, m bivariate random samples, each consisting of m units, are quantified from the parent population. These m units are ranked within each set regarding the auxiliary variable as it is hypothesized that the study variable has some missing values. Now, from the first sample, the smallest ranked unit of X along with the correlated Y is selected. From the second sample, the second smallest ranked unit of X along with the correlated Y is selected. The above procedure is continued in the same mode until the mth sample from the highest ranked unit of X along with the correlated Y is selected. Compatible to the study variable from the first cycle, m units can provide a response for the measurement of the element out of the selected m units such that m > m . The whole procedure is repeated k times until responses from n units out of n selected units is obtained, where n > n .

Notations

Let μ y = N 1 i = 1 N Y i be the mean of the finite population Ω of N identifiable units with values Y i , i Ω . Let a ranked set sample s of size n = m k be quantified from Ω to estimate the population mean μ y . Let m be the number of responding units out of the sampled m units. Let P be the probability that the ith respondent belongs to the responding group A and ( 1 P ) be the probability that the ith respondent belongs to the non-responding group A ¯ such that s = A A ¯ . The value Y i , i A is observed for every unit, but for the units i A ¯ the values are missing and need imputation to build the complete structure of the data to draw a valid conclusion. The auxiliary variable X assists in the execution of imputation of missing values. Let X i be the value of X for the unit i which is positive and known ∀ i s such that X s = X i ; i s are known. Let X ¯ r , r s s = i = 1 m j = 1 k X ( i : i ) j / m k P and Y ¯ r , r s s = i = 1 m j = 1 k Y [ i : i ] j / m k P possess the unbiased estimator of population means μ x and μ y , respectively. Here, X ( i : i ) j and Y [ i : i ] j are the ith order statistics and ith judgement order in the ith sample, respectively, of size m in cycle j for variable X and Y. For the sake of simplicity, we denote X ( i : i ) j and Y [ i : i ] j by X ( i ) and Y [ i ] , respectively. Let P be the probability of determining the response, then E ( r j ) = { E ( r ) } j , which provides the variance as
E { V ( Y ¯ r , r s s ) } = σ y 2 m k P 1 m 2 k P i = 1 m τ y [ i ] 2
then E ( j 1 ) = { E ( j ) } 1 = n P
The proof of (1) and (2) can be viewed in [20].
To tabulate the bias and mean square error ( M S E ), the following notations and results are used throughout this paper. Let Y ¯ r , r s s = μ y ( 1 + ϵ 0 ) , X ¯ r , r s s = μ x ( 1 + ϵ 1 ) , and X ¯ n , r s s = μ x ( 1 + ϵ 2 ) , where ϵ 0 , ϵ 1 , and ϵ 2 are the error terms, such that E ( ϵ 0 ) = E ( ϵ 1 ) = E ( ϵ 2 ) = 0 and
E ( ϵ 0 2 ) = C y 2 m k P 1 m 2 k P i = 1 m τ y [ i ] 2 μ y 2 = γ * C y 2 W y 2 * E ( ϵ 1 2 ) = C x 2 m k P 1 m 2 k P i = 1 m τ x ( i ) 2 μ x 2 = γ * C x 2 W x 2 * E ( ϵ 2 2 ) = C x 2 m k 1 m 2 k i = 1 m τ x ( i ) 2 μ x 2 = γ C x 2 W x 2 E ( ϵ 0 , ϵ 1 ) = ρ x y C x C y m k P 1 m 2 k P i = 1 m τ x y [ i ] μ x μ y = γ * ρ x y C x C y W x y * E ( ϵ 0 , ϵ 2 ) = ρ x y C x C y m k 1 m 2 k i = 1 m τ x y [ i ] μ x μ y = γ ρ x y C x C y W x y E ( ϵ 1 , ϵ 2 ) = C x 2 m k 1 m 2 k i = 1 m τ x ( i ) 2 μ x 2 = γ C x 2 W x 2
where γ * = 1 / m k P , γ = 1 / m k , W y 2 = 1 i = 1 m τ y [ i ] 2 / m 2 k P μ y 2 , W x 2 = 1 i = 1 m τ x ( i ) 2 / m 2 k μ x 2 , W x 2 * = 1 i = 1 m τ x ( i ) 2 / m 2 k P μ x 2 , W x y = 1 i = 1 m τ x y [ i ] / m 2 k μ x μ y , W x y * = 1 i = 1 m τ x y [ i ] / m 2 k p μ x μ y , τ y [ i ] = ( μ y [ i ] μ y ) , τ x ( i ) = ( μ x ( i ) μ x ) , τ x y = ( μ x ( i ) μ x ) ( μ y [ i ] μ y ) , C x = S x / μ x , C y = S y / μ y , μ y = E ( Y ) , μ x = E ( X ) , μ y [ i ] = E ( Y [ i ] ) , and μ x ( i ) = E ( X ( i ) ) .
Here, S x and S y are the population standard deviations due to the auxiliary variable X and study variable Y, respectively, C x and C y are the population coefficients of variation due to the auxiliary variable X and study variable Y, respectively, and ρ x y is the population correlation coefficient between the auxiliary variable X and study variable Y. Moreover, we would also like to annotate that the quantities μ x ( i ) and μ y [ i ] consist of order statistics from some particular distributions and can be easily determined from [23].

3. Review of Imputation Methods under RSS

3.1. Mean Imputation Method

The method of imputation is
y . i = Y i for i A Y ¯ r , r s s for i A ¯
The consequent estimator is
t m = Y ¯ r , r s s
The imputation methods are categorized into three strategies under the consideration of the availability of auxiliary information.
S t r a t e g y I : When μ x is known and X ¯ n , r s s is used.
S t r a t e g y I I : When μ x is known and X ¯ r , r s s is used.
S t r a t e g y I I I : When μ x is unknown and X ¯ n , r s s and X ¯ r , r s s are used.

3.2. The Al-Omari and Bouza Imputation Method

To improve the efficiency of the estimators in the presence of missing data, [9,21] suggested some regression-cum-ratio-type estimators under RSS as
S t r a t e g y I
y ¯ K C 1 = Y ¯ r , r s s + b ( μ x X ¯ n , r s s ) μ x X ¯ n , r s s
S t r a t e g y I I
y ¯ K C 2 = Y ¯ r , r s s + b ( μ x X ¯ r , r s s ) μ x X ¯ r , r s s
S t r a t e g y I I I
y ¯ K C 3 = Y ¯ r , r s s + b ( X ¯ n , r s s X ¯ r , r s s ) X ¯ n , r s s X ¯ r , r s s
where b = S x y / S x 2 is the regression coefficient of Y on X.

