3.1 The 2SCS with non-response

The 2SCS method selects a sample through two stages. The population comprises N₁ PSUs, each with N_2,i SSUs. Let Y_i,j denote the characteristic value for the jth SSU in the ith PSU, where N_2,i is the total number of SSUs. Additionally, and represent the units in the response and non-response groups, such that . Based on this, G(y) under 2SCS is expressed as follows:

(5)

where,

Here, , and and correspond to the average cluster size and CDF, for the response group and non-response groups in the ith PSU, respectively.

To estimate finite population CDF for 2SCS while accounting for non-response, we select a sample of size n₁ from N₁ PSUs with SRS, and then from ith PSU, a sub-sample of size n_2,i is chosen. It is observed that out of n_2,i SSUs, units respond and units do not respond. Following [7]’s sub-sampling technique of non-respondents, we select a subsample of from the non-respondent units of using SRS scheme from the ith PSU selected in the sample such that , where and then interview all units. It should be noted that the samples are chosen using SRS without replacement on all stages of sampling in the 2SCS scheme. For the 2SCS with non-response, we propose an estimator of G(y) given by:

(6)

where,

(7)

Here, and are the weights and and are the CDF based on and units, respectively.

In next Lemmas, we study the mathematical properties of .

Lemma 1: Under the 2SCS, is an unbiased estimator of G(y).

Proof: By assigning indices “1”, and “2” to represent the first, and second stages of sampling, respectively, we can express it as follows:

(8)

Note that may be observed as a simple random sample of size n₁ from in the first steps of sampling. For more details, see [10] and the references cited therein.

Lemma 2: The variances of is given by

where

(9)

Proof: We know that

(10)

From Eq (8), . Thus by using this result:

(11)

(12)

For more details, the readers may see [33,9] and the references cited therein.

3.2 The S2SCS with non-response

Consider a finite population that is divided into L strata, with each stratum comprising N_1,h PSUs, where . Each PSU contains N_2,i,h SSUs for and . Let Y_i,j,h represent the characteristic value for the jth SSU within the ith PSU of the hth stratum, where . Furthermore, let and denote the counts of units in the response and non-response groups, respectively, such that . Utilizing this information under the S2SCS framework, the CDF for finite population can be expressed as:

(13)

where

are the hth stratum’s average cluster sizes and CDF, respectively.

To estimate G(y) with S2SCS under non-response, n_1,h PSUs is chosen from the hth stratum. The sample sizes n_1,h are allotted according to an allocation system, such as equal, Neyman or proportional allocations. Further, a sub-sample of size n_2,i,h SSUs from ith selected PSU of hth stratum is selected. It should be noted that on both stages of sampling under S2SCS, the samples are taken using SRS without replacement. It is observed that out of n_2,i,h SSUs, units respond and units do not respond. Adopting [7] technique of sub-sampling of non-respondent, the unbiased estimator of G(y) with S2SCS can be written as:

(14)

where

In the subsequent Lemmas, we will examine the mathematical properties of estimator with non-response under S2SCS.

Lemma 3: is an unbiased estimators of G(y). The variances of is given by

(15)

The proof is similar to Lemmas 1 and 2. The mathematical expressions of is similar to with the exception that earlier is computed from the hth stratum.

3.3 3SCS with non-response

The 3SCS extends the 2SCS method by adding a third stage. In 3SCS, the population U consists of N₁ PSUs, each containing N_2,i SSUs, which in turn include N_3,ij TSUs. Let define Y_ijk as the characteristic value associated with the kth TSU located within the jth SSU of the ith PSU, where the indices are defined as , , and . Furthermore, it is established that , with and retaining their conventional definitions. Under 3SCS with non-response, the finite population CDF may be written as

(16)

where

(17)

In this context, and represent the weights, while signifies the average size of the cluster. Additionally, let and denote the CDF derived from the jth SSU of the ith PSU for the response group and the non-response group, respectively.

To estimate G(y) under 3SCS with non-response, an unbiased estimator of population CDF can be derived as follow. First, a sample of PSUs, n₁, is chosen. Next, from each selected PSU, a sample of SSUs, n_2,i, is drawn. Finally, within each selected SSU, a sample of TSUs, n_3,ij, is selected. Further, let where and have their usual meanings. A sub-sample of size where units is drawn from units. Under the 3SCS with non-response at third stage, an unbiased estimator of G(y) is given by:

(18)

where

(19)

and , , have their usual meanings. and are the CDF computed from response and non-response group, respectively. It should be noted that under 3SCS with non-response, SRS without replacement is used in all stages of sampling. For more detail, see [33,1] and references cited therein.

