An Efficient Ratio-Cum-Exponential Estimator for Estimating the Population Distribution Function in the Existence of Non-Response Using an SRS Design

Abstract

To gain insight into various phenomena of interest, cumulative distribution functions (CDFs) can be used to analyze survey data. The purpose of this study was to present an efficient ratiocum-exponential estimator for estimating a population CDF using auxiliary information under two scenarios of non-response. Up to first-order approximation, expressions for the bias and mean squared error (MSE) were derived. The proposed estimator was compared theoretically and empirically, with the modified estimators. The proposed estimator was found to be better than the modified estimators based on present-relative efficiency PRE and MSE criteria under the specific conditions.

1. Introduction

It is a well-accepted fact in survey sampling that, under certain conditions, auxiliary information can provide precise estimates of population parameters such as the mean, median, standard deviation, totals, quantiles, and the cumulative distribution function (CDF), etc. If a linear and higher correlation is observed between the study variable Y and auxiliary variable X, researchers often use a traditional estimator for the mean, like ratio, product, and regression estimators, to estimate a population mean. The literature includes a significant amount of work for estimating different parameters of a population, for example, see [1,2,3,4,5,6,7].These studies propose improved ratio-, product-, and regression-type estimators for estimating the mean and variance of a population using auxiliary variables.

Non-response is an issue that cannot be avoided in complex sample surveys, and it can be found in surveys that involve human responses. Language problems, inaccurate return addresses, a lack of information, and the sensitivity of the survey question(s), among many other reasons, can play a part in causing this issue. For example, an individual may be reluctant to provide salary information. Non-response in sample surveys is more prevalent and pervasive in postal surveys than in special canvasser surveys. Therefore, the term non-response refers to the inability to measure part of the units in a sample survey. Non-response can compromise estimator accuracy and increase its bias.

To cope with the non-response problem, several measures are proposed by various researchers, such as the weighting adjustment approach as, suggested by Oh and Scheuren [8]; imputation techniques provided by Kalton [9] and Kalton and Maligalig [10]; and the approach of sub-sampling non-respondents as recommended by Hansen and Hurwitz [11]. Various researchers have attempted to reduce the bias and to improve the effectiveness of the estimators of a population mean in the existence of non-response. Some significant references on estimating the population mean utilizing auxiliary variables in the existence of non-response include [12,13,14,15,16,17], etc.

Although there is extensive literature on estimating different estimators, it is noted that an auxiliary information-based estimation of a population CDF is less emphasized. It is becoming increasingly significant in survey sampling when statisticians are frequently interested in the proportion of a variable’s domain under examination. For example, policymakers may want to know where the percentage of educated women is higher or equal to 50% in Pakistan, the proportion of individuals having a weekly income of 100 USD or more in a developing country, etc. Similarly, a psychiatrist may be interested in knowing how many children spend one or more hours with their mobile activities or what proportion of children spend one or more hours on their phones. Several studies have revealed that spending more than an hour a day on phones or smart devices has a significant relation with psychological problems among childern, such as anxiety, loneliness, and depression.

Hence, it has become necessary to estimate the finite population CDF. Therefore, Singh et al. [18], Muñoz et al. [19], Yaqub and Shabbir [20,21], Hussain et al. [22], and Hussain et al. [23] have put their efforts on estimating the population CDF using auxiliary information.

2. Sampling Design and Notations

2.1. Notations for the CDF under SRS

Consider a finite population $\mathcal{U}=\{{\mathcal{U}}_1,{\mathcal{U}}_2,{\mathcal{U}}_3,\dots ,{\mathcal{U}}_N\}$ of N distinct units, let $(y_i,x_i)ϵ{\mathcal{U}}_i$ be the values of research variable Y and auxiliary variable X, respectively, on the $i^{th}$ unit. For every index $t_y$ and $t_x$ where ($t_y$, $t_x$) $ϵ\mathcal{U}$, the population CDFs of Y and X are defined, respectively, by,

$$\begin{array}{ccc}\hfill F_Y\left(t_Y\right)={\displaystyle \frac{{\sum }_{i=1}^{N}I(Y_i\le t_y)}{N}},& \hfill & \hfill F_X\left(t_X\right)={\displaystyle \frac{{\sum }_{i=1}^{N}I(X_i\le t_x)}{N}},\end{array}$$

where I(.) is an indicator variable. It is an average of the Bernoulli distributed variable, such that

$$I(Y_i\le t_y)=\left\{\begin{array}{cc}1\hfill & \mathrm{for}(Y_i\le t_y)\hfill \\ 0\hfill & \mathrm{and}(Y_i>t_y).\hfill \end{array}\right.$$

Theorem 1. 

In SRS, $\ell {\widehat{F}}_y\left(t_y\right)={\sum }_{i=1}^{\ell }I(y_i\le t_y)$ = $I(Y\le t_y)$ is a hyper-geometrically distributed variable with expected mean $E(.)$ and variance $V(.)$ for Y, respectively,

$$\begin{array}{cc}\hfill & E\left({\widehat{F}}_y\left(t_y\right)\right)=F_Y\left(t_y\right),\hfill \\ \hfill & V\left({\widehat{F}}_y\left(t_y\right)\right)={\displaystyle \frac{N-\ell }{\ell (N-1)}}\left[F_y\left(t_y\right)\left(1-F_1\left(t_y\right)\right)\right],\hfill \\ \hfill & \mathrm{Cov}\left({\widehat{F}}_y\left(t_y\right),{\widehat{F}}_x\left(t_x\right)\right)={\displaystyle \frac{N-\ell }{\ell N}}\left({\displaystyle \frac{N_{11}{N}_{22}-N_{12}{N}_{21}}{N^2}}\right),\hfill \end{array}$$

where we have the following:

$N_{11}=$ the number of units in the population that belong to $I(Y_i\le t_y)$ and $I(X_i\le t_x)$;

$N_{12}=$ the number of units in the population that belong to $I(Y_i\le t_y)$ and $I(X_i>t_x)$;

$N_{21}=$ the number of units in the population with $I(Y_i>t_y)$ and $I(X_i\le t_x)$; and

$N_{22}=$ the number of units in the population that belong to $I(Y_i>t_y)$ and $I(X_i>t_x)$.