3.3. The Sohail, Shabbir and Ahmed Imputation Methods

Following [20,21,22], we examined the ratio-type estimators of [8] using RSS for the imputation of missing values. These imputation methods are
S t r a t e g y I
y . i s 1 = Y i for i A 1 n r n Y ¯ r , r s s μ x X ¯ n , r s s β 1 r Y ¯ r , r s s for i A ¯ y . i s 4 = Y i for i A 1 n r n Y ¯ r , r s s μ x β 4 X ¯ n , r s s + ( 1 β 4 ) μ x r Y ¯ r , r s s for i A ¯
S t r a t e g y I I
y . i s 2 = Y i for i A 1 n r n Y ¯ r , r s s μ x X ¯ r , r s s β 2 r Y ¯ r , r s s for i A ¯ y . i s 5 = Y i for i A 1 n r n Y ¯ r , r s s μ x β 5 X ¯ r , r s s + ( 1 β 5 ) μ x r Y ¯ r , r s s for i A ¯
S t r a t e g y I I I
y . i s 3 = Y i for i A 1 n r n Y ¯ r , r s s X ¯ n , r s s X ¯ r , r s s β 3 r Y ¯ r , r s s for i A ¯
y . i s 6 = Y i for i A 1 n r n Y ¯ r , r s s X ¯ n , r s s β 6 X ¯ r , r s s + ( 1 β 6 ) X ¯ n , r s s r Y ¯ r , r s s for i A ¯
The consequent estimators are
t s 1 = y ¯ r , r s s μ x X ¯ n , r s s β 1
t s 2 = y ¯ r , r s s μ x X ¯ r , r s s β 2
t s 3 = y ¯ r , r s s X ¯ n , r s s X ¯ r , r s s β 3
t s 4 = y ¯ r , r s s μ x β 4 X ¯ n , r s s + ( 1 β 4 ) μ x
t s 5 = y ¯ r , r s s μ x β 5 X ¯ r , r s s + ( 1 β 5 ) μ x
t s 6 = y ¯ r , r s s X ¯ n , r s s β 6 X ¯ r , r s s + ( 1 β 6 ) X ¯ n , r s s
where β i ; i = 1 , 2 , , 6 are appropriately chosen optimizing scalars.
The M S E values of the consequent estimators consisting of different imputation methods are given in Appendix A for quick reference and further analytical comparison.

4. The Proposed Imputation Methods

The crux of the present article is:
  • To provide efficient imputation methods for mean estimation.
  • To access the impact of the skewness and kurtosis coefficients on the choice of imputation procedures.
Motivated by the works of [11,12], we propose nine new imputation methods under the three strategies discussed earlier, defined as
S t r a t e g y I
y . i 1 = α 1 Y i for i A α 1 Y ¯ r , r s s + n θ 1 n r ( X ¯ n , r s s μ x ) for i A ¯ y . i 4 = Y i for i A 1 n r n α 4 Y ¯ r , r s s μ x X ¯ n , r s s θ 4 r Y ¯ r , r s s for i A ¯ y . i 7 = Y i for i A 1 n r n α 7 Y ¯ r , r s s μ x μ x + θ 7 ( X ¯ n , r s s μ x ) r Y ¯ r , r s s for i A ¯
S t r a t e g y I I
y . i 2 = α 2 Y i for i A α 2 Y ¯ r , r s s + n θ 2 n r ( X ¯ r , r s s μ x ) for i A ¯ y . i 5 = Y i for i A 1 n r n α 5 Y ¯ r , r s s μ x X ¯ r , r s s θ 5 r Y ¯ r , r s s for i A ¯ y . i 8 = Y i for i A 1 n r n α 8 Y ¯ r , r s s μ x μ x + θ 8 ( X ¯ r , r s s μ x ) r Y ¯ r , r s s for i A ¯
S t r a t e g y I I I
y . i 3 = α 3 Y i for i A α 3 Y ¯ r , r s s + n θ 3 n r ( X ¯ r , r s s X ¯ n , r s s ) for i A ¯ y . i 6 = Y i for i A 1 n r n α 6 Y ¯ r , r s s X ¯ n , r s s X ¯ r , r s s θ 6 r Y ¯ r , r s s for i A ¯ y . i 9 = Y i for i A 1 n r n α 9 Y ¯ r , r s s X ¯ n , r s s X ¯ n , r s s + θ 9 ( X ¯ r , r s s X ¯ n , r s s ) r Y ¯ r , r s s for i A ¯
Under the above strategies, the consequent estimators are
T 1 = α 1 Y ¯ r , r s s + θ 1 ( X ¯ n , r s s μ x )
T 2 = α 2 Y ¯ r , r s s + θ 2 ( X ¯ r , r s s μ x )
T 3 = α 3 Y ¯ r , r s s + θ 3 ( X ¯ r , r s s X ¯ n , r s s )
T 4 = α 4 Y ¯ r , r s s μ x X ¯ n , r s s θ 4
T 5 = α 5 Y ¯ r , r s s μ x X ¯ r , r s s θ 5
T 6 = α 6 Y ¯ r , r s s X ¯ n , r s s X ¯ r , r s s θ 6
T 7 = α 7 Y ¯ r , r s s μ x μ x + θ 7 ( X ¯ n , r s s μ x )
T 8 = α 8 Y ¯ r , r s s μ x μ x + θ 8 ( X ¯ r , r s s μ x )
T 9 = α 9 Y ¯ r , r s s X ¯ n , r s s X ¯ n , r s s + θ 9 ( X ¯ r , r s s X ¯ n , r s s )
where α i and θ i ; i = 1 , 2 , , 9 are suitably chosen scalars.
Theorem 1.
The M S E of the consequent estimators comprising the suggested imputation methods are
M S E ( T 1 ) = ( α 1 1 ) 2 μ y 2 + α 1 2 μ y 2 ( γ * C y 2 W y 2 * ) + θ 1 2 μ x 2 ( γ C x 2 W x 2 ) + 2 α 1 θ 1 μ x μ y ( γ ρ x y C x C y W x y )
M S E ( T 2 ) = ( α 2 1 ) 2 μ y 2 + α 2 2 μ y 2 ( γ * C y 2 W y 2 * ) + θ 2 2 μ x 2 ( γ * C x 2 W x 2 * ) + 2 α 2 θ 2 μ x μ y ( γ * ρ x y C x C y W x y * )
M S E ( T 3 ) = ( α 3 1 ) 2 μ y 2 + α 3 2 μ y 2 ( γ * C y 2 W y 2 * ) + θ 3 2 μ x 2 ( γ * C x 2 W x 2 * γ C x 2 + W x 2 ) + 2 α 3 θ 3 μ x μ y ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y )
M S E ( T 4 ) = μ y 2 1 + α 4 2 1 + γ * C y 2 W y 2 * + θ 4 ( 2 θ 4 + 1 ) ( γ C x 2 W x 2 ) 4 θ 4 ( γ ρ x y C x C y W x y ) 2 α 4 1 θ 4 ( γ ρ x y C x C y W x y ) + θ 4 ( θ 4 + 1 ) 2 ( γ C x 2 W x 2 )
M S E ( T 5 ) = μ y 2 1 + α 5 2 1 + γ * C y 2 W y 2 * + θ 5 ( 2 θ 5 + 1 ) ( γ * C x 2 W x 2 * ) 4 θ 5 ( γ * ρ x y C x C y W x y * ) 2 α 5 1 θ 5 ( γ * ρ x y C x C y W x y * ) + θ 5 ( θ 5 + 1 ) 2 ( γ * C x 2 W x 2 * )
M S E ( T 6 ) = μ y 2 1 + α 6 2 1 + γ * C y 2 W y 2 * + θ 6 ( 2 θ 6 + 1 ) ( γ * C x 2 W x 2 * γ C x 2 + W x 2 ) 4 θ 6 ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) 2 α 6 1 θ 6 ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) + θ 6 ( θ 6 + 1 ) 2 ( γ * C x 2 W x 2 * γ C x 2 + W x 2 )
M S E ( T 7 ) = μ y 2 1 + α 7 2 1 + γ * C y 2 W y 2 * + 3 θ 7 2 ( γ C x 2 W x 2 ) 4 θ 7 ( γ ρ x y C x C y W x y ) 2 α 7 1 + θ 7 2 ( γ C x 2 W x 2 ) θ 7 ( γ ρ x y C x C y W x y )
M S E ( T 8 ) = μ y 2 1 + α 8 2 1 + γ * C y 2 W y 2 * + 3 θ 8 2 ( γ * C x 2 W x 2 * ) 4 θ 8 ( γ * ρ x y C x C y W x y * ) 2 α 8 1 + θ 8 2 ( γ * C x 2 W x 2 * ) θ 8 ( γ * ρ x y C x C y W x y * )
M S E ( T 9 ) = μ y 2 1 + α 9 2 1 + γ * C y 2 W y 2 * + 3 θ 9 2 ( γ * C x 2 W x 2 * γ C x 2 + W x 2 ) 4 θ 9 ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) 2 α 9 1 + θ 9 2 ( γ * C x 2 W x 2 * γ C x 2 + W x 2 ) θ 9 ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y )
Proof. 
The precis of the derivations are given in Appendix B for quick reference. □
Corollary 1.
The minimum M S E of the consequent estimators comprising the suggested imputation methods are given by
m i n M S E ( T i ) = μ y 2 ( 1 α i ( o p t ) ) ; i = 1 , 2 , 3 , 7 , 8 , 9
m i n M S E ( T j ) = μ y 2 1 α j ( o p t ) A j ; j = 4 , 5 , 6
Proof. 
A summary of the derivations and the definition of the parametric function A j are given in Appendix B. □