In the subsequent Lemmas, we will examine the mathematical properties of .

Lemma 4: Under 3SCS, is an unbiased estimator of G(y).

Proof. By assigning indices “1”, “2”, and “3” to represent the first, second, and third stages of sampling, respectively, we can express it as follows:

(20)

Lemma 5: The variances of is given by

(21)

where

(22)

Proof:

(23)

From Eq (20), we can write

(24)

and

(25)

Also from Eq (20), we have

(26)

Now take expectations of Eq (20) to get

(27)

Now Eq (21) follows by replacing Eq (24), Eq (25) and Eq (27) in Eq (23), which completes the proof.

3.4 S3SCS under Non-Response

S3SCS begins by dividing the entire population U into L strata, based on certain characteristics such as different regions or income levels. Within the hth stratum, the population contains N_1,h PSUs, each with N_2,i,h SSUs, each of which has N_3,ij,h TSUs for . Let Y_ijk,h be the characteristics under study obtained for the kth TSUs in the jth SSUs drawn from the ith PSUs within the hth stratum for , , and . Further, let where and have their usual meanings. Under S3SCS with non-response the population CDF may be written as:

(28)

In estimating the population CDF using S3SCS under non-response conditions, the sampling process consists of several stages within each stratum. First, a sample of PSUs is selected from N_1,h, resulting in a size of n_1,h. From each selected PSU, SSUs are sampled from N_2,i,h, yielding a size of n_2,i,h. Lastly, TSUs are chosen from each sampled SSU, with a size of n_3,ij,h from N_3,ij,h. It is observed that out of n_3,ij,h, units respond and units do not respond, such that . Adopting [8] technique of sub-sampling of non-respondent, the unbiased estimator of population CDF, G(y), under S3SCS with non-response can be written as:

(29)

where

Here, and have there usual meanings.

In the forthcoming Lemmas, we examine the characteristics of the CDF estimator in the context of non-response within the S3SCS framework.

Lemma 6: is an unbiased estimators of G(y). The variances of is given by

(30)

Proof. The proof has a resemblance to Lemmas 4 and 5. The proof of is completed by observing that its mathematical expressions are similar to , except that the earlier is assessed from the hth stratum.

4.1 Two-stage and stratified two-stage cluster sampling

In a finite population U, we assume that X represents the auxiliary, and Y denotes the primary variable under study. Additionally, to approximate (G(y),G(x)) within the frameworks of 2SCS and S2SCS, let and represent the corresponding CDF estimators derived from variables (Y,X). It should be noted that the CDF estimators on the basis of X can be computed on the similar lines of Y.

Lemma 7: Under 2SCS scheme, the covariance between and is given by

(31)

where

(32)

(33)

(34)

Proof. Here, , , and retain their standard interpretations. The demonstration of this Lemma can be found in the S3 Appendix.

Lemma 8: Under S2SCS scheme, the covariance between and is given by

(35)

Proof. The proof has a resemblance to Lemmas 7. The proof of is completed by observing that its mathematical expressions is similar to , except that the aforesaid is assessed from the hth stratum.

4.2 Three-stage and stratified three-stage cluster sampling

Under 3SCS and S3SCS with non response, let and be the respective CDF estimators of (G(y),G(x)) that are based on (Y,X), respectively.

Lemma 9: Under 3SCS scheme, the covariance between and is given by

(36)

where

(37)

(38)

(39)

(40)

and .

Proof. Here, , , and retain their standard interpretations. The demonstration of this Lemma can be accessed in the S3 Appendix.

Lemma 10: Under S3SCS scheme, the covariance between and is given by

(41)

Proof. The proof has a resemblance to Lemmas 9. The proof of is completed by observing that its mathematical expressions is similar to , except that the earlier is computed from the hth stratum.

5.1 First proposed family of CDF estimators

We suggest a class of ratio and product-type estimators, analogous to those introduced by [1] and [14], for estimating the population CDF under non-response.

(44)

where a is different from 0 and b as real numbers or functions of the known parameters of the supplementary variable X such as coefficient of variation , correlation coefficient , coefficient of skewness and coefficient of kurtosis etc. The selected scalars, and , minimize the MSE of . Different estimators of may be developed by selecting appropriate a, b, , and values. In Table 1, we display some members of for various values of a, b, , and .