Theorem 1. can be proved easily along the lines of García et al. [24].

Lemma 1. 

For a large sample size $\ell ,\mathrm{the}\mathrm{variance}\mathrm{of}{\widehat{F}}_y\left(t_y\right)$ is defined as

$$\begin{array}{c}\hfill V\left({\widehat{F}}_y\left(t_y\right)\right)={\displaystyle \frac{N-\ell }{\ell N}}\left[F_y\left(t_y\right)\left(1-F_y\left(t_y\right)\right)\right].\end{array}$$

Let us consider that $S_{F_{Y\left(t_y\right)}}^2=$$\left[F_Y\left(t_y\right)\left(1-F_Y\left(t_y\right)\right)\right]$ and $S_{F_{X\left(t_x\right)}}^2=$$\left[F_X\left(t_x\right)\left(1-F_X\left(t_x\right)\right)\right]$ are the population variances of $I(Y\le t_y)$ and $I(X\le t_x)$, respectively.

Let $S_{(FY\left(t_y\right),F_{X\left(t_x\right)})}=\mathrm{Cov}\left(F_Y\left(t_y\right),F_X\left(t_x\right)\right)$ be the population covariance between $I(Y\le t_y)$ and $I(X\le t_x)$, then we have the following:

$C_{F_Y\left(t_y\right)}={\displaystyle \frac{S_{F_Y\left(t_y\right)}}{F_Y\left(t_y\right)}}$ is the population coefficient of variation of $I(Y\le t_y)$, and

$C_{F_Y\left(t_y\right)}^2={\displaystyle \frac{1-F_Y\left(t_y\right)}{F_Y\left(t_y\right)}}$;

$C_{F_X\left(t_x\right)}={\displaystyle \frac{S_{F_X\left(t_x\right)}}{F_X\left(t_x\right)}}$ is the population coefficient of variation of $I(X\le t_x)$, and

$C_{F_X\left(t_x\right)}^2={\displaystyle \frac{1-F_X\left(t_x\right)}{F_X\left(t_x\right)}}$;

${\rho }_{(F_Y\left(t_y\right),F_X\left(t_x\right))}={\displaystyle \frac{S_{(F\left(t_y\right),F\left(t_x\right))}}{\left(S_{F_Y\left(t_y\right)}\right)\left(S_{F_X\left(t_x\right)}\right)}}$ is the phi-coefficient of correlation between $I(Y\le t_y)$ and $I(X\le t_x)$.

2.2. Notation for the CDF with Non-Response under an SRS Design

Consider the case where a finite population of N units is divided into two groups: a respondent’s group of $N_1$ units and another non-respondent’s group of $N_2$ units, where $N=N_1+N_2$. Consider the case where a sample of size ℓ is drawn from a target population using SRSWOR, and it is further assumed that only ${\ell }_1$ out of ℓ units respond, while ${\ell }_2=\ell -{\ell }_1$ units do not. Now, a sub-sample, also referred to as the 2nd phase sample, of $q={\ell }_2/k$ units, where $k>1$, is taken from the group of non-respondents of size ${\ell }_2$ for interviewing. This way of dealing with non-respondents to obtain responses from them is also referred to as the canvasser method. Hence, the total number of responses is $({\ell }_1+q)$, collected from ℓ units, and only $({\ell }_2-q)$ units are left as non-respondents who are not selected in the 2nd phase sample. Following Hansen and Hurwitz [11] a population CDF in the existence of non-response can be defined as follows:

$$F_Y^{*}\left(t_y\right)={\mathcal{W}}_1{F}_{Y\left(1\right)}\left(t_y\right)+{\mathcal{W}}_2{F}_{Y\left(2\right)}\left(t_y\right).$$

(1)

Similarly, let

$$F_X^{*}\left(t_x\right)={\mathcal{W}}_1{F}_{X\left(1\right)}\left(t_x\right)+{\mathcal{W}}_2{F}_{X\left(2\right)}\left(t_x\right),$$

where ${\mathcal{W}}_j={\displaystyle \frac{N_j}{N}}$ and $j=1,2.$ In addition, we have the following:

$F_{Y\left(1\right)}\left(t_y\right)={\sum }_{i=1}^{N_1}I(Y_i\le t_y)/N_1$ is the population CDF of $I(Y\le t_y)$ for the response group;

$F_{Y\left(2\right)}\left(t_y\right)={\sum }_{i=1}^{N_2}I(Y_i\le t_y)/N_2$ is the population CDF of $I(Y\le t_y)$ for the non-response group;

$F_{X\left(1\right)}\left(t_x\right)={\sum }_{i=1}^{N_1}I(X_i\le t_x)/N_1$ is the population CDF of $I(X\le t_x)$ for the response group;

$F_{X\left(2\right)}\left(t_x\right)={\sum }_{i=1}^{N_2}I(X_i\le t_x)/N_2$ is the population CDF of $I(X\le t_x)$ for the non-response group.

Yaqub and Shabbir [20] briefly studied the unbiased estimator of the population CDF of the research variable when there was non-response in the sample.

Let the sample CDF $\left({\widehat{F}}_y^{*}\left(t_y\right),{\widehat{F}}_x^{*}\left(t_x\right)\right)$ be the unbiased estimators of the population CDF $\left(F_Y^{*}\left(t_y\right),F_X^{*}\left(t_x\right)\right)$, based on ℓ units in the existence of non-response.

