Enhanced exponential ratio-cum-ratio estimator in ranked set sampling using transformed auxiliary information

Lakhkar Khan; Sohaib Ahmad; Abdullah Mohammed Alomair; Mohammed Ahmed Alomair

doi:10.1016/j.heliyon.2024.e27522

. 2024 Mar 7;10(6):e27522. doi: 10.1016/j.heliyon.2024.e27522

Enhanced exponential ratio-cum-ratio estimator in ranked set sampling using transformed auxiliary information

Lakhkar Khan ^a, Sohaib Ahmad ^b, Abdullah Mohammed Alomair ^c,^⁎, Mohammed Ahmed Alomair ^c

PMCID: PMC10944216 PMID: 38496881

Abstract

Estimation of population mean is a determined subject issue in sampling surveys and many efforts have been paid by various researchers to enhance the precision of the estimates by utilizing the correlated auxiliary information. In connection with this, we suggest an improved exponential ratio-cum-ratio estimator using transformed auxiliary variables under ranked set sampling scheme. Theoretical comparison between estimators is made in terms of mean square errors ( $M S E s$ ), percentage relative efficiencies ( $P R E s$ ), and percentage relative root mean squared error ( $P R R M S E$ ). The numerical expression for the bias and $M S E$ of the suggested estimator is derived up to first order of approximation. Based on the results of actual data sets and a simulation study, it is found that the suggested estimator perform well as compared to its existing counterparts.

Keywords: Ratio estimator, Exponential estimator, Transformed auxiliary variables, MSE, PRE

1. Introduction

Ranked set sampling $(R S S)$ is the modification of simple random sampling for bringing better precision in estimation. In sample survey, major concern of the Researchers is cost-effective sampling, particularly, when the features of variable of interest is time consuming and costly. In view of [[1], [2], [3]], sampling methods based on ranking are designed to utilize extra information from easily obtained and inexpensive sources to collection best representative samples compared to other methods. The authors in Ref. [4] adapted the efficient ratio estimators in rank set sampling. The authors in Ref. [5] recommended an unbiased estimators for population mean using ranked set sampling. The authors in Refs. [6,7] recommended an efficient class of estimators using idea of ranked set sampling. In this paper the properties of the proposed estimator are investigated against usual RSS mean estimator in Refs. [[8], [9], [10]]. The authors in Ref. [11] recommended a family of estimators for population proportion using idea of rank. The [12] was the first who placed his consistent efforts to introduce this method for efficient estimating the average pasture yield in forest. The [13] suggested a modified estimator for population mean utilizing ranked set sampling. The [14,15] adapted the efficient ratio estimators in $R S S$ scheme for mean and compared it to its counterpart estimators under simple random sampling and proved that RSS ratio estimators are more efficient than the later. The [16] was the first who used auxiliary information for the purpose of perfect ordering in $R S S$ . The [17] catered the mathematical theory for the ranked set sampling for the first time. The [18] give idea of ratio and product type estimators, when there is positive or negative correlation occur between the study and the auxiliary variable.

The [19] suggested a novel estimator for population mean using idea of rank under stratified random sampling. The [20] discussed a new family of estimators using rank idea for population distribution function. Some latest works done using rank set sampling see Refs. [[21], [22], [23], [24], [25], [26]]. In this article we have suggested an enhanced exponential ratio-cum-ratio estimator in ranked set sampling using transformed auxiliary information. In terms of mean squared errors, our suggested estimator has minimum mean squared error and higher percentage relative efficiency as compared to existing estimators.

2. Ranked set sampling in case of two auxiliary variables and its algorithm for sample selection

Ranked set sampling is a procedure either it is the case of one auxiliary variable or two auxiliary variables. It the first stage, units are selected randomly through SRS and then it 2nd stage, the units are ranked by any suited concomitant variable. Finally, units are drawn from the selected units.

Under this case, it first stage, from a tri-variate finite population, draw independent m² random samples, each of volume $m$ , through SRS without replacement along with equal probability. Then in each set, order the units either by correlated variable X or Z, where X and Z are concomitant variables, but the common practice is done according to the variable X. RSS procedure is then encounter to take the desired sample of size n = rm for the actual measurement. From the first set m, draw the unit of Y and Z attached with the smallest unit of X. From the next sample, take the value of Y and Z attached with the second smallest unit of X. Following the same approach, the procedure is maintained until the units of Y and Z attached with the highest unit of X from the set m. This make up one cycle of the procedure. The procedure is replicate r times to achieve the desired sample size of n-rm units for analysis.

In bullet form, the procedure in case of two auxiliary variables may be elaborated as under:

a.
Select randomly m² tri-variate sample units from the tri-variate finite population where m is set size that denotes the ranked number of samples in each set.
b.
Allocate randomly m² chosen units into m sets having size m.
c.
Within each set rank the units by either the auxiliary variable X or Z correlated with the variable of interest Y, but here is done according to variable X.
d.
Now start choosing units of Y and Z as choose the smallest ordered unit X in the set 1, then 2nd smallest X in set 2 along-with the associated values of Y and Z, and so on until the largest ordered unit X is selected for the last set. This make up one cycle.
e.
Replicate the above - mentioned steps for r cycles until the needed sample size i.e. n = mr is achieved.