5. Efficiency Conditions

By successively comparing the M S E s of the suggested imputation methods y i 1 to y i 9 regarding the other existing imputation methods proposed by [21,22], we obtain the following efficiency conditions.
(i).
From (31) and (A1)
α i ( o p t ) > 1 γ * C y 2 + W y 2 * ; i = 1 , 2 , 3 , 7 , 8 , 9
(ii).
From (32) and (A1)
α j ( o p t ) > 1 A j 1 γ * C y 2 + W y 2 * ; j = 4 , 5 , 6
(iii).
From (31) and (A2)
α i ( o p t ) > 1 γ * C y 2 + W y 2 * 1 B R 2 ( γ C x 2 W x 2 ) ; i = 1 , 2 , 3 , 7 , 8 , 9
(iv).
From (32) and (A2)
α j ( o p t ) > 1 A j 1 γ * C y 2 + W y 2 * 1 B R 2 ( γ C x 2 W x 2 ) ; j = 4 , 5 , 6
(v).
From (31) and (A3)
α i ( o p t ) > 1 γ * C y 2 + W y 2 * γ * C x 2 + W x 2 * + B R ( γ * ρ x y C x C y W x y * ) ; i = 1 , 2 , 3 , 7 , 8 , 9
(vi).
From (32) and (A3)
α j ( o p t ) > 1 A j 1 γ * C y 2 + W y 2 * γ * C x 2 + W x 2 * + B R ( γ * ρ x y C x C y W x y * ) ; j = 4 , 5 , 6
(vii).
From (31) and (A4)
α i ( o p t ) > 1 γ * C y 2 + W y 2 * 1 + B R 2 ( γ * C x 2 W x 2 * ) + 2 1 + B R ( γ * ρ x y C x C y W x y * ) ; i = 1 , 2 , 3 , 7 , 8 , 9
(viii).
From (32) and (A4)
α j ( o p t ) > 1 A j 1 γ * C y 2 + W y 2 * 1 + B R 2 ( γ * C x 2 W x 2 * ) + 2 1 + B R ( γ * ρ x y C x C y W x y * ) ; j = 4 , 5 , 6
(ix).
From (31) and (A8)
α i ( o p t ) > 1 γ * C y 2 + W y 2 * + ( γ ρ x y C x C y W x y ) 2 ( γ C x 2 W x 2 ) ; i = 1 , 2 , 3 , 7 , 8 , 9
(x).
From (32) and (A8)
α j ( o p t ) > 1 A j 1 γ C y 2 + W y 2 + ( γ ρ x y C x C y W x y ) 2 ( γ * C x 2 W x 2 * ) ; j = 4 , 5 , 6
(xi).
From (31) and (A9)
α i ( o p t ) > 1 γ * C y 2 + W y 2 * + ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) 2 ( γ * C x 2 W x 2 * γ C x 2 + W x 2 ) ; i = 1 , 2 , 3 , 7 , 8 , 9
(xii).
From (32) and (A9)
α j ( o p t ) > 1 A j 1 γ * C y 2 + W y 2 * + ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) 2 ( γ * C x 2 W x 2 * γ C x 2 + W x 2 ) ; j = 4 , 5 , 6
It is notable that only under these conditions, we can ensure the efficiency of the suggested imputation methods. Furthermore, we observed that these conditions are usually satisfied in various populations.

6. Computational Study

To enhance the soundness of the efficiency conditions obtained in the previous section, a computational study was designed in three subsections, namely, a numerical analysis based on a real population, a simulation analysis based on an artificially generated population, and a discussion of the computational findings.

6.1. Numerical Study

In this subsection, a numerical study is performed and the performance of the proposed imputation methods is compared with existing imputation methods. The numerical analysis was accomplished on four real datasets. Population 1 was taken from [24], where the level of apple production was taken as the study variable and the number of apple trees taken as the auxiliary variable in 69 villages of the South Anatolia region of Turkey in 1999. Population 2 was taken from [25], where the population (in millions) in 1983 was considered as the study variable and the export (in millions of U.S. dollars) was considered the auxiliary variable. Population 3 was taken from [26], where the amount (in U.S. dollars) of real estate farm loans in different states during 1997 was considered as the study variable and the amount (in U.S. dollars) of non-real estate farm loans in different states during 1997 was considered the auxiliary variable. Population 4 was taken from [25], where the total number of seats in the municipal council in 1982 was considered the study variable and the number of conservative seats in the municipal council in 1982 was considered the auxiliary variable. The necessary values of the parameters for all four populations are reported in Table 1.
The percent relative efficiency (PRE) of the proposed imputation methods regarding the conventional imputation methods was calculated using the following formula:
P R E = M S E ( t m ) M S E ( T ) × 100
where T = t m , t s i , i = 1 , 2 , , 6 , and T i , i = 1 , 2 , , 9 . The results of the numerical analysis are summarized in Table 2 and depicted in Figure 1 under strategies I, II, III and III for each population.