By using the Hansen and Hurwitz [11] approach, ${\widehat{F}}_y^{*}\left(t_y\right)$ is defined as

$${\widehat{F}}_y^{*}\left(t_y\right)={\omega }_1{\widehat{F}}_{y\left(1\right)}\left(t_y\right)+{\omega }_2{\widehat{F}}_{y\left(2\mathbf{q}\right)}\left(t_y\right),$$

(2)

where ${\omega }_1={\displaystyle \frac{{\ell }_1}{\ell }}$ and ${\omega }_2={\displaystyle \frac{{\ell }_2}{\ell }}$. In addition, we have the following:

${\widehat{F}}_{y\left(1\right)}\left(t_y\right)={\sum }_{i=1}^{{\ell }_1}I(Y_i\le t_y)/{\ell }_1$ denotes the sample CDF based on ${\ell }_1$ responding units out of ℓ units;

${\widehat{F}}_{\left(2q\right)}\left(t_y\right)={\sum }_{i=1}^{q}I(Y_i\le t_y)/q$ denotes the sample CDF based on q responding units out of ${\ell }_2$ non-response units.

Theorem 2. 

The mean and variance of ${\widehat{F}}_y^{*}\left(t_y\right)$ is defined as follows:

• ${\widehat{F}}_y^{*}\left(t_y\right)$ is an unbiased estimator of $F_Y\left(t_y\right)$, i.e., $E\left({\widehat{F}}_y^{*}\left(t_y\right)\right)=F_Y\left(t_y\right)$;
• The variance of ${\widehat{F}}_y^{*}\left(t_y\right)$ is defined by

  $$\mathit{Var}\left({\widehat{F}}_Y^{*}\left(t_y\right)\right)=\left[{\delta }_1{F}_{Y\left(1\right)}\left(t_y\right)\left(1-F_{Y\left(1\right)}\left(t_y\right)\right)+{\delta }_2{F}_{Y\left(2\right)}\left(t_y\right)\left(1-F_{Y\left(2\right)}\left(t_y\right)\right)\right],$$

  (3)

where ${\delta }_1={\displaystyle \frac{(N-\ell )}{N\ell }}$ and ${\delta }_2={\displaystyle \frac{{\mathcal{W}}_2(k-1)}{\ell }}$. Theorem 2. can be proved along the lines of [20].

Similarly, for the supplemental variable X, the estimator ${\widehat{F}}_x^{*}\left(t_x\right)$ is defined as

$${\widehat{F}}_x^{*}\left(t_x\right)={\omega }_1{\widehat{F}}_{x\left(1\right)}\left(t_x\right)+{\omega }_2{\widehat{F}}_{x\left(2\mathbf{q}\right)}\left(t_x\right).$$

In addition, we have the following:

${\widehat{F}}_{x\left(1\right)}\left(t_x\right)={\sum }_{i=1}^{{\ell }_1}I(X_i\le t_x)/{\ell }_1$ denotes the sample CDF based on ${\ell }_1$ responding units out of ℓ units;

${\widehat{F}}_{\left(2q\right)}\left(t_x\right)={\sum }_{i=1}^{q}I(X_i\le t_x)/q$ denotes the sample CDF based on q responding units out of ${\ell }_2$ non-response units.

Lemma 2. 

On the lines of Theorem 2, the mean and variance of ${\widehat{F}}_x^{*}\left(t_x\right)$ are defined as follows:

• ${\widehat{F}}_x^{*}\left(t_x\right)$ is an unbiased estimator of $F_X\left(t_x\right)$, i.e., $E\left({\widehat{F}}_x^{*}\left(t_x\right)\right)=F_X\left(t_x\right)$;
• The variance of ${\widehat{F}}_x^{*}\left(t_x\right)$ is defined by

  $$\begin{array}{c}\hfill Var\left({\widehat{F}}_x^{*}\left(t_x\right)\right)=\left[{\delta }_1{F}_{X\left(1\right)}\left(t_x\right)\left(1-F_{X\left(1\right)}\left(t_x\right)\right)+{\delta }_2{F}_{X\left(2\right)}\left(t_{x_2}\right)\left(1-F_{X\left(2\right)}\left(t_{x_2}\right)\right)\right];\end{array}$$
  .
• The covariance between ${\widehat{F}}_y^{*}\left(t_y\right)$ and ${\widehat{F}}_x^{*}\left(t_x\right)$ is given by

  $$\mathrm{Cov}\left({\widehat{F}}_y^{*}\left(t_y\right),{\widehat{F}}_x^{*}\left(t_x\right)\right)=\left[{\delta }_1\left({\displaystyle \frac{N_{11}{N}_{22}-N_{12}{N}_{21}}{N^2}}\right)+{\delta }_2\left({\displaystyle \frac{N_{11\left(2\right)}{N}_{22\left(2\right)}-N_{12\left(2\right)}{N}_{21\left(2\right)}}{N_{\left(2\right)}^2}}\right)\right].$$

  (For the proof see [20]).