3. General procedure for obtaining bias and mean square error

To derive the bias and MSE of the suggested estimators, we proceed as:

\begin{array}{c} {\overline{y}}_{[r s s]} = \overline{Y} (1 + e_{0}), & {\overline{x}}_{(r s s)} = \overline{X} (1 + e_{1}), {\overline{z}}_{(r s s)} = \overline{Z} (1 + e_{2}), \end{array}

Such that $(e_{p}) = 0,$ $(p = 0, 1, 2)$ ,

and

\begin{array}{c} E (e_{0}^{2}) = γ C_{y}^{2} - W_{(y)}^{2} = V_{200} & E (e_{1}^{2}) = γ C_{x}^{2} - W_{(x)}^{2} = V_{020}, \end{array}

\begin{array}{c} E (e_{2}^{2}) = γ C_{z}^{2} - W_{(z)}^{2} = V_{002} & E (e_{0} e_{1}) = γ C_{y x} - W_{(y x)} = V_{110} \end{array},

\begin{array}{c} E (e_{0} e_{2}) = γ C_{y z} - W_{(y z)} = V_{101} & E (e_{1} e_{2}) = γ C_{x z} - W_{(x z)} = V_{011}, \end{array}

\begin{array}{c} W_{(y)}^{2} = \frac{1}{m^{2} r {\overline{Y}}^{2}} \sum_{i = 1}^{m} τ_{y (i : m)}^{2} & W_{(x)}^{2} = \frac{1}{m^{2} r {\overline{X}}^{2}} \sum_{i = 1}^{m} τ_{x (i : m)}^{2}, \end{array}

\begin{array}{c} W_{(z)}^{2} = \frac{1}{m^{2} r {\overline{Z}}^{2}} \sum_{i = 1}^{m} τ_{z (i : m)}^{2} & \begin{array}{c} W_{(y x)} = \frac{1}{m^{2} r \overline{Y} \overline{X}} \sum_{i = 1}^{m} τ_{y x (i : m)} & W_{(y z)} = \frac{1}{m^{2} r \overline{Z} \overline{Y}} \sum_{i = 1}^{m} τ_{y z (i : m)} & \begin{array}{c} W_{(x z)} = \frac{1}{m^{2} r \overline{X} \overline{Z}} \sum_{i = 1}^{m} τ_{x z (i : m)} & τ_{y (i : m)} = (μ_{y (i : m)} - \overline{Y}) & τ_{x (i : m)} = (μ_{x (i : m)} - \overline{X}), \end{array} \end{array} \end{array}

\begin{array}{c} τ_{z (i : m)} = (μ_{z (i : m)} - \overline{Z}) & τ_{y x (i : m)} = (μ_{y (i : m)} - \overline{Y}) (μ_{x (i : m)} - \overline{X}), \end{array}

$τ_{y z (i : m)} = (μ_{y (i : m)} - \overline{Y}) (μ_{z (i : m)} - \overline{Z})$ , $τ_{x z (i : m)} = (μ_{z (i : m)} - \overline{Z}) (μ_{x (i : m)} - \overline{X})$ , $γ = (\frac{1}{m r})$ , $C_{y x} = ρ C_{y} C_{x}$ , $C_{y z} = ρ C_{y} C_{z}$ & $C_{x z} = ρ C_{x} C_{z}$ , where $C_{x}$ , $C_{y}$ and $C_{z}$ are the coefficients of variation of $X$ , $Y$ and $Z$ respectively. The values of $μ_{y (i : m)}$ , $μ_{z (i : m)}$ and $μ_{x (i : m)}$ depend on order statistics from some specific distributions (See Ref. [3]).

4. Existing estimators

The usual per unit mean estimator with its MSE is given as under:

{\overline{y}}_{1 (R S S)} = \frac{1}{m r} \sum_{j = 1}^{r} \sum_{i = 1}^{m} y_{[i : m] j},

The MSE of ${\overline{y}}_{1 (R S S)}$ is given in equation (1):

Equation 1.

(1)

Some of the members of [9] under RSS using two auxiliary variables as under:

{\overline{y}}_{2 (R S S)} = {\overline{y}}_{[r s s]} (\frac{\overline{X}}{{\overline{x}}_{(r s s)}}) \exp (\frac{\overline{Z} - {\overline{z}}_{(r s s)}}{\overline{Z} + {\overline{z}}_{(r s s)}}),

The MSE of ${\overline{y}}_{2 (R S S)}$ is given in equation (2):

Equation 2.

(2)

{\overline{y}}_{3 (R S S)} = {\overline{y}}_{[r s s]} (\frac{\overline{Z}}{{\overline{z}}_{(r s s)}}) \exp (\frac{\overline{X} - {\overline{x}}_{(r s s)}}{\overline{X} + {\overline{x}}_{(r s s)}}),

The MSE of ${\overline{y}}_{3 (R S S)}$ is given in equation (3):

Equation 3.

(3)

The authors in Ref. [10] also developed the following estimators under RSS in case of two auxiliary variables:

{\overline{y}}_{4 (R S S)} = {\overline{y}}_{[r s s]} \exp (\frac{\overline{X} - {\overline{x}}_{(r s s)}}{\overline{X} + {\overline{x}}_{(r s s)}}) \exp (\frac{\overline{Z} - {\overline{z}}_{(r s s)}}{\overline{Z} + {\overline{z}}_{(r s s)}}),

The MSE of ${\overline{y}}_{4 (R S S)}$ is given in equation (4):

Equation 4.