6.2. Simulation Analysis

To assess the performance of the suggested imputation methods, following [27], simulation experiments were conducted over three parent populations, namely, Normal, Gamma, and Weibull, of size N = 1000 units with variables X and Y, expressed by
Y = 2.9 + ( 1 ρ x y 2 ) Y * + ρ x y S y S x X * X = 2.5 + X *
where X * and Y * are independent variables of the corresponding parent population. The sampling methodology of Section 2 was used to draw an RSS of size 12 units with set size 3 from each parent population. Using 20,000 iterations, the PRE of the consequent estimators compared to the conventional mean estimator were computed as
P R E = 1 20 , 000 i = 1 20 , 000 ( t m μ y ) 2 1 20 , 000 i = 1 20 , 000 ( T μ y ) 2 × 100
The outcomes of the simulation experiments are reported in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 by their PRE for each sensibly opted values of response probability P = 0.2 , 0.3 , 0.4 , 0.5 , 0.6 , 0.7 , 0.8 with corresponding correlation coefficient ρ x y = 0.6 , 0.7 , 0.8 , 0.9 .

6.3. Discussion of Computational Findings

After carefully observing the findings reported in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, we discuss the following points:
(i).
From the findings of Table 2, the proposed imputation methods y . i j , j = 1 , 2 , , 9 outperform the mean imputation methods, ref. [21] imputation methods and ref. [22] imputation methods in each real population. Furthermore, the proposed imputation methods y . i j , j = 1 , 3 , 7 , 9 were superior among the proposed imputation methods in population 1, whereas the proposed imputation methods y . i j , j = 4 , 5 , 6 were superior among the proposed imputation methods in populations 2–4. This is easily observed in Figure 1.
(ii).
From the findings of Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, the proposed imputation methods y . i j , j = 1 , 2 , , 9 are also better than the mean imputation, ref. [21] imputation methods and ref. [22] imputation methods under both the symmetric and asymmetric populations for different correlation coefficients ρ x y , coefficients of skewness β 1 and coefficients of kurtosis β 2 .
(iii).
When the parent population was normal (symmetric) and Weibull (asymmetric), the proposed ratio-type imputation methods y . i j , j = 4 , 5 , 6 always performed better than the competitors as well as within the proposed class of imputation methods under strategies I, II and III.
(iv).
When the parent population was Gamma (asymmetric), the proposed difference- and ratio-type imputation methods y . i j , j = 1 , 2 , 3 , 7 , 8 , 9 were equally efficient and outperformed the conventional methods and performed better in comparison with the proposed imputation methods under strategies I, II and III.
(v).
The suggested imputation methods performed better in strategy II compared to strategies I and III in the real and artificially generated populations.
(vi).
It can be easily seen that the PRE decreases with the increase in asymmetry and peakedness for asymmetric distributions such as Gamma and Weibull.
(vii).
Moreover, the numerical analysis is summarized in Table 2 and Figure 1 under strategies I, II, and III for real populations 1–4. The PRE of the consequent estimators for the remaining simulation results in Table 3, Table 4, Table 5, Table 6 and Table 7 exhibit the same pattern and can be easily presented as line diagrams, if required.

7. Conclusions

In this manuscript, we proposed efficient difference- and ratio-type imputation methods for the estimation of the population mean in the presence of missing data. The efficiency conditions have been derived and sustained with computational analysis on some real and hypothetically generated symmetric and asymmetric populations. The computational and theoretical results show that the proposed imputation methods y . i j , j = 1 , 2 , , 9 outperformed the mean imputation method y . i , ref. [21] imputation methods y ¯ K C i , i = 1, 2, 3, and ref. [22] imputation methods y . i s j , j = 1 , 2 , , 6 .
In the simulation analysis, we considered one family of a symmetric population, namely, Normal, and two families of asymmetric populations, namely, Gamma and Weibull, to ascertain the effect of the correlation coefficient for a symmetric population and the effect of skewness and kurtosis for asymmetric populations.
It is worth mentioning that among the asymmetric populations, all imputation methods exhibited a decreasing trend in PRE as the coefficient of skewness and kurtosis increased. Although, in such cases, the proposed estimators fared better than their conventional counterparts. These results are in agreement with the results of [17,28,29], where these authors took skewed distributions and reported that the efficiency of the estimators decreased with an increase in skewness and kurtosis. The same was also true for imputation as well.
Lastly, the proposed imputation methods currently provide the best possible imputation methods for the estimation of a population mean in the presence of the missing data.
Furthermore, the proposed imputation strategies can be defined using multi-auxiliary information, which our future research with investigate.

Author Contributions

Supervision, S.B.; conceptualization, S.B. and A.K.; methodology, S.B. and A.K.; software, A.K.; validation, S.B.; writing—original draft preparation, A.K.; writing—review and editing, A.K., S.B., T.Z. and A.A.M.; Funding, T.Z. and A.A.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research is supported by researchers supporting project number (PNURSP2023R368), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Data Availability Statement

The article includes all data utilized for this investigation.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R368), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

The expressions of M S E , minimum M S E , and the optimum scalar values of the existing resultant estimators is reported below.
V ( t m ) = μ y 2 ( γ * C y 2 W y 2 * )
M S E ( y ¯ K C 1 ) μ y 2 ( γ * C y 2 W y 2 * ) + ( R 2 B 2 ) μ x 2 ( γ C x 2 W x 2 )
M S E ( y ¯ K C 2 ) = μ y 2 ( γ * C y 2 W y 2 * ) + μ x 2 ( γ * C x 2 W x 2 * ) B μ x μ y ( γ * ρ x y C x C y W x y * )
M S E ( y ¯ K C 3 ) = μ y 2 ( γ * C y 2 W y 2 * ) + ( R + B ) 2 μ x 2 ( γ * C x 2 W x 2 * ) 2 ( R + B ) μ x μ y ( γ * ρ x y C x C y W x y * )
M S E ( t s i ) = μ y 2 γ * C y 2 W y 2 * + β i 2 ( γ C x 2 W x 2 ) 2 β i ( γ ρ x y C x C y W x y ) , i = 1 , 4
M S E ( t s i ) = μ y 2 γ * C y 2 W y 2 * + β i 2 ( γ * C x 2 W x 2 * ) 2 β i ( γ * ρ x y C x C y W x y * ) , i = 2 , 5
M S E ( t s i ) = μ y 2 γ * C y 2 W y 2 * + β i 2 γ * C x 2 W x 2 * γ C x 2 + W x 2 2 β i γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y , i = 3 , 6
m i n M S E ( t s i ) = μ y 2 γ * C y 2 W y 2 * ( γ ρ x y C x C y W x y ) 2 ( γ C x 2 W x 2 ) ; i = 1 , 4
m i n M S E ( t s i ) = μ y 2 γ * C y 2 W y 2 * ( γ * ρ x y C x C y W x y * ) 2 ( γ * C x 2 W x 2 * ) ; i = 2 , 5
m i n M S E ( t s i ) = μ y 2 γ * C y 2 W y 2 * γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y 2 γ * C x 2 W x 2 * γ C x 2 + W x 2 ; i = 3 , 6
To obtain the minimum M S E s , the optimum scalar values associated with the estimators discussed in Section 3 are given below.
B = S x y S x 2 , β 1 ( o p t ) = β 4 ( o p t ) = γ ρ x y C x C y W x y γ C x 2 W x 2 , β 2 ( o p t ) = β 5 ( o p t ) = γ * ρ x y C x C y W x y * γ * C x 2 W x 2 * , β 3 ( o p t ) = β 6 ( o p t ) = ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) ( γ * C x 2 W x 2 * γ C x 2 + W x 2 )