In addition, let us define the following:

$S_{F_{Y\left(1\right)}\left(t_y\right)}^2=F_{Y\left(1\right)}\left(t_y\right)\left(1-F_{Y\left(1\right)}\left(t_y\right)\right)$ is the population variance of $I(Y\le t_y)$ for the response group;

$S_{F_{Y\left(2\right)}\left(t_y\right)}^2=F_{Y\left(2\right)}\left(t_y\right)\left(1-F_{Y\left(2\right)}\left(t_y\right)\right)$ is the population variance of $I(Y\le t_y)$ for the non-response group;

$S_{F_{X\left(1\right)}\left(t_x\right)}^2=F_{X\left(1\right)}\left(t_x\right)\left(1-F_{X\left(1\right)}\left(t_x\right)\right)$ is the population variance of $I(X\le t_x)$ for the response group;

$S_{F_{X\left(2\right)}\left(t_x\right)}^2=F_{X\left(2\right)}\left(t_x\right)\left(1-F_{X\left(2\right)}\left(t_x\right)\right)$ is the population variance of $I(X\le t_x)$ for the non-response group;

$C_{F_{Y\left(1\right)}\left(t_y\right)}=S_{F_{Y\left(1\right)}\left(t_y\right)}/F_{Y\left(1\right)}\left(t_y\right)$ is the population coefficient of variation of $I(Y\le t_y)$ for the response group;

$C_{F_{Y\left(1\right)}\left(t_y\right)}=S_{F_{Y\left(1\right)}\left(t_y\right)}/F_{Y\left(1\right)}\left(t_y\right)$ is the population coefficient of variation of $I(Y\le t_y)$ for the response group;

$C_{F_{X\left(2\right)}\left(t_x\right)}=S_{F_{X\left(2\right)}\left(t_x\right)}/F_{X\left(2\right)}\left(t_x\right)$ is the population coefficient of variation of $I(X\le t_x)$ for the non-response group;

$C_{F_{X\left(2\right)}\left(t_x\right)}=S_{F_{X\left(2\right)}\left(t_x\right)}/F_{X\left(2\right)}\left(t_x\right)$ is the population coefficient of variation of $I(X\le t_x)$ for the non-response group;

$S_{F_{Y\left(1\right)}\left(t_y\right)F_{X\left(1\right)}\left(t_x\right)}=\mathrm{Cov}\left(F_{Y\left(1\right)}\left(t_y\right),F_{X\left(1\right)}\left(t_x\right)\right)$ is the population covariance between $I(Y\le t_y)$ and $I(X\le t_x)$ for the response group;

$S_{F_{Y\left(2\right)}\left(t_y\right)F_{X\left(2\right)}\left(t_x\right)}=\mathrm{Cov}\left(F_{Y\left(2\right)}\left(t_y\right),F_{X\left(2\right)}\left(t_x\right)\right)$ is the population covariance between $I(Y\le t_y)$ and $I(X\le t_x)$ for the non-response group.

The following relative error terms are taken into account to determine the biases and MSEs of the existing and proposed estimators:

$$\begin{array}{ccccc}\hfill & e_0^{*}={\displaystyle \frac{{\widehat{F}}_y^{*}\left(t_y\right)-F_Y\left(t_y\right)}{F_Y\left(t_y\right)}},\hfill & \hfill e_1^{*}={\displaystyle \frac{{\widehat{F}}_x^{*}\left(t_x\right)-F_X\left(t_x\right)}{F_X\left(t_x\right)}},\mathrm{and}& \hfill & \hfill e_2={\displaystyle \frac{\widehat{F_x}\left(t_x\right)-F_X\left(t_x\right)}{F_X\left(t_x\right)}},\end{array}$$

such that $E\left(e_i^{*}\right)=0=E\left(e_2\right)$ for $i=0,1$, where $E(·)$ is mathematical expectation. Utilizing approximation up to the first order we have the following:

$$\begin{array}{cc}\hfill E\left(e_0^{*2}\right)\cong & {\displaystyle \frac{1}{F_Y^2\left(t_y\right)}}\left[{\delta }_1{F}_{Y\left(1\right)}\left(t_y\right)\left(1-F_{Y\left(1\right)}\left(t_y\right)\right)+{\delta }_2{F}_{Y\left(2\right)}\left(t_y\right)\left(1-F_{Y\left(2\right)}\left(t_y\right)\right)\right];\hfill \\ \hfill E\left(e_0^{*2}\right)\cong & {\displaystyle \frac{1}{F_Y^2\left(t_y\right)}}\left[{\delta }_1{S}_{F_{Y\left(1\right)}\left(t_y\right)}^2+{\delta }_2{S}_{F_{Y\left(2\right)}\left(t_y\right)}^2\right]\cong V_{200}^{*};\hfill \\ \hfill E\left(e_1^{*2}\right)\cong & {\displaystyle \frac{1}{F_X^2\left(t_x\right)}}\left[{\delta }_1{F}_{X\left(1\right)}\left(t_x\right)\left(1-F_{X\left(1\right)}\left(t_x\right)\right)+{\delta }_2{F}_{X\left(2\right)}\left(t_x\right)\left(1-F_{X\left(2\right)}\left(t_x\right)\right)\right];\hfill \\ \hfill E\left(e_1^{*2}\right)\cong & {\displaystyle \frac{1}{F_X^2\left(t_x\right)}}\left[{\delta }_1{S}_{F_{X\left(1\right)}\left(t_x\right)}^2+{\delta }_2{S}_{F_{X\left(2\right)}\left(t_x\right)}^2\right]\cong V_{020}^{*};\hfill \end{array}$$