(4)

{\overline{y}}_{5 (R S S)} = {\overline{y}}_{[r s s]} \exp (\frac{\overline{X} - {\overline{x}}_{(r s s)}}{\overline{X}}) \exp (\frac{\overline{Z} - {\overline{z}}_{(r s s)}}{\overline{Z} + {\overline{z}}_{(r s s)}}),

The MSE of ${\overline{y}}_{5 (R S S)}$ is given in equation (5):

Equation 5.

(5)

{\overline{y}}_{6 (R S S)} = {\overline{y}}_{[r s s]} \exp (\frac{\overline{X} - {\overline{x}}_{(r s s)}}{\overline{X} + {\overline{x}}_{(r s s)}}) \exp (\frac{\overline{Z} - {\overline{z}}_{(r s s)}}{\overline{Z}}),

The MSE of ${\overline{y}}_{6 (R S S)}$ is given in equation (6):

Equation 6.

(6)

5. Suggested estimator

In this article we have suggested exponential ratio-cum-ratio estimator in ranked set sampling using transformed auxiliary information.

Let {\overline{T}}_{x (r s s)} = \frac{N \overline{X} - n {\overline{x}}_{(r s s)}}{N - n},

{\overline{T}}_{z (r s s)} = \frac{N \overline{Z} - n {\overline{z}}_{(r s s)}}{N - n},

{\overline{T}}_{x (r s s)} = (1 + g) \overline{X} - g {\overline{x}}_{(r s s)},

{\overline{T}}_{z (r s s)} = (1 + g) \overline{Z} - g {\overline{z}}_{(r s s)},

where $g = \frac{n}{N - n}$ .

Using the transformation ${\overline{T}}_{x (r s s)}$ and ${\overline{T}}_{z (r s s)}$ , the suggested exponential ratio-cum-ratio estimator in RSS is given in equation (7):

Equation 7.

(7)

where

\begin{array}{c} {\overline{y}}_{(r s s]} = \frac{\sum_{j = 1}^{r} \sum_{i = 1}^{m} y_{[i : m] j}}{m r} & {\overline{x}}_{(r s s)} = \frac{\sum_{j = 1}^{r} \sum_{i = 1}^{m} x_{[i : m] j}}{m r} & {\overline{z}}_{(r s s)} = \frac{\sum_{j = 1}^{r} \sum_{i = 1}^{m} z_{[i : m] j}}{m r} \end{array} .

Here $a$ and $b$ are some suitable chosen scalars whose values are to be determined, so that $M S E$ of ${\overline{y}}_{L (R S S)}$ is minimized. Also, $\overline{X}, \overline{Y}$ and $\overline{Z}$ are the population means of $X, Y$ and $Z$ respectively.

In error terms, we have the expression for equation (7), up to first order of approximation: ${\overline{y}}_{L (R S S)} = \overline{Y} (1 + e_{0}) {1 - \frac{1}{2} a g e_{1} - \frac{1}{4} a g^{2} e_{1}^{2} - \frac{1}{2} b g e_{2} - \frac{1}{4} b g^{2} e_{2}^{2} + \frac{1}{8} a^{2} g^{2} e_{1}^{2} + \frac{1}{8} b^{2} g^{2} e_{2}^{2} + \frac{1}{4} a b g^{2} e_{1} e_{2}}$ .

The simplified form is given in equation (8):

Equation 8.

(8)

The bias of ${\overline{y}}_{L (R S S)}$ , is given in equation (9):

Equation 9.

(9)

Squaring and taking expectation of equation (8):

Equation 10.

(10)

To get the $a$ and $b$ optimal values, differentiate equation (10) with respect to $a$ and $b$ respectively and equate to zero i.e

a_{(o p t)} = \frac{2 (V_{002} V_{110} - V_{011} V_{101})}{g (V_{020} V_{002} - V_{011}^{2})},

and

b_{(o p t)} = \frac{2 (V_{020} V_{101} - V_{011} V_{110})}{g (V_{020} V_{002} - V_{011}^{2})}

Putting the optimum values of $a$ and $b$ in equation (10), we get the optimum $M S E ({\overline{y}}_{L (R S S)})$ , which is given by:

M S E {({\overline{y}}_{L (R S S)})}_{(o p t)} = {\overline{Y}}^{2} [V_{200} + \frac{Ω_{1} - Ω_{2} + Ω_{3} - Ω_{4} - Ω_{5} + Ω_{6}}{{(V_{020} V_{002} - V_{011}^{2})}^{2}}] .

where

\begin{array}{c} Ω_{1} = V_{002} V_{110}^{2} V_{011}^{2} & Ω_{2} = V_{002} V_{020}^{2} V_{101}^{2} & \begin{array}{c} Ω_{3} = V_{020} V_{101}^{2} V_{011}^{2} & Ω_{4} = V_{020} V_{002}^{2} V_{110}^{2} & \begin{array}{c} Ω_{5} = 2 V_{110} V_{101} V_{011}^{3} and & Ω_{6} = 2 V_{110} V_{101} V_{011} V_{020} V_{002} \end{array} \end{array} \end{array}

6. Special cases of the proposed estimator

If $a = 0$ and $b = 0$ in equation (7), the proposed estimator reduced to per unit mean estimator of population mean under RSS i.e:

{\overline{y}}_{(r s s)} = \frac{1}{m r} \sum_{j = 1}^{r} \sum_{i = 1}^{m} y_{[i : m] j},

The MSE of ${\overline{y}}_{(r s s)}$ is given in equation (11):

Equation 11.