Appendix B

In this section, we outline the proof of Theorem 1 and Corollary 1.
Under strategy I, consider the estimator
T 1 = α 1 Y ¯ r , r s s + θ 1 ( X ¯ n , r s s μ x )
Using the notations discussed in the earlier section, we obtain
T 1 μ y = ( α 1 1 ) μ y + α 1 μ y ϵ 0 + θ 1 μ x ϵ 1
Squaring both sides of (A11) and taking the expectation, we obtain the M S E of the estimator as
M S E ( T 1 ) = ( α 1 1 ) 2 μ y 2 + α 1 2 μ y 2 ( γ * C y 2 W y 2 * ) + θ 1 2 μ x 2 ( γ C x 2 W x 2 ) + 2 α 1 θ 1 μ x μ y ( γ ρ x y C x C y W x y )
The optimum values of α 1 and θ 1 can be obtained by minimizing (A12) with respect to α 1 and θ 1 as
α 1 ( o p t ) = 1 1 + γ * C y 2 W y 2 * ( γ ρ x y C x C y W x y ) 2 ( γ C x 2 W x 2 ) = α 7 ( o p t )
and θ 1 ( o p t ) = μ y μ x ( γ ρ x y C x C y W x y ) ( γ C x 2 W x 2 ) α 1 ( o p t )
Introducing α 1 ( o p t ) and θ 1 ( o p t ) into (A12), we obtain the minimum M S E as
m i n M S E ( T 1 ) = μ y 2 ( 1 α 1 ( o p t ) )
Similarly, we can obtain the optimum values of constants and minimum M S E s of other proposed estimators, which are
α 2 ( o p t ) = 1 1 + ( γ * C y 2 W y 2 * ) ( γ * ρ x y C x C y W x y * ) 2 ( γ * C x 2 W x 2 * ) = α 8 ( o p t )
θ 2 ( o p t ) = μ y μ x ( γ * ρ x y C x C y W x y * ) ( γ * C x 2 W x 2 * ) α 2 ( o p t )
α 3 ( o p t ) = 1 1 + γ * C y 2 W y 2 * ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) 2 ( γ C x 2 W x 2 γ * C x 2 + W x 2 * ) = α 9 ( o p t )
θ 3 ( o p t ) = μ y μ x γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y γ C x 2 W x 2 γ * C x 2 + W x 2 * α 3 ( o p t )
α j ( o p t ) = A j B j ; j = 4 , 5 , 6
θ j ( o p t ) = ( γ ρ x y C x C y W x y ) ( γ C x 2 W x 2 ) ; j = 4 , 7
θ j ( o p t ) = ( γ * ρ x y C x C y W x y * ) ( γ * C x 2 W x 2 * ) ; j = 5 , 8
θ j ( o p t ) = ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) ( γ C x 2 W x 2 γ * C x 2 + W x 2 * ) ; j = 6 , 9
where
A 4 = 1 + ( γ ρ x y C x C y W x y ) 2 ( γ ρ x y C x C y W x y ) 2 2 ( γ C x 2 W x 2 ) , B 4 = 1 + γ * C y 2 W y 2 * + γ ρ x y C x C y W x y 2 ( γ ρ x y C x C y W x y ) 2 ( γ C x 2 W x 2 ) , A 5 = 1 + ( γ * ρ x y C x C y W x y * ) 2 ( γ * ρ x y C x C y W x y * ) 2 2 ( γ * C x 2 W x 2 * ) , B 5 = 1 + γ * C y 2 W y 2 * + γ * ρ x y C x C y W x y * 2 ( γ * ρ x y C x C y W x y * ) 2 ( γ * C x 2 W x 2 * ) , A 6 = 1 1 2 ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) 2 ( γ * C x 2 W x 2 * γ C x 2 + W x 2 ) + 1 2 ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) , B 6 = 1 + γ * C y 2 W y 2 * + γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y 2 ( γ * ρ x y C x C y W x y * γ ρ x y C x C y + W x y ) 2 ( γ * C x 2 W x 2 * γ C x 2 + W x 2 ) .