$$\begin{array}{cc}\hfill & E\left(e_0^{*}{e}_1^{*}\right)\cong {\displaystyle \frac{1}{F_Y\left(t_y\right)F_X\left(t_x\right)}}\left[{\delta }_1\left({\displaystyle \frac{N_{11}{N}_{22}-N_{12}{N}_{21}}{N^2}}\right)+{\delta }_2\left({\displaystyle \frac{N_{11\left(2\right)}{N}_{22\left(2\right)}-N_{12\left(2\right)}{N}_{21\left(2\right)}}{N_{\left(2\right)}^2}}\right)\right];\hfill \\ \hfill & E\left(e_0^{*}{e}_1^{*}\right)\cong {\displaystyle \frac{1}{F_Y\left(t_y\right)F_X\left(t_x\right)}}\left[{\delta }_1{S}_{F_{Y\left(1\right)}\left(t_y\right)F_{X\left(1\right)}\left(t_x\right)}+{\delta }_2{S}_{F_{Y\left(2\right)}\left(t_y\right)F_{X\left(2\right)}\left(t_x\right)}\right]\cong V_{110}^{*};\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left(e_2^2\right)& \cong {\displaystyle \frac{1}{F_X^2\left(t_x\right)}}\left[{\delta }_1{F}_X\left(t_x\right)\left(1-F_X\left(t_x\right)\right)\right]\cong V_{002};\mathrm{and}\hfill \\ \hfill E\left(e_0^{*}{e}_2\right)& \cong {\displaystyle \frac{1}{F_Y\left(t_y\right)F_X\left(t_x\right)}}\left[{\delta }_1{S}_{\left(1\right)F\left(t_y\right)F\left(t_x\right)}\right]\cong V_{101}^{{*}^{\prime }}.\hfill \end{array}$$

There are two scenarios under consideration in the existence of non-response:

• Scenario I refers to non-response on both the study and auxiliary variables, whereas
• Scenario II solely refers to non-response on the study variable.

3. Some Modified Estimators for the CDF under Non-Response

3.1. Modified Estimators in Scenario I

In this section, some existing estimators for population mean estimation are modified for the estimation of a population CDF using SRS under Scenario I, i.e., non-response is present in both the study and the auxiliary variables. Furthermore, the biases and MSEs of the modified estimators are derived to the first order of approximation.

1.

The Cochran [25] ratio estimator is modified for $F_Y^{*}\left(t_y\right)$, and given by

$$\begin{array}{c}\hfill {\widehat{F}}_R^{*}\left(t_y\right)=\left({\displaystyle \frac{{\widehat{F}}_y^{*}\left(t_y\right)}{{\widehat{F_x}}^{*}\left(t_x\right)}}\right)F_X\left(t_x\right).\end{array}$$

(4)

  To the first order of approximation, the bias and MSE of Equation (4) are

$$\mathrm{Bias}\left({\widehat{F}}_R^{*}\left(t_y\right)\right)\cong F_Y\left(t_y\right)\left(V_{020}^{*}-V_{110}^{*}\right)\mathrm{and}$$

$$\begin{array}{c}\hfill \mathrm{MSE}\left({\widehat{F}}_R\left(t_y^{*}\right)\right)\cong F_Y^2\left(t_y\right)\left(V_{200}^{*}+V_{020}^{*}-2V_{110}^{*}\right).\end{array}$$

(5)

2.

Singh et al. [26] proposed an exponential estimator under non-response on both the study and auxiliary variables along the lines of Bahl and Tuteja [27]. The modified form of [26] for estimating the CDF is given by

$$\begin{array}{c}\hfill \widehat{F_{S_1}}\left(t_y^{*}\right)={\widehat{F}}_y^{*}\left(t_y\right)exp\left({\displaystyle \frac{F_X\left(t_x\right)-{\widehat{F_x}}^{*}\left(t_x\right)}{F_X\left(t_x\right)+{\widehat{F_x}}^{*}\left(t_x\right)}}\right).\end{array}$$

(6)

  The bias and MSEs of Equation (6) to the first order of approximation are given as

$$\begin{array}{cc}\hfill \mathrm{Bias}\left(\widehat{F_{S_1}}\left(t_x\right)\right)\cong & F_Y\left(t_y\right)\left({\displaystyle \frac{3}{8}}{V}_{020}^{*}-{\displaystyle \frac{1}{2}}{V}_{110}^{*}\right),\mathrm{and}\hfill \\ \hfill \mathrm{MSE}\left(F_{S_1}\left(t_x\right)\right)\cong & {\displaystyle \frac{F_Y^2\left(t_y\right)}{4}}\left(4V_{200}^{*}+V_{020}^{*}+4V_{110}^{*}\right).\hfill \end{array}$$

(7)

3.

The modified regression estimator for $F_Y^{*}\left(t_y\right)$ is provided as

$$\begin{array}{c}\hfill {\widehat{F}}_{Reg}\left(t_y\right)={\widehat{F_Y}}^{*}\left(t_y\right)+{\mathcal{B}}^{*}\left(F_X\left(t_x\right)-{\widehat{F_x}}^{*}\left(x\right)\right),\end{array}$$

(8)

where ${\mathcal{B}}^{*}$ is the regression co-efficient. Moreover, Equation (8) is an unbiased estimator of $F_Y\left(t_y\right)$.

  In addition, at the optimum value ${\mathcal{B}}_{\left(\mathrm{opt}\right)}^{*}=\left(S_{F_{Y\left(t_y\right)}{F}_{X\left(t_x\right)}}^{*}\right)/\left(S_{F_{X\left(t_x\right)}}^{*2}\right)$, the minimum variance of ${\widehat{F}}_{Reg}^{*}\left(t_y\right)$ is given as

$${\mathrm{MSE}}_{\min }\left(F_{Reg}^{*}\left(t_x\right)\right)=F_Y^2\left(t_y\right)V_{200}^{*}\left(1-{\displaystyle \frac{V_{110}^{*2}}{V_{200}^{*}{V}_{020}^{*}}}\right).$$

(9)

3.2. Modified Estimators in Scenario II

In this section, some of the existing estimators used to estimate the mean of a population, are modified for the estimation of the population CDF under Scenario II, i.e., non-response is present only on the study variable. Furthermore, the biases and MSEs of these modified estimators are obtained to their first order approximation. Let ${\widehat{F}}_{(.)}^{\prime }\left(t_y\right)$ be the estimator of the population CDF under Scenario II.

1.