(11)

If $a = 1$ and $b = 1$ in equation (7), the following exponential ratio-cum ratio estimator may be obtained

{\overline{y}}_{L_{1} (R S S)} = {\overline{y}}_{[r s s]} \exp [(\frac{{\overline{T}}_{x (r s s)} - \overline{X}}{{\overline{T}}_{x (r s s)} + \overline{X}}) + (\frac{{\overline{T}}_{z (r s s)} - \overline{Z}}{{\overline{T}}_{z (r s s)} + \overline{Z}})],

The bias of ${\overline{y}}_{L_{1} (R S S)}$ is given by:

B i a s ({\overline{y}}_{L_{1} (R S S)}) = \overline{Y} [- \frac{1}{4} g^{2} V_{020} - \frac{1}{4} g^{2} V_{002} - \frac{1}{2} g V_{110} - \frac{1}{2} g V_{101} + \frac{1}{4} g^{2} V_{011}],

The $M S E$ of ${\overline{y}}_{L_{1} (R S S)}$ is given in equation (12):

Equation 12.

(12)

If $a = 1$ and $b = 0$ in equation (7), the following exponential ratio estimator may be obtained

{\overline{y}}_{L_{2} (R S S)} = {\overline{y}}_{[r s s]} \exp [(\frac{{\overline{T}}_{x (r s s)} - \overline{X}}{{\overline{T}}_{x (r s s)} + \overline{X}})],

The bias of ${\overline{y}}_{L_{2} (R S S)}$ is given by:

B i a s ({\overline{y}}_{L_{2} (R S S)}) = \overline{Y} [- \frac{1}{8} g^{2} V_{020} - \frac{1}{2} V_{110}]

The $MSE$ of ${\overline{y}}_{L_{2} (R S S)}$ is given in equation (13) by:

Equation 13.

(13)

If $a = 0$ and $b = 1$ in equation (7), the following exponential ratio estimator may be obtained

{\overline{y}}_{L_{3} (R S S)} = {\overline{y}}_{[r s s]} \exp [(\frac{{\overline{T}}_{z (r s s)} - \overline{Z}}{{\overline{T}}_{z (r s s)} + \overline{Z}})],

The bias of ${\overline{y}}_{L_{3} (R S S)}$ is given by:

B i a s ({\overline{y}}_{L_{3} (R S S)}) = \overline{Y} [- \frac{1}{8} g^{2} V_{002} - \frac{1}{2} g V_{101}],

The $M S E$ of ${\overline{y}}_{L_{3} (R S S)}$ is given in equation (14):

Equation 14.

(14)

7. Theoretical conditions

In this section, we compare the proposed estimator with some existing estimators which are considered in this article, is given by.

1.
By comparing equation (1) and equation (12),

M S E ({\overline{y}}_{L_{1} (R S S)}) < M S E ({\overline{y}}_{1 (r s s)}), if

\frac{g^{2} V_{020} + g^{2} V_{002}}{4 g V_{110} + 4 g V_{101} + 2 g^{2} V_{011}} < 1

2.
By comparing equation (2) and equation (12),

M S E ({\overline{y}}_{L_{1} (R S S)}) < M S E ({\overline{y}}_{2 (R S S)}), if

\frac{(g^{2} - 4) V_{020} + (g^{2} - 1) V_{002}}{4 (g - 2) V_{110} + 4 (g - 1) V_{101} + 2 (2 - g^{2}) V_{011}} < 1

3.
By comparing equation (3) and equation (12),

M S E ({\overline{y}}_{L_{1} (R S S)}) < M S E ({\overline{y}}_{3 (R S S)}), if

\frac{(g^{2} - 1) V_{020} + (g^{2} - 4) V_{002}}{4 (g - 1) V_{110} + 4 (g - 2) V_{101} + 2 (2 - g^{2}) V_{011}} < 1

4.
By comparing equation (4) and equation (12),

M S E ({\overline{y}}_{L_{1} (R S S)}) < M S E ({\overline{y}}_{4 (R S S)}), if

\frac{(g + 1) V_{020} + (g + 1) V_{002}}{4 V_{110} + 4 V_{101} - 2 (g + 1) V_{011})} < 1

5.
By comparing equation (5) and equation (12),

M S E ({\overline{y}}_{L_{1} (R S S)}) < M S E ({\overline{y}}_{5 (R S S)}), if

\frac{(g^{2} - 4) V_{020} + (g^{2} - 1) V_{002}}{4 (g - 2) V_{110} + 4 (g - 1) V_{101} + 2 (2 - g^{2}) V_{011}} < 1

6.
By comparing equation (6) and equation (12),

M S E ({\overline{y}}_{L_{1} (R S S)}) < M S E ({\overline{y}}_{6 (R S S)}), if

\frac{(g^{2} - 1) V_{020} + (g^{2} - 4) V_{002}}{4 (g - 1) V_{110} + 4 (g - 2) V_{101} + 2 (2 - g^{2}) V_{011}} < 1

8. Simulation study

To check the efficiency of the suggested estimator, a simulation study is designed to estimate $M S E s$ , $P R E s$ , $P R B s$ and $P R R M S E s$ . With respect to the correlated auxiliary variable X, rank is executed. Tri-variate random observations $(X, Y, Z)$ , were produced from a multivariate normal distribution with known population correlation coefficients $ρ_{y x} = 0.90$ , $ρ_{y z} = 0.80$ and $ρ_{x z} = 0.70$ . Using $10, 000$ simulations, estimates of $M S E s$ , $P R E s$ , $P R R M S E s$ and $P R B s$ are computed using R-Language and presented in Table 1, Table 2, Table 3, Table 4.