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Figure 1. PRE results of the consequent estimators under (a) strategy I, (b) strategy II, and (c) strategy III for the real populations reported in Table 2.
Figure 1. PRE results of the consequent estimators under (a) strategy I, (b) strategy II, and (c) strategy III for the real populations reported in Table 2.
Axioms 12 00558 g001
Table 1. Description of the population parameters.
Table 1. Description of the population parameters.
ParametersPopulation 1Population 2Population 3Population 4
N6912450284
n12121212
m3333
r4444
P0.50.30.70.6
μ y 71.3436.65555.4347.50
μ x 3165.0214276.03878.169.05
S y 110.85116.80584.8211.06
S x 3965.2431431.8110846774.95
ρ x y 0.910.230.800.66
Table 2. PRE of the proposed estimators for real populations.
Table 2. PRE of the proposed estimators for real populations.
EstimatorsPopulation 1Population 2Population 3Population 4
t m 100.00100.00100.00100.00
Strategy I
T 1 200.23379.07213.14266.61
T 4 198.93388.40216.47276.07
T 7 200.23379.07213.14266.61
t s i , i = 1, 4169.41102.59199.16233.60
y ¯ k c 1 111.9389.6072.2578.77
Strategy II
T 2 584.83385.66360.332169.67
T 5 576.85421.37369.292459.06
T 8 584.83385.66360.332169.67
t s i , i = 2, 5554.02109.18346.352136.65
y ¯ k c 2 118.1572.9968.4469.78
Strategy III
T 3 220.06380.63126.93167.17
T 6 218.39395.85127.34169.46
T 9 223.56368.69125.09160.86
t s i , i = 3, 6189.24104.14112.94134.15
y ¯ k c 3 98.9179.9989.1272.58
Table 3. PRE of the proposed estimators at P = 0.20.
Table 3. PRE of the proposed estimators at P = 0.20.
ρ xy 0.60.70.80.9
t m 100100100100
X * N ( 20 , 25 )
Y * N ( 30 , 35 )
Strategy I
T 1 108.2314110.1882113.1969118.9193
T 4 108.9253110.9860114.1003119.9397
t s i , i = 1, 4102.4440102.7442102.8370102.7567
y ¯ k c 1 56.336562.528070.055878.6946
Strategy II
T i , i = 2, 8119.3313122.8567126.3605131.6545
T 5 123.7161127.9222132.0179137.8729
t s i , i = 2, 5113.5438115.4126116.0005115.492
y ¯ k c 2 20.586025.172032.157643.0168
Strategy III
T i , i = 3 , 9 116.3368119.4054122.7634128.1837
T 6 119.6366123.2108127.0276132.9048
t s i , i = 3, 6110.5493111.9614112.4035112.0212
y ¯ k c 3 21.769927.379835.986449.5031
X * G a m m a ( 5.6 , 0.9 )
Y * G a m m a ( 6.9 , 0.9 )
Skewness of Y0.60460.61340.65710.7487
Kurtosis of Y3.41843.43083.53933.7750
Strategy I
T i , i = 1 , 7 104.1361104.0201103.8752103.7148
T 4 103.9699103.8591103.7210103.5712
t s i , i = 1, 4102.2484102.1062101.8782101.5178
y ¯ k c 1 84.483884.586785.039986.0938
Strategy II
T i , i = 2, 8114.2407113.4139112.1508110.2765
T 5 113.4229112.6215111.3919109.5685
t s i , i = 2, 5112.3531111.5000110.1539108.0795
y ¯ k c 2 52.544452.736553.602655.6966
Strategy III
T i , i = 3 , 9 111.5319110.9071109.9584108.5579
T 6 110.8735110.2693109.3479107.9888
t s i i = 3, 6109.6442108.9931107.9614106.3608
y ¯ k c 3 50.072950.187850.977852.9983
X * W b ( 0.945 , 1.0 )
Y * W b ( 0.953 , 0.99 )
Skewness of Y1.35611.35261.47141.7607
Kurtosis of Y5.22765.28446.05357.8079
Strategy I
T i , i = 1, 7107.3833107.2255107.1537107.1449
T 4 107.5074107.3496107.2772107.2674
t s i , i = 1, 4105.6587105.0803104.5253103.9469
y ¯ k c 1 80.593383.948386.620288.7951
Strategy II
T i , i = 2, 8138.2957134.0250130.2555126.6323
T 5 139.1752134.8628131.0549127.3948
t s i , i = b2, 5136.5711131.8799127.6271123.4344
y ¯ k c 2 46.431552.364457.804262.7748
Strategy III
T i , i = 3 , 9 128.9875126.1204123.5728121.1061
T 6 129.6256126.7374124.1689121.6812
t s i , i = 3, 6127.2628123.9752120.9444117.9082
y ¯ k c 3 55.502762.580668.795374.1058
Bold values indicate maximum PRE.
Table 4. PRE of the proposed estimators at P = 0.30.
Table 4. PRE of the proposed estimators at P = 0.30.
ρ xy 0.60.70.80.9
t m 100100100100
X * N ( 20 , 25 )
Y * N ( 30 , 35 )
Strategy I
T i , i = 1, 7107.5696109.1362111.2233114.9679
T 4 108.2824109.9559112.1497116.0109
t s i , i = 1, 4103.7113104.1735104.3167104.1929
y ¯ k c 1 46.241252.661360.932971.1186
Strategy II
T i , i = 2, 8117.4021120.3753122.9071126.2670
T 5 120.3133123.7369126.6603130.3914
t s i , i = 2, 5113.5438115.4126116.0005115.4920
y ¯ k c 2 20.586025.172032.157643.0168
Strategy III
T i , i = 3 , 9 112.9688115.2747117.5940121.1378
T 6 114.8323117.4214120.0030123.8138
t s i , i = 3, 6109.1105110.3120110.6873110.3628
y ¯ k c 6 21.769927.379835.986449.5031
X * G a m m a ( 5.6 , 0.9 )
Y * G a m m a ( 6.9 , 0.9 )
Skewness of Y0.60460.61340.65710.7487
Kurtosis of Y3.41843.43083.53933.7750
Strategy I
T i , i = 1, 7104.6694104.4689104.1753103.7588
T 4 104.5033104.3080104.0213103.6153
t s i , i = 1, 4103.4110103.1930102.8440102.2941
y ¯ k c 1 78.401478.534479.121680.4968
Strategy II
T i , i = 2, 8113.6115112.7759111.4852109.5442
T 5 113.0660112.2474110.9790109.0720
t s i , i = 2, 5112.3531111.5000110.1539108.0795
y ¯ k c 2 52.544452.736553.602655.6966
Strategy III
T i , i = 3 , 9 109.5966109.0575108.2289106.9865
T 6 109.2114108.6844107.8719106.6538
t s i , i = 3, 6108.3381107.7815106.8976105.5218
y ¯ k c 3 50.072950.187850.977852.9983
X * W b ( 0.945 , 1.0 )
Y * W b ( 0.953 , 0.99 )
Skewness of Y1.35611.35261.47141.7607
Kurtosis of Y5.22765.28446.05357.8079
Strategy I
T i , i = 1, 7109.8849109.2492108.6973108.1715
T 4 110.0138109.3775108.8245108.2972
t s i , i = 1, 4108.7352107.8191106.9450106.0396
y ¯ k c 1 73.464877.711381.188884.0843
Strategy II
T i , i = 2, 8137.7209133.3100129.3794125.5664
T 5 138.3068133.8682129.9120126.0744
t s i , i = 2, 5136.5711131.8799127.6271123.4344
y ¯ k c 2 46.431552.364457.804262.7748
Strategy III
T i , i = 3 , 9 124.2186121.7980119.6111117.4585
T 6 124.574122.1439119.9472117.7845
t s i , i = 3, 6123.0688120.3679117.8588115.3266
y ¯ k c 3 55.502762.580668.795374.1058
Bold values indicate maximum PRE.
Table 5. PRE of the proposed estimators at P = 0.40.