The Cochran [25] ratio estimator is modified for $F_Y^{*}\left(t_y\right)$ under Scenario II, and given as

$$\begin{array}{c}\hfill {\widehat{F}}_R^{\prime }\left(t_x\right)=\left({\displaystyle \frac{{\widehat{F}}_y^{*}\left(t_y\right)}{\widehat{F_x}\left(t_x\right)}}\right)F_X\left(t_x\right).\end{array}$$

(10)

  Up to the first order of approximation, the bias and MSE of Equation (10) are

$$\mathrm{Bias}\left({\widehat{F}}_R^{\prime }\left(t_y\right)\right)\cong F_Y\left(t_y\right)\left(V_{020}^{*}-V_{101}^{{*}^{\prime }}\right),\mathrm{and}$$

$$\begin{array}{c}\hfill \mathrm{MSE}\left({\widehat{F}}_R^{\prime }\left(t_y\right)\right)\cong F_Y^2\left(t_y\right)\left(V_{200}^{*}+V_{020}-2V_{101}^{{*}^{\prime }}\right).\end{array}$$

(11)

2.

The exponential estimator of Singh et al. [26] is modified for estimating $F_Y^{*}\left(t_y\right)$ and is provided as

$$\begin{array}{c}\hfill {\widehat{F_{S_2}}}^{\prime }\left(t_y\right)={\widehat{F}}_y^{*}\left(t_y\right)exp\left({\displaystyle \frac{F_X\left(t_x\right)-\widehat{F_x}\left(t_x\right)}{F_X\left(t_x\right)+\widehat{F_x}\left(t_x\right)}}\right).\end{array}$$

(12)

  The bias and MSEs of Equation (12) up to the first order of approximation are given as

$$\begin{array}{cc}\hfill \mathrm{Bias}\left(\widehat{F_{S_2}^{\prime }}\left(t_y\right)\right)\cong & F_Y\left(t_y\right)\left({\displaystyle \frac{3}{8}}{V}_{002}-{\displaystyle \frac{1}{2}}{V}_{101}^{{*}^{\prime }}\right),\mathrm{and}\hfill \\ \hfill \mathrm{MSE}\left(F_{S_2}^{\prime }\left(t_y\right)\right)\cong & {\displaystyle \frac{F_Y^2\left(t_y\right)}{4}}\left(4V_{200}^{*}+V_{002}+4V_{101}^{{*}^{\prime }}\right).\hfill \end{array}$$

(13)

3.

The modified regression estimator for $F_Y^{*}\left(t_y\right)$ in Scenario II is provided as

$$\begin{array}{c}\hfill {\widehat{F^{\prime }}}_{Reg}\left(t_y\right)={\widehat{F_y}}^{*}\left(t_y\right)+{\mathcal{B}}^{\prime }\left(F_X\left(t_x\right)-\widehat{F_x}\left(t_x\right)\right),\end{array}$$

(14)

where ${\mathcal{B}}^{\prime }$ is said to be the regression coefficient. Moreover, Equation (14) is an unbiased estimator of ${\widehat{F_Y}}^{\prime }\left(t_y\right)$. In addition, at the optimum value ${\mathcal{B}}_{\left(\mathrm{opt}\right)}^{\prime }=\left(S_{F_{Y\left(t_y\right)}{F}_{X\left(t_x\right)}}^{*}\right)/\left(S_{F_{X\left(t_x\right)}}^2\right)$, the minimum variance of ${\widehat{F}}_{Reg}^{\prime }\left(t_y\right)$ is given as

$${\mathrm{Var}}_{\min }\left({\widehat{F_{Reg}}}^{\prime }\left(t_y\right)\right)=F_Y^2\left(t_y\right)V_{200}^{*}\left(1-{\displaystyle \frac{V_{101}^{{*}^{\prime }2}}{V_{200}^{*}{V}_{002}}}\right),$$

(15)

or

$${\mathrm{Var}}_{\min }\left(\widehat{F_{Reg}^{\prime }}\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left({\delta }_1{C}_{F_{Y\left(1\right)\left(t_y\right)}}^2(1-{\rho }_{F_y\left(t_y\right)F_x\left(t_y\right)}^2)+{\delta }_2\left(C_{F_{Y\left(2\right)}\left(t_y\right)}^2\right)\right).$$

4. Proposed Estimators

4.1. The Proposed Estimator in Scenario I

Following [17], an estimator for estimating a population CDF of the study variable under Scenario I is defined as

$${\widehat{F}}_{prop1}^{*}\left(t_y\right)={\widehat{F_y}}^{*}\left(t_y\right){\left({\displaystyle \frac{{\widehat{F}}_x^{*}\left(t_x\right)}{F_X\left(t_x\right)}}\right)}^{\alpha }exp\left({\displaystyle \frac{\left(F_X\left(t_x\right)-{\widehat{F_x}}^{*}\left(t_x\right)\right)}{\left(F_X\left(t_x\right)+{\widehat{F_x}}^{*}\left(t_x\right)\right)}}\right),$$

(16)

where $\alpha$$(-\infty <\alpha <+\infty )$ is unknown and needs to be estimated such that the MSE is minimum.

Theorem 3. 

The bias and MSE of (16) are given, respectively, as follows:

$$Bias\left({\widehat{F}}_{prop1}^{*}\left(t_y\right)\right)=F_Y\left(t_y\right)\left[\left(\frac{{\alpha }^2}{2}-\alpha +\frac{3}{8}\right)V_{020}^{*}+\left(\alpha -\frac{1}{2}\right)V_{110}^{*}\right],\mathrm{and}$$

$$MSE\left({\widehat{F}}_{prop1}^{*}\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left[V_{200}^{*}+\left({\alpha }^2-\alpha +\frac{1}{4}\right)V_{020}^{*}+\left(2\alpha -1\right)V_{110}^{*}\right].$$

(17)

Proof. 