Table 1.

The simulated $M S E s$ of different estimators.

$m$	$r$	$n$	${\overline{y}}_{1 (R S S)}$	${\overline{y}}_{2 (R S S)}$	${\overline{y}}_{3 (R S S)}$	${\overline{y}}_{4 (R S S)}$	${\overline{y}}_{5 (R S S)}$	${\overline{y}}_{6 (R S S)}$	${\overline{y}}_{L (R S S)}$
3	3	9	0.1807	0.0832	0.1124	0.0449	0.0914	0.1350	0.0393
	4	12	0.1379	0.0605	0.0819	0.0330	0.0649	0.0924	0.0295
	5	15	0.1104	0.0487	0.0668	0.0259	0.0521	0.0748	0.0228
	10	30	0.0555	0.0240	0.0333	0.0130	0.0247	0.0354	0.0114
	15	45	0.0373	0.0162	0.0227	0.0088	0.0165	0.0235	0.0077
	20	60	0.0274	0.0122	0.0171	0.0066	0.0125	0.0177	0.0058
4	3	12	0.1203	0.0557	0.0797	0.0332	0.0590	0.0909	0.0292
	4	16	0.0903	0.0421	0.0593	0.0248	0.0439	0.0647	0.0219
	5	20	0.0701	0.0336	0.0489	0.0196	0.0349	0.0525	0.0172
	10	40	0.0353	0.0171	0.0249	0.0101	0.0175	0.0259	0.0087
	15	60	0.0239	0.0112	0.0163	0.0065	0.0113	0.0166	0.0057
	20	80	0.0180	0.0084	0.0122	0.0048	0.0085	0.0124	0.0042
5	3	15	0.0853	0.0429	0.0640	0.0268	0.0443	0.0696	0.0233
	4	20	0.0647	0.0313	0.0466	0.0195	0.0319	0.0498	0.0172
	5	25	0.0514	0.0254	0.0386	0.0156	0.0260	0.0408	0.0134
	10	50	0.0258	0.0123	0.0188	0.0076	0.0125	0.0194	0.0067
	15	75	0.0170	0.0081	0.0123	0.0051	0.0082	0.0126	0.0044
	20	100	0.0127	0.0062	0.0095	0.0038	0.0062	0.0096	0.0033

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Table 2.

The Simulated $P R E s$ of Different Estimators with respect to usual $R S S$ mean estimator.

$m$	$r$	$n$	${\overline{y}}_{1 (R S S)}$	${\overline{y}}_{2 (R S S)}$	${\overline{y}}_{3 (R S S)}$	${\overline{y}}_{4 (R S S)}$	${\overline{y}}_{5 (R S S)}$	${\overline{y}}_{6 (R S S)}$	${\overline{y}}_{L (R S S)}$
3	3	9	100	217	160	402	197	133	459
	4	12	100	227	168	416	212	149	466
	5	15	100	226	165	425	211	147	483
	10	30	100	230	166	425	224	156	484
	15	45	100	229	164	420	226	158	481
	20	60	100	223	159	412	219	154	471
4	3	12	100	216	150	361	203	132	412
	4	16	100	214	152	364	205	139	410
	5	20	100	208	143	356	200	133	406
	10	40	100	206	141	350	201	136	406
	15	60	100	212	146	367	210	143	420
	20	80	100	214	147	370	211	144	425
5	3	15	100	198	133	317	192	122	365
	4	20	100	206	138	330	202	129	374
	5	25	100	202	133	328	197	125	381
	10	50	100	209	137	336	206	132	385
	15	75	100	208	138	333	206	135	380
	20	100	100	204	134	330	203	131	378

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Table 3.

The simulated $P R R M S E s$ of different estimators.