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Table 1. Several members of the suggested families of CDF estimators under non-response

https://doi.org/10.1371/journal.pone.0322660.t001

The bias and MSE of can be approximated mathematically by writing and . The expression on right-hand side (RHS) of Eq (47) can be written in terms of ’s:

(45)

where  +  . By expanding the RHS of Eq (48) and considering only the terms of order second power in ’s, we have

(46)

To obtain bias of , subtract G(y) from both sides of Eq (49) and consider only the terms up-to first order of approximation, given as:

(47)

Retain terms up-to first order of approximation, we can write Eq (49) as follows:

(48)

To obtain MSE of , take square and then its expectation on both sides of the Eq (51) and retain terms up-to first order, we have:

(49)

The minimal MSE of is given by

(50)

(51)

where, is the correlation coefficient between and . The minimum MSE of can be obtained by utilizing the optimum value of , say .

5.2 Second proposed family of CDF estimators

We propose another class of exponential ratio/product estimators similar to those proposed by [1] and [28] to estimate the population CDF G(y) under non-response given as:

(52)

where a, b, and have their usual meanings. Some members of for various values of a, b, , and can be seen in Table 1. The bias and MSE of can be obtained by expressing the Eq (55) in terms of ’s as:

(53)

where . By expanding the RHS of Eq (56) and considering only the terms of order second power in ’s, we have

(54)

To obtain bias of , subtract G(y) from both sides of Eq (57) and consider only the terms up-to first order of approximation, given as:

(55)

Retain terms up-to first order of approximation, we can write Eq (57) as follows:

(56)

Under the first order of approximation, we obtain the MSE of by taking the square on both sides of Eq (59).

(57)

The minimal MSE of can be expressed by utilizing the optimum value of , say , is given by

(58)

which is equivalent to that of .

A variety of estimators can be derived from the suggested families and , as shown in Eqs (47) and (55), by selecting appropriate values for the parameters a, b, , and . By imputing specific values of , , a, and b into Eqs (50), (52), (58), and (60), we can derive the first-order approximations of the bias and MSE/variance for the respective members of the families and .

5.3 Difference CDF Estimator

Additionally, supplementary information about the variance and covariance of the estimators can be used to improve the precision of CDF estimators. Let denotes the ratio of covariance between to the variance of , under an S sampling scheme, i.e.

(59)

For situations involving non-response, the difference estimator for the population CDF, say , which relies on , , G(x), and , is expressed as:

(60)

where expressed as a linear aggregator of and , and G(x) denotes the population CDF under the S sampling scheme. It can be demonstrated that the is an unbiased estimator of G(y), and variance of can be obtained by expressing the Eq (63) in terms of ’s as:

(61)

To calculate the variance of , square both sides of Eq (64) and apply expectation.

(62)

The simplified equation for the variance of may be obtained by substituting , given by

(63)

which is identical to the minimal MSE of and .

6.1 Population I

A dataset sourced from the Centers for Disease Control (CDC), is associated with the Second National Health and Nutrition Examination Survey (NHANES-II). It comprises 10,351 units, representing the entire non-institutionalized civilian (NIC) population of the United States (US), including all 50 states and the District of Columbia. The data is separated into four geographic regions (REGs): midwestern, southern, northeastern, and western, each farther subdivided into specific locations (LOCs). With a success probability of 0.50, random numbers are generated from a Bernoulli distribution to stratify the dataset into two strata, with 0 indicating Stratum-I and 1 indicating Stratum-II. The dataset can be found at: https://www.stata-press.com/data/r15/svy.html. Our goal is to determine the percentage of underweight individuals in the NIC US population using body mass index (BMI) as the study variable indices by “Y” and weight as the ancillary variable indices by “X”. An individual is classified as underweight if their BMI is less than or equal to y = 18.50, with the average weight of the NIC US population being x = 71.8975. For the 2SCS/S2SCS design, REG and BMI are treated as PSU and SSU. In the 3SCS/S3SCS design, REG, LOC, and BMI are considered as PSU, SSU, and TSUs, respectively. Below are the constants for the population.

To demonstrate what we have aforesaid, the values of based on an S sampling scheme are computed, which are given below.

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Table 2. The values based on an S sampling scheme under non-response utilizing Population I

https://doi.org/10.1371/journal.pone.0322660.t002

Note: Stratifying is abbreviated as .