Equation (16) is expressed in error terms as

$${\widehat{F}}_{prop1}^{*}\left(t_y\right)=F_Y\left(t_y\right)(1+e_0^{*}){\left(1+e_1^{*}\right)}^{\alpha }exp\left(\frac{(-e_1^{*})}{(2+e_1^{*})}\right).$$

(18)

Expanding Equation (18) up to the first order of approximation, yields

$$\begin{array}{c}\hfill {\widehat{F}}_{Prop1}^{*}\left(t_y\right)=F_Y\left(t_y\right)(1+e_0^{*})\left(1+\alpha e_1^{*}+\frac{\alpha (\alpha -1)}{2}{e}_1^{*2}\right)\left(1-\frac{1}{2}{e}_1^{*}+\frac{3}{8}{e}_1^{*2}-\cdots \right).\end{array}$$

(19)

Keeping the terms up to the second power and extending the above equation, we get the following:

$$\begin{array}{cc}\hfill {\widehat{F}}_{prop1}^{*}\left(t_y\right)=& F_Y\left(t_y\right)\left[1-\frac{e_1^{*}}{2}+\frac{3e_1^{*2}}{8}+\alpha e_1^{*}-\frac{\alpha }{2}{e}_1^{*2}+\frac{{\alpha }^2}{2}{e}_1^{*2}-\frac{\alpha }{2}{e}_1^{*2}+e_0^{*}-\frac{e_0^{*}{e}_1^{*}}{2}+\alpha e_0^{*}{e}_1^{*}\right]\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\widehat{F}}_{prop1}^{*}\left(t_y\right)-F_Y\left(t_y\right)=F_Y\left(t_y\right)\left[e_0^{*}+e_1^{*}\left(\alpha -\frac{1}{2}\right)+\frac{3e_1^{*2}}{8}-\alpha e_1^{*2}+\frac{{\alpha }^2}{2}{e}_1^{*2}-\frac{e_0^{*}{e}_1^{*}}{2}+\alpha e_0^{*}{e}_1^{*}\right].\end{array}$$

(20)

After simplifying the expectation on both sides of Equation (20), we obtain the bias of (16):

$$\mathrm{Bias}\left({\widehat{F}}_{prop1}^{*}\left(t_y\right)\right)=F_Y\left(t_y\right)\left[\left(\frac{{\alpha }^2}{2}-\alpha +\frac{3}{8}\right)V_{020}^{*}+\left(\alpha -\frac{1}{2}\right)V_{110}^{*}\right].$$

Squaring (20) and applying the expectation yield MSE of ${\widehat{F}}_{prop1}^{*}\left(t_y\right)$ we obtain

$$\begin{array}{c}\hfill {\left({\widehat{F}}_{prop1}^{*}\left(t_y\right)-F_Y\left(t_y\right)\right)}^2=F_Y^2\left(t_y\right)\left(e_0^{*2}+{\alpha }^2{e}_1^{*2}+\frac{1}{4}{e}_1^{*2}+2\alpha e_0^{*}{e}_1^{*}-e_0^{*}{e}_1^{*}-\alpha e_1^{*2}\right).\end{array}$$

The MSE of (16) is obtained as

$$\mathrm{MSE}\left({\widehat{F}}_{prop1}^{*}\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left[V_{200}^{*}+\left({\alpha }^2-\alpha +\frac{1}{4}\right)V_{020}^{*}+\left(2\alpha -1\right)V_{110}^{*}\right].$$

(21)

□

Hence the theorem is proved.

Theorem 4. 

The minimum MSE of ${\widehat{F}}_{prop1}^{*}\left(t_y\right)$ is given as follows:

$${MSE}_{min}\left({\widehat{F}}_{prop1}^{*}\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left[V_{200}^{*}-{\displaystyle \frac{{\left(V_{110}^{*}\right)}^2}{V_{020}^{*}}}\right].$$

(22)

Proof. 

Differentiating Equation (17) with respect to $\alpha$ and simplifying it to obtain the optimal value of $\alpha$ for minimal $MSE$, we get

$$\begin{array}{c}\hfill {\alpha }_{\left(\mathrm{opt}\right)}=\frac{V_{020}^{*}-2V_{110}^{*}}{2V_{020}^{*}}.\end{array}$$

Substituting ${\alpha }_{\left(\mathit{opt}\right)}$ into (17) we obtain the minimal MSE of ${\widehat{F}}_{prop1}^{*}\left(t_y\right)$, such that

$${\mathrm{MSE}}_{min}\left(F_{prop1}^{*}\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left[V_{200}^{*}-{\displaystyle \frac{{\left(V_{110}^{*}\right)}^2}{V_{020}^{*}}}\right].$$

(23)

□

Hence the theorem is proved.

4.2. Proposed Estimator in Scenario II

Motivated by [28], to estimate the population CDF in the presence of non-response under Scenario II, an estimator is proposed as

$$F_{prop2}^{\prime }\left(t_y\right)={\widehat{F_y}}^{*}\left(t_y\right){\left({\displaystyle \frac{{\widehat{F}}_x\left(t_x\right)}{F_X\left(t_x\right)}}\right)}^{{\alpha }_1}exp\left({\displaystyle \frac{\left(F_X\left(t_x\right)-\widehat{F_x}\left(t_x\right)\right)}{\left(F_X\left(t_x\right)+\widehat{F_x}\left(t_x\right)\right)}}\right),$$

(24)

where ${\alpha }_1$$(-\infty <{\alpha }_1<+\infty )$ is an unknown and needs to be estimated such that the MSE is minimum.

Theorem 5. 