$m$	$r$	$n$	${\overline{y}}_{1 (R S S)}$	${\overline{y}}_{2 (R S S)}$	${\overline{y}}_{3 (R S S)}$	${\overline{y}}_{4 (R S S)}$	${\overline{y}}_{5 (R S S)}$	${\overline{y}}_{6 (R S S)}$	${\overline{y}}_{L (R S S)}$
3	3	9	21.503	14.593	16.962	10.933	15.453	18.832	9.037
	4	12	18.787	12.444	14.482	9.208	12.972	15.496	7.014
	5	15	16.814	11.170	13.075	8.160	11.621	13.951	6.028
	10	30	11.919	7.849	9.246	6.782	7.991	9.576	4.440
	15	45	9.781	6.453	7.625	5.972	6.519	7.792	4.070
	20	60	8.377	5.603	6.629	4.626	5.668	6.747	3.139
4	3	12	17.551	11.94	14.285	9.236	12.391	15.403	7.319
	4	16	15.203	10.37	12.321	7.969	10.653	12.954	6.862
	5	20	13.395	9.282	11.190	7.190	9.487	11.641	6.181
	10	40	9.514	6.624	7.997	5.084	6.704	8.156	4.737
	15	60	7.834	5.375	6.467	4.089	5.408	6.542	3.018
	20	80	6.802	4.648	5.597	3.534	4.679	5.655	2.303
5	3	15	14.776	10.48	12.802	8.299	10.697	13.432	7.762
	4	20	12.868	8.953	10.927	7.088	9.089	11.355	5.667
	5	25	11.472	8.071	9.942	6.338	8.197	10.278	4.882
	10	50	8.138	5.625	6.952	5.435	5.676	7.076	3.149
	15	75	6.613	4.579	5.625	3.620	4.605	5.688	2.392
	20	100	5.709	3.996	4.931	3.140	4.011	4.977	2.934

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Table 4.

The simulated $P R B s$ of different estimators.

$m$	$r$	$n$	${\overline{y}}_{2 (R S S)}$	${\overline{y}}_{3 (R S S)}$	${\overline{y}}_{4 (R S S)}$	${\overline{y}}_{5 (R S S)}$	${\overline{y}}_{6 (R S S)}$	${\overline{y}}_{L (R S S)}$
3	3	9	2.1983	3.0396	0.5515	0.0261	0.2753	−0.8576
	4	12	1.4248	1.8640	0.3369	−0.2062	−0.1097	−0.4044
	5	15	1.2970	1.7598	0.3450	−0.0342	0.1337	−0.1356
	10	30	0.6725	0.9914	0.2649	0.0276	0.2011	−0.5135
	15	45	0.4879	0.7140	0.1700	0.0579	0.1912	−0.3642
	20	60	0.3789	0.4786	0.0888	0.0581	0.0872	−0.2824
4	3	12	1.5350	2.1025	0.4256	0.1779	0.2599	−0.1355
	4	16	1.0077	1.4225	0.2233	−0.0138	0.0949	−0.9594
	5	20	0.8045	1.0704	0.2317	0.0073	−0.0025	−0.6733
	10	40	0.3755	0.4896	0.0515	−0.0279	−0.0331	−0.4061
	15	60	0.3001	0.3356	0.0716	0.0309	−0.0131	−0.2204
	20	80	0.2153	0.2777	0.0365	0.0108	0.0152	−0.2032
5	3	15	1.0073	1.5071	0.3741	0.0822	0.1725	−0.7652
	4	20	0.8611	1.2349	0.3336	0.1687	0.2443	−0.5282
	5	25	0.7337	1.0110	0.3036	0.1747	0.2032	−0.3855
	10	50	0.3676	0.4971	0.1431	0.0905	0.0978	−0.2008
	15	75	0.2230	0.2954	0.0753	0.0404	0.0388	−0.1473
	20	100	0.2266	0.2764	0.0991	0.0902	0.0803	−0.0685

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The following formulae are used for point estimations. The expression for $P R B ({\overline{y}}_{P (R S S)})$ , $M S E ({\overline{y}}_{P (R S S)})$ , $P R R M S E_{(P)}$ and $PR E_{(P)}$ are given in equation (15), equation (16) and equation (17):

Equation 15.

(15)

Equation 16.

(16)

Equation 17.

(17)

and

Equation 18.

(18)

P = (1, 2 . . . 6, L)

9. Real data set

In addition to simulation study, to observe the performance of the proposed estimator in real life application, a real data set is used. The data set contains three variables, whereas Y is variable of interest while X and Z are correlated auxiliary variables. From this real population set, number of samples were taken for comparing the efficiency of newly developed estimator with the existing counterpart estimators available in the literature stock in case of two auxiliary variables under RSS. Via R-Language, MSEs and PREs are obtained, using equation (16) and equation (18), for the proposed and exiting estimators, which is given in Table 5.

Table 5.

$M S E s$ and $(P R E s)$ of different estimators.

Sample	${\overline{y}}_{1 (R S S)}$	${\overline{y}}_{2 (R S S)}$	${\overline{y}}_{3 (R S S)}$	${\overline{y}}_{4 (R S S)}$	${\overline{y}}_{5 (R S S)}$	${\overline{y}}_{6 (R S S)}$	${\overline{y}}_{L (R S S)}$
1	1732.91	105.74	301.57	114.76	449.90	302.65	93.23
1	(100)	(1641)	(574)	(1508)	(385)	(573)	(1846)
2	1808.90	98.57	307.18	106.65	461.67	306.44	88.41
2	(100)	(1835)	(588)	(1696)	(391)	(590)	(2046)
3	1575.82	108.11	281.65	121.05	417.99	283.45	103.22
3	(100)	(1457)	(559)	(1301)	(377)	(557)	(1526)
4	1736.13	113.57	291.35	123.70	436.34	291.16	99.23
4	(100)	(1528)	(596)	(1403)	(398)	(596)	(1749)
5	1784.78	109.25	308.13	122.23	462.64	308.96	97.64
5	(100)	(1633)	(579)	(1460)	(387)	(578)	(1825)

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9.1. Population [source: [18]]