6.2 Population II

This data is taken from the study conducted by [27] in Multan district, southern Punjab, Pakistan, spans from January 2020 to March 2020. The dataset includes 1040 units, categorized into two strata: Stratum-I (males) and Stratum-II (females). In this study, BMI and weight are used as study and auxiliary variables, respectively. The objective is to estimate the proportion of underweight children by utilizing the auxiliary variable, with x = 35.76 representing the average weight, under the S sampling scheme. For the 2SCS/S2SCS design, socioeconomic status (SES) and BMI are treated as PSU and SSU. In the 3SCS/S3SCS design, SES, Age, and BMI are considered as PSU, SSU, and TSUs, respectively. The population constants are as follows:

To demonstrate what we have aforesaid, the values of primarily based on an S sampling scheme are computed below.

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Table 3. The values based on an S sampling scheme under non-response utilizing Population II

https://doi.org/10.1371/journal.pone.0322660.t003

We present the following expressions to compute REs of the proposed CDF estimators of finite population distribution function under S sampling schemes with non-response relative to .

where . Table 4 and 5 represent the REs of these CDF estimators.

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Table 4. Evaluating REs of proposed CDF estimator of G(y) under non-response relative to utilizing Population I

https://doi.org/10.1371/journal.pone.0322660.t004

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Table 5. Evaluating REs of the proposed CDF estimator of G(y) under non-response relative to utilizing Population-II

https://doi.org/10.1371/journal.pone.0322660.t005

Based on the S sampling scheme, the REs of the proposed CDF estimators were estimated with the distinctive values of n₁, n_2,i, and n_3,ij using the aforementioned datasets. It can be observed that proposed CDF estimators under S sampling schemes that utilizing supplementary information appear to be marginally more efficient than do not, as indicated by RE values greater than one. Furthermore, CDF estimators for S sampling schemes with stratification can also be seen slightly more efficient than those without stratification. Efficiency tends to increase with the number of sampling stages, although the impact of increasing the sample size remains uncertain. Typically, RE tends to increase by increasing the sample size at various sampling stages and vice varsa.

6.3 Simulation Study

To further validate our findings, we estimated the MSE through a simulation approach by computing the squared difference between the estimator and the true parameter. The CDF of the proposed family of estimators under different non-response rates and sample sizes were simulated, and their corresponding MSEs were calculated using a simulation-based approach. The MSE/Variance of the CDF estimators under non-response are computed by drawing 10 thousand samples from Population I under a given sampling scheme. The simulation MSE of estimators for both families under S-sampling scheme are computed as:

(64)

(65)

(66)

where . The relative precision (RP) of and with respect to under an S-sampling scheme is given by:

(67)

We then calculated the RP between the estimators and found that our proposed classical ratio/product-type, exponential ratio/product-type, and difference CDF estimators performed well compared to existing methods. Additionally, we extended our simulation analysis by computing the MSE for different sample sizes of PSUs, SSUs, and TSUs under 2SCS/S2SCS, and 3SCS/S3SCS schemes. To assess the robustness of our estimators under varying levels of non-response, we evaluated their efficiency at response rates of 75% and 80%. For brevity of discussion, we considered c = 1 in our calculations.

While estimating the MSE through simulation, we adopted this structured sampling design for selecting PSUs, SSUs, and TSUs under different sampling schemes. This framework was applied across 2SCS, 3SCS, and their stratified counterparts, ensuring a systematic and realistic representation of hierarchical data selection and can bee seen in Table 6. This structured approach allowed us to analyze the performance of the proposed estimators under varying levels of non-response effectively.

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Table 6. The structured sampling design for the estimation of MSE through simulation under non-response utilizing Population I

https://doi.org/10.1371/journal.pone.0322660.t006

Overall, the proposed families of estimators performed well under different sampling conditions. However, the proposed difference estimator demonstrated slightly higher efficiency compared to both the estimators without auxiliary information and the other proposed families of estimators. Although the efficiency gains of the difference estimator were modest, it consistently outperformed other estimators across all sampling schemes. Notably, the maximum gain in precision was found in the ratio/product-type estimators, particularly under the 3SCS scheme, where it reached 15% when considering a sampling structure of n₁ = 2, n_2,i = 12, and n_3,ij = 20.

The consistency between the theoretical and simulation-based MSE results further validates the reliability of the proposed estimators in handling non-response within complex survey sampling frameworks. The detailed results of these simulation studies under various complex survey sampling schemes and non-response rates can be found in the supplementary file (S1 Tables), from Table 1 to Table 16.