Table 5. PRE of the proposed estimators at P = 0.40.
ρ xy 0.60.70.80.9
t m 100100100100
X * N ( 20 , 25 )
Y * N ( 30 , 35 )
Strategy I
T i , i = 1 , 7 107.9041109.3652111.0196113.7510
T 4 108.6366110.2077111.9699114.8175
t s i , i = 1 , 4 105.0104105.6432105.8396105.6698
y ¯ k c 1 39.214245.484153.912264.8732
Strategy II
T i , i = 2 , 8 116.4375119.1347121.1805123.5733
T 5 118.6164121.6501123.9885126.6586
t s i , i = 2 , 5 113.5438115.4126116.0005115.4920
y ¯ k c 2 20.586025.172032.157643.0168
Strategy III
T i , i = 3 , 9 110.6024112.4326114.2028116.8341
T 6 111.7649113.7708115.7070118.5106
t s i , i = 3 , 6 107.7087108.7106109.0228108.7528
y ¯ k c 3 21.769927.379835.986449.5031
X * G a m m a ( 5.6 , 0.9 )
Y * G a m m a ( 6.9 , 0.9 )
Skewness of Y0.60460.61340.65710.7487
Kurtosis of Y3.41843.43083.53933.7750
Strategy I
T i , i = 1 , 7 105.5441105.2600104.8268104.1809
T 4 105.3782105.0993104.6730104.0376
t s i , i = 1 , 4 104.6003104.3031103.8283103.0824
y ¯ k c 1 73.135973.290373.973475.5831
Strategy II
T i , i = 2 , 8 113.2969112.4569111.1524109.1780
T 5 112.8877112.0605110.7727108.8238
t s i , i = 2 , 5 112.3531111.5000110.1539108.0795
y ¯ k c 2 52.544452.736553.602655.6966
Strategy III
T i , i = 3 , 9 108.0067107.5535106.8530105.7945
T 6 107.7585107.3131106.6230105.5802
t s i , i = 3 , 6 107.0628106.5965105.8546104.6959
y ¯ k c 3 50.072950.187850.977852.9983
X * W b ( 0.945 , 1.0 )
Y * W b ( 0.953 , 0.99 )
Skewness of Y1.35611.35261.47141.7607
Kurtosis of Y5.22765.28446.05357.8079
Strategy I
T i , i = 1 , 7 112.8585111.7770110.7937109.8172
T 4 112.9926111.9098110.9247109.9461
t s i , i = 1 , 4 111.9962110.7044109.4795108.2182
y ¯ k c 1 67.494872.336976.398379.8482
Strategy II
T i , i = 2 , 8 137.4334132.9525128.9413125.0334
T 5 137.8727133.3710129.3406125.4143
t s i , i = 2 , 5 136.5711131.8799127.6271123.4344
y ¯ k c 2 46.431552.364457.804262.7748
Strategy III
T i , i = 3 , 9 120.0047118.0372116.2409114.4546
T 6 120.2233118.2512116.4501114.6584
t s i , i = 3 , 6 119.1424116.9646114.9267112.8556
y ¯ k c 3 55.502762.580668.795374.1058
Bold values indicate maximum PRE.
Table 6. PRE of the proposed estimators at P = 0.50.
Table 6. PRE of the proposed estimators at P = 0.50.
ρ xy 0.60.70.80.9
t m 100100100100
X * N ( 20 , 25 )
Y * N ( 30 , 35 )
Strategy I
T i , i = 1 , 7 108.6574110.1325111.5516113.6541
T 4 109.4103110.9988112.5270114.7449
t s i , i = 1 , 4 106.3424107.1549107.4076107.1891
y ¯ k c 1 34.041240.028648.342259.6361
Strategy II
T i , i = 2 , 8 115.8588118.3903120.1445121.9570
T 5 117.5997120.3998122.3876124.4215
t s i , i = 2 , 5 113.5438115.4126116.0005115.4920
y ¯ k c 2 20.586025.172032.157643.0168
Strategy III
T i , i = 3 , 9 108.6574110.1325111.5516113.6541
T 6 109.4103110.9988112.5270114.7449
t s i , i = 3 , 6 106.3424107.1549107.4076107.1891
y ¯ k c 3 21.769927.379835.986449.5031
X * G a m m a ( 5.6 , 0.9 )
Y * G a m m a ( 6.9 , 0.9 )
Skewness of Y0.60460.61340.65710.7487
Kurtosis of Y3.41843.43083.53933.7750
Strategy I
T i , i = 1 , 7 106.5723106.2029105.6304104.7617
T 4 106.4066106.0424105.4768104.6186
t s i , i = 1 , 4 105.8172105.4373104.8316103.8829
y ¯ k c 1 68.533268.702769.454371.2347
Strategy II
T i , i = 2 , 8 113.1081112.2655110.9527108.9583
T 5 112.7807111.9483110.6489108.6749
t s i , i = 2 , 5 112.3531111.5000110.1539108.0795
y ¯ k c 2 52.544452.736553.602655.6966
Strategy III
T i , i = 3 , 9 106.5723106.2029105.6304104.7617
T 6 106.4066106.0424105.4768104.6186
t s i , i = 3 , 6 105.8172105.4373104.8316103.8829
y ¯ k c 3 50.072950.187850.977852.9983
X * W b ( 0.945 , 1.0 )
Y * W b ( 0.953 , 0.99 )
Skewness of Y1.35611.35261.47141.7607
Kurtosis of Y5.22765.28446.05357.8079
Strategy I
T i , i = 1 , 7 116.1487114.6065113.1884113.1884
T 4 116.2883114.7441113.3234113.3234
t s i , i = 2 , 4 115.4588113.7484112.1370112.1370
y ¯ k c 1 62.422267.657972.141672.1416
Strategy II
T i , i = 2 , 8 137.2610132.7379128.6785128.6785
T 5 137.6123133.0727128.9979128.9979
t s i , i = 2 , 5 136.5711131.8799127.6271127.6271
y ¯ k c 2 46.431552.364457.804257.8042
Strategy III
T i , i = 3 , 9 116.1487114.6065113.1884111.7674
T 6 116.2883114.7441113.3234111.8997
t s i , i = 3 , 6 115.4588113.7484112.137110.4883
y ¯ k c 3 55.502762.580668.795368.7953
Bold values indicate maximum PRE.
Table 7. PRE of the proposed estimators at P = 0.60.
Table 7. PRE of the proposed estimators at P = 0.60.
ρ xy 0.60.70.80.9
t m 100100100100
X * N ( 20 , 25 )
Y * N ( 30 , 35 )
Strategy I
T i , i = 1 , 7 109.6378111.1919112.4761114.1403
T 4 110.4120112.0830113.4776115.2564
t s i , i = 1 , 4 107.7087108.7106109.0228108.7528
y ¯ k c 1 30.073935.741743.815455.1814
Strategy II
T i , i = 2 , 8 115.4729117.8940119.4538120.8795
T 5 116.9226119.5671121.3212122.9311
t s i , i = 2 , 5 113.5438115.4126116.0005115.4920
y ¯ k c 2 20.586025.172032.157643.0168
Strategy III
T i , i = 3 , 9 106.9395108.1245109.2929111.0573
T 6 107.4275108.6858109.926111.7676
t s i , i = 3 , 6 105.0104105.6432105.8396105.6698
y ¯ k c 3 21.769927.379835.986449.5031
X * G a m m a ( 5.6 , 0.9 )
Y * G a m m a ( 6.9 , 0.9 )
Skewness of Y0.60460.61340.65710.7487
Kurtosis of Y3.41843.43083.53933.7750
Strategy I
T i , i = 1 , 7 107.6921107.2345106.5202105.4283
T 4 107.5266107.0743106.3668105.2854
t s i , i = 1 , 4 107.0628106.5965105.8546104.6959
y ¯ k c 1 64.475664.655665.455567.3595
Strategy II
T i , i = 2 , 8 112.9823112.1380110.8195108.8119
T 5 112.7094111.8736110.5664108.5756
t s i , i = 2 , 5 112.3531111.5000110.1539108.0795
y ¯ k c 2 52.544452.736553.602655.6966
Strategy III
T i , i = 3 , 9 105.2295104.9411104.4940103.8147
T 6 105.1188104.8339104.3914103.7192
t s i , i = 3 , 6 104.6003104.3031103.8283103.0824
y ¯ k c 3 50.072950.187850.977852.9983
X * W b ( 0.945 , 1.0 )