The summary statistics for the aforementioned population as follows:

y: Output for 80 factories in a region

x: Number of workers

z: Fixed capital

\begin{array}{c} N = 80 & m = 4 & r = 5 \end{array}

\begin{array}{c} n = 20 & \overline{Y} = 285.125 & \overline{X} = 1126.463 \end{array}

\begin{array}{c} \overline{Z} = 5182.637 & C_{y} = 0.9484 & C_{x} = 0.7507 \end{array}

\begin{array}{c} C_{z} = 0.3542 & ρ_{x y} = 0.98 & ρ_{z y} = 0.91 \end{array}

ρ_{x z} = 0.94

10. Conclusion

From Table 1, Table 5 and it is shown that the suggested class of estimator ${\overline{y}}_{L (R S S)}$ , have minimum $M S E$ values as to ${\overline{y}}_{(r s s)}$ , [9,10]. Also, with increasing the sample size the values of $M S E$ decreases. The proposed estimator has reasonable biases, since the values of $P R B$ listed in Table 4 are all below 1 $%$ in absolute form. Amongst the competitor estimators, as shown in Table 2, Table 5, ${\overline{y}}_{L (R S S)}$ has highest $P R E s$ . Similarly from Table 3 it is shown that the proposed estimator has minimum $P R R M S E$ amongst counterpart estimators. It is noticed that the efficiency of the proposed estimator increases in real life survey. So, we reason out that, the developed estimator is strongly preferable over [6,7] estimators in real life application for better estimation of population mean. The present work can be extended to estimate population mean under stratified random sampling, probability proportional to size using predicative approach and non-response using ranked set sampling.

Data availability

Data will be made available on request.

CRediT authorship contribution statement

Lakhkar Khan: Software, Formal analysis. Sohaib Ahmad: Writing – original draft, Conceptualization. Abdullah Mohammed Alomair: Validation, Data curation. Mohammed Ahmed Alomair: Investigation, Formal analysis.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. 5123]

Contributor Information

Lakhkar Khan, Email: l.khan@stat.qau.edu.pk.

Sohaib Ahmad, Email: sohaib_ahmad@awkum.edu.pk.

Abdullah Mohammed Alomair, Email: ama.alomair@kfu.edu.sa.

Mohammed Ahmed Alomair, Email: ma.alomair@kfu.edu.sa.

References

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21.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. [Google Scholar]
22.Bhushan S., Kumar A. Novel predictive estimators using ranked set sampling. Concurrency Comput. Pract. Ex. 2023;35(3) [Google Scholar]
23.Bhushan S., Kumar A. Predictive estimation approach using difference and ratio type estimators in ranked set sampling. J. Comput. Appl. Math. 2022;410 [Google Scholar]
24.Bhushan S., Kumar A., Lone S.A. On some novel classes of estimators using ranked set sampling. Alex. Eng. J. 2022;61(7):5465–5474. [Google Scholar]
25.Alomair A.M., Shahzad U. Optimizing mean estimators with calibrated minimum covariance determinant in median ranked set sampling. Symmetry. 2023;15(8):1581. [Google Scholar]
26.Abbasi A.M., Yousaf Shad M. Estimation of population proportion using concomitant based ranked set sampling. Commun. Stat. Theor. Methods. 2022;51(9):2689–2709. [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Data Availability Statement

Data will be made available on request.

[bib1] 1.Al-Omari A.I. Ratio estimation of the population mean using auxiliary information in simple random sampling and median ranked set sampling. Stat. Probab. Lett. 2012;82(11):1883–1890. [Google Scholar]

[bib2] 2.Amiri S., Jozani M.J., Modarres R. Randomly selected order statistics in ranked set sampling: a less expensive comparable alternative to simple random sampling. Environ. Ecol. Stat. 2018;25(2):237–256. [Google Scholar]

[bib3] 3.Arnold B.C., Balakrishnan N., Nagaraja H.N. vol. 54. Siam; 1992. (A First Course in Order Statistics). [Google Scholar]

[bib4] 4.Kadilar C., Unyazici Y., Cingi H. Ratio estimator for the population mean using ranked set sampling. Stat. Pap. 2009;50(2):301–309. [Google Scholar]

[bib5] 5.Khan L., Shabbir J. A class of Hartley-Ross type unbiased estimators for population mean using ranked set sampling. Hacettepe Journal of Mathematics and Statistics. 2015;44(3):1–12. [Google Scholar]

[bib6] 6.Khan L., Shabbir J. An efficient class of estimators for the finite population mean in ranked set sampling. Open J. Stat. 2016;6(3):426–435. [Google Scholar]

[bib7] 7.Khan L., Shabbir J. Hartley-Ross type unbiased estimators using ranked set sampling and stratified ranked set sampling. The North Carolina Journal of Mathematics and Statistics. 2016;2:10–22. [Google Scholar]

[bib8] 8.Khan L., Shabbir J. Improved ratio-type estimators of population mean in ranked set sampling using two concomitant variables. Pak. J. Statistics Oper. Res. 2016;12(3):507–518. [Google Scholar]

[bib9] 9.Khan L., Shabbir J., Gupta S. Unbiased ratio estimators of mean in stratified ranked set sampling. Hacettepe Journal of Mathematics and Statistics. 2016;46(6):1151–1158. [Google Scholar]

[bib10] 10.Khan L., Shabbir J. Generalized exponential-type ratio-cum ratio estimators of population mean in ranked set and stratified ranked set sampling. J. Stat. Manag. Syst. 2017;20(1):133–151. [Google Scholar]

[bib11] 11.Ahmad S., Zahid E., Shabbir J., Aamir M., Emam W., Tashkandy Y., Ahmad S. A simulation study: an enhanced generalized class of estimators for estimation of population proportion using twofold auxiliary attribute. Heliyon. 2023;9 doi: 10.1016/j.heliyon.2023.e17269. 2405-844. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib12] 12.McIntyre G. A method for unbiased selective sampling, using ranked sets. Aust. J. Agric. Res. 1952;3(4):385–390. [Google Scholar]

[bib13] 13.Mehta N., Mandowara V. A modified ratio-cum-product estimator of finite population mean using ranked set sampling. Commun. Stat. Theor. Methods. 2016;45(2):267–276. [Google Scholar]

[bib14] 14.Murray R., Ridout M., Cross J. The use of ranked set sampling in spray deposit assessment. Aspect Appl. Biol. 2000;57:141–146. [Google Scholar]

[bib15] 15.Samawi H.M., Muttlak H.A. Estimation of ratio using rank set sampling. Biom. J. 1996;38(6):753–764. [Google Scholar]

[bib16] 16.Stokes S.L. Ranked set sampling with concomitant variables. Commun. Stat. Theor. Methods. 1977;6(12):1207–1211. [Google Scholar]

[bib17] 17.Takahasi K., Wakimoto K. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Ann. Inst. Stat. Math. 1968;20(1):1–31. [Google Scholar]

[bib18] 18.Murthy M.N. 1967. Sampling Theory and Methods. Sampling theory and methods. [Google Scholar]

[bib19] 19.Ahmad S., Hussain S., Yasmeen U., Aamir M., Shabbir J., El-Morshedy M., Ahmad Z. A simulation study: using dual ancillary variable to estimate population mean under stratified random sampling. PLoS One. 2022;17(11) doi: 10.1371/journal.pone.0275875. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib20] 20.Ahmad S., Ullah K., Zahid E., Shabbir J., Aamir M., Alshanbari H.M., El-Bagoury A.A.A.H. A new improved generalized class of estimators for population distribution function using auxiliary variable under simple random sampling. Sci. Rep. 2023;13(1):5415. doi: 10.1038/s41598-023-30150-9. [DOI] [PMC free article] [PubMed] [Google Scholar]

[bib21] 21.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. [Google Scholar]

[bib22] 22.Bhushan S., Kumar A. Novel predictive estimators using ranked set sampling. Concurrency Comput. Pract. Ex. 2023;35(3) [Google Scholar]

[bib23] 23.Bhushan S., Kumar A. Predictive estimation approach using difference and ratio type estimators in ranked set sampling. J. Comput. Appl. Math. 2022;410 [Google Scholar]

[bib24] 24.Bhushan S., Kumar A., Lone S.A. On some novel classes of estimators using ranked set sampling. Alex. Eng. J. 2022;61(7):5465–5474. [Google Scholar]

[bib25] 25.Alomair A.M., Shahzad U. Optimizing mean estimators with calibrated minimum covariance determinant in median ranked set sampling. Symmetry. 2023;15(8):1581. [Google Scholar]

[bib26] 26.Abbasi A.M., Yousaf Shad M. Estimation of population proportion using concomitant based ranked set sampling. Commun. Stat. Theor. Methods. 2022;51(9):2689–2709. [Google Scholar]

PERMALINK

Enhanced exponential ratio-cum-ratio estimator in ranked set sampling using transformed auxiliary information

Lakhkar Khan

Sohaib Ahmad

Abdullah Mohammed Alomair

Mohammed Ahmed Alomair

Abstract

1. Introduction

2. Ranked set sampling in case of two auxiliary variables and its algorithm for sample selection

3. General procedure for obtaining bias and mean square error

4. Existing estimators

5. Suggested estimator

6. Special cases of the proposed estimator

7. Theoretical conditions

8. Simulation study

Table 1.

Table 2.

Table 3.

Table 4.

9. Real data set

Table 5.

9.1. Population [source: [18]]

10. Conclusion

Data availability

CRediT authorship contribution statement

Declaration of competing interest

Acknowledgments

Contributor Information

References

Associated Data

Data Availability Statement

ACTIONS

PERMALINK

RESOURCES

Cite

Add to Collections

PERMALINK

Enhanced exponential ratio-cum-ratio estimator in ranked set sampling using transformed auxiliary information

Lakhkar Khan

Sohaib Ahmad

Abdullah Mohammed Alomair

Mohammed Ahmed Alomair

Abstract

1. Introduction

2. Ranked set sampling in case of two auxiliary variables and its algorithm for sample selection

3. General procedure for obtaining bias and mean square error

4. Existing estimators

5. Suggested estimator

6. Special cases of the proposed estimator

7. Theoretical conditions

8. Simulation study

Table 1.

Table 2.

Table 3.

Table 4.

9. Real data set

Table 5.

9.1. Population [source: [18]]

10. Conclusion

Data availability

CRediT authorship contribution statement

Declaration of competing interest

Acknowledgments

Contributor Information

References

Associated Data

Data Availability Statement

ACTIONS

PERMALINK

RESOURCES

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