Y * W b ( 0.953 , 0.99 )
Skewness of Y1.35611.35261.47141.7607
Kurtosis of Y5.22765.28446.05357.8079
Strategy I
T i , i = 1 , 7 119.7173117.6797115.8029113.9216
T 4 119.8630117.8223115.9423114.0574
t s i , i = 1 , 4 119.1424116.9646114.9267112.8556
y ¯ k c 1 58.058863.547468.334372.5392
Strategy II
T i , i = 2 , 8 137.1460132.5949128.5032124.5004
T 5 137.4387132.8739128.7694124.7542
t s i , i = 2 , 5 136.5711131.8799127.6271123.4344
y ¯ k c 2 46.431552.364457.804262.7748
Strategy III
T i , i = 3 , 9 112.571111.4195110.3556109.2842
T 6 112.6604111.5080110.4430109.3701
t s i , i = 3 , 6 111.9962110.7044109.4795108.2182
y ¯ k c 3 55.502762.580668.795374.1058
Bold values indicate maximum PRE.
Table 8. PRE of the proposed estimators at P = 0.70.
Table 8. PRE of the proposed estimators at P = 0.70.
ρ xy 0.60.70.80.9
t m 100100100100
X * N ( 20 , 25 )
Y * N ( 30 , 35 )
Strategy I
T i , i = 1 , 7 110.7640112.4389113.6473114.9807
T 4 111.5604113.3559114.6761116.1231
t s i , i = 1 , 4 109.1105110.3120110.6873110.3628
y ¯ k c 1 26.934832.284140.063851.3460
Strategy II
T i , i = 2 , 8 115.1973117.5395118.9605120.1099
T 5 116.4391118.9727120.5600121.8671
t s i , i = 2 , 5 113.5438115.4126116.0005115.4929
y ¯ k c 2 20.586025.172032.157643.0168
Strategy III
T i , i = 3 , 9 105.3648106.3004107.2767108.8108
T 6 105.6700106.6513107.6731109.2571
t s i , i = 3 , 6 103.7113104.1735104.3167104.1929
y ¯ k c 3 21.769927.379835.986449.5031
X * G a m m a ( 5.6 , 0.9 )
Y * G a m m a ( 6.9 , 0.9 )
Skewness of Y0.60460.61340.65710.7487
Kurtosis of Y3.41843.43083.53933.7750
Strategy I
T i , i = 1 , 7 108.8775108.3284107.4682106.1495
T 4 108.7123108.1684107.3151106.0069
t s i , i = 1 , 4 108.3381107.7815106.8976105.5218
y ¯ k c 1 60.871561.058861.892163.8842
Strategy II
T i , i = 2 , 8 112.8924112.0468110.7245108.7072
T 5 112.6585111.8202110.5074108.5048
t s i , i = 2 , 5 112.3531111.5000110.1539108.0795
y ¯ k c 2 52.544452.736553.602655.6966
Strategy III
T i , i = 3 , 9 103.9503103.7398103.4146102.9218
T 6 103.8791103.6708103.3486102.8603
t s i , i = 3 , 6 103.4110103.1930102.844102.2941
y ¯ k c 3 50.072950.187850.977852.9983
X * W b ( 0.945 , 1.0 )
Y * W b ( 0.953 , 0.99 )
Skewness of Y1.35611.35261.47141.7607
Kurtosis of Y5.22765.28446.05357.8079
Strategy I
T i , i = 1 , 7 123.5616120.9808118.6098116.2403
T 4 123.7138121.1290118.7538116.3799
t s i , i = 1 , 4 123.0688120.3679117.8588115.3266
y ¯ k c 1 54.265559.907764.908769.3645
Strategy II
T i , i = 2 , 8 137.0639132.4928128.3781124.3481
T 5 137.3148132.7318128.6062124.5657
t s i , i = 2 , 5 136.5711131.8799127.6271123.4344
y ¯ k c 2 46.431552.364457.804262.7748
Strategy III
T i , i = 3 , 9 109.2279108.432107.696106.9533
T 6 109.2831108.4869107.7505107.0071
t s i , i = 3 , 6 108.7352107.8191106.945106.0396
y ¯ k c 3 55.502762.580668.795374.1058
Bold values indicate maximum PRE.
Table 9. PRE of the proposed estimators at P = 0.80.
Table 9. PRE of the proposed estimators at P = 0.80.
ρ xy 0.60.70.80.9
t m 100100100100
X * N ( 20 , 25 )
Y * N ( 30 , 35 )
Strategy I
T i , i = 1 , 7 111.9961113.8224114.9935116.0618
T 4 112.8155114.7665116.0507117.2317
t s i i = 1 , 4 110.5493111.9614112.4035112.0212
y ¯ k c 1 24.389129.436536.904048.0091
Strategy II
T i , i = 2 , 8 114.9907117.2736118.5905119.5327
T 5 116.0767118.5271119.9893121.0694
t s i i = 2 , 5 113.5438115.4126116.0005115.4920
y ¯ k c 2 20.586025.172032.157643.0168
Strategy III
T i , i = 3 , 9 103.8908104.6052105.427106.7974
T 6 104.0641104.8043105.6524107.052
t s i , i = 3 , 6 102.444102.7442102.8370102.7567
y ¯ k c 3 21.769927.379835.986449.5031
X * G a m m a ( 5.6 , 0.9 )
Y * G a m m a ( 6.9 , 0.9 )
Skewness of Y0.60460.61340.65710.7487
Kurtosis of Y3.41843.43083.53933.7750
Strategy I
T i , i = 1 , 7 110.1161109.4716108.4607106.9101
T 4 109.9513109.3120108.3079106.7677
t s i , i = 1 , 4 109.6442108.9931107.9614106.3608
y ¯ k c 1 57.649157.841158.696760.7498
Strategy II
T i , i = 2 , 8 112.8250111.9785110.6531108.6288
T 5 112.6203111.7802110.4632108.4516
t s i , i = 2 , 5 112.3531111.5000110.1539108.0795
y ¯ k c 2 52.544452.736553.602655.6966
Strategy III
T i , i = 3 , 9 102.7203102.5847102.3774102.0671
T 6 102.6788102.5444102.3389102.0311
t s i , i = 3 , 6 102.2484102.1062101.8782101.5178
y ¯ k c 3 50.072950.187850.977852.9983
X * W b ( 0.945 , 1.0 )
Y * W b ( 0.953 , 0.99 )
Skewness of Y1.35611.35261.47141.7607
Kurtosis of Y5.22765.28446.05357.8079
Strategy I
T i , i = 1 , 7 127.6940124.5115121.6015118.7077
T 4 127.8533124.6656121.7504118.8513
t s i , i = 1 , 4 127.2628123.9752120.9444117.9082
y ¯ k c 1 50.937556.662461.810166.4561
Strategy II
T i , i = 2 , 8 137.0023132.4162128.2842124.2339
T 5 137.2218132.6253128.4838124.4243
t s i , i = 2 , 5 136.5711131.8799127.6271123.4344
y ¯ k c 2 46.431552.364457.804262.7748
Strategy III
T i , i = 3 , 9 106.0898105.6166105.1824104.7464
T 6 106.1208105.6476105.2132104.777
t s i , i = 3 , 6 105.6587105.0803104.5253103.9469
y ¯ k c 3 55.502762.580668.795374.1058
Bold values indicate maximum PRE.
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Bhushan, S.; Kumar, A.; Zaman, T.; Al Mutairi, A. Efficient Difference and Ratio-Type Imputation Methods under Ranked Set Sampling. Axioms 2023, 12, 558. https://doi.org/10.3390/axioms12060558

AMA Style

Bhushan S, Kumar A, Zaman T, Al Mutairi A. Efficient Difference and Ratio-Type Imputation Methods under Ranked Set Sampling. Axioms. 2023; 12(6):558. https://doi.org/10.3390/axioms12060558

Chicago/Turabian Style

Bhushan, Shashi, Anoop Kumar, Tolga Zaman, and Aned Al Mutairi. 2023. "Efficient Difference and Ratio-Type Imputation Methods under Ranked Set Sampling" Axioms 12, no. 6: 558. https://doi.org/10.3390/axioms12060558

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