The $Bias\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)$ and $MSE\left({\widehat{F^{\prime }}}_{prop2}\left(t_y\right)\right)$ are given, respectively, by the following:

$$Bias\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)=F_Y\left(t_y\right)\left[\left(\frac{{\alpha }_1^2}{2}-{\alpha }_1+\frac{3}{8}\right)V_{002}+\left({\alpha }_1-\frac{1}{2}\right)V_{101}^{{*}^{\prime }}\right]\mathrm{and}$$

$$MSE\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left[V_{200}^{*}+\left({\alpha }_1^2-{\alpha }_1+\frac{1}{4}\right)V_{002}+\left(2{\alpha }_1-1\right)V_{101}^{{*}^{\prime }}\right].$$

(25)

Proof. 

In error terms Equation (24) can be expressed as

$${\widehat{F}}_{prop2}^{\prime }\left(t_y\right)=F_Y\left(t_y\right)(1+e_0^{*}){\left(1+e_2\right)}^{{\alpha }_1}exp\left(\frac{(-e_2)}{(2+e_2)}\right).$$

(26)

Expanding Equation (26) to the first order of approximation yields

$$\begin{array}{cc}\hfill & {\widehat{F}}_{prop2}^{\prime }\left(t_y\right)=F_Y\left(t_y\right)(1+e_0^{*})\left(1+{\alpha }_1{e}_2+\frac{{\alpha }_1({\alpha }_1-1)}{2}{e}_2^2\right)\left(1-\frac{1}{2}{e}_2+\frac{3}{8}{e}_2^2-\cdots \right).\hfill \end{array}$$

(27)

Keeping the terms up to the second power and extending the above equation, we get

$$\begin{array}{cc}\hfill {\widehat{F}}_{prop2}^{\prime }\left(t_y\right)=& F_Y\left(t_y\right)\left[1-\frac{1}{2}{e}_2+\frac{3}{8}{e}_2^2+{\alpha }_1{e}_2-\frac{{\alpha }_1}{2}{e}_2^2+\frac{{\alpha }_1^2}{2}{e}_2^2-\frac{{\alpha }_1}{2}{e}_2^2+e_0^{*}-\frac{e_0^{*}{e}_2}{2}+{\alpha }_1{e}_0^{*}{e}_2\right]\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\widehat{F}}_{prop2}^{\prime }\left(t_y\right)-F_Y\left(t_y\right)=F_Y\left(t_y\right)\left[e_0^{*}+\left({\alpha }_1-\frac{1}{2}\right)e_2+\left(\frac{{\alpha }_1^2}{2}-{\alpha }_1+\frac{3}{8}\right)e_2^2+\left({\alpha }_1-\frac{1}{2}\right)e_0^{*}{e}_2\right].\end{array}$$

(28)

After simplifying the expectation on both sides of Equation (28), we obtain the bias of (24) as

$$\mathrm{Bias}\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)=F_Y\left(t_y\right)\left[\left(\frac{{\alpha }_1^2}{2}-{\alpha }_1+\frac{3}{8}\right)V_{002}+\left({\alpha }_1-\frac{1}{2}\right)V_{101}^{{*}^{\prime }}\right].$$

Squaring (28) and applying the expectation after simplification we obtain

$$\begin{array}{c}\hfill E{\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)-F_Y\left(t_y\right)\right)}^2=F_Y^2\left(t_y\right)E\left(e_0^{*2}+{\alpha }_1^2{e}_2^2+\frac{1}{4}{e}_2^2+2{\alpha }_1{e}_0^{*}{e}_2-e_0^{*}{e}_2-{\alpha }_1{e}_2^2\right).\end{array}$$

(29)

The MSE of ${\widehat{F}}_{prop2}^{\prime }\left(t_y\right)$ is determined as

$$\mathrm{MSE}\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left[V_{200}^{*}+\left({\alpha }_1^2-{\alpha }_1+\frac{1}{4}\right)V_{002}+\left(2{\alpha }_1-1\right)V_{101}^{{*}^{\prime }}\right].$$

□

Hence the theorem is proved.

Theorem 6. 

The Minimum MSE of ${\widehat{F}}_{prop2}^{\prime }\left(t_y\right)$ is given as

$${MSE}_{min}\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left[V_{200}^{*}-{\displaystyle \frac{{\left(V_{101}^{{*}^{\prime }}\right)}^2}{V_{002}}}\right].$$

(30)

Proof. 

Differentiating Equation (25) with respect to ${\alpha }_1$, equating it to zero, and simplifying it to obtain the optimal value of ${\alpha }_1$ for the minimum MSE, we get

$$\begin{array}{c}\hfill {\alpha }_{1\left(\mathit{opt}\right)}=\frac{V_{002}-2V_{110}^{{*}^{\prime }}}{2V_{002}}.\end{array}$$

(31)

Substituting ${\alpha }_{1\left(\mathit{opt}\right)}$ into (25) we obtain the minimum MSE of ${\widehat{F}}_{prop2}^{\prime }\left(t_y\right)$ as

$${\mathrm{MSE}}_{min}\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)=F_Y^2\left(t_y\right)\left[V_{200}^{*}-{\displaystyle \frac{{\left(V_{101}^{{*}^{\prime }}\right)}^2}{V_{002}}}\right].$$

(32)

□

Hence the theorem is proved.

5. Efficiency Comparisons

The MSEs of the modified and proposed estimators ${\widehat{F}}_{prop1}^{*}\left(t_y\right)$ are compared in this section.

5.1. Efficiency Comparisons under Scenario I

The proposed estimator under Scenario I is more efficient if we have the following:

(a)

From (5) and (22),

$$\begin{array}{cc}\hfill \mathrm{MSE}\left({\widehat{F^{*}}}_R\left(t_y\right)\right)& >{\mathrm{MSE}}_{min}\left({\widehat{F^{*}}}_{prop1}\left(t_y\right)\right)\mathrm{if}\hfill \\ \hfill \mathrm{MSE}\left({\widehat{F^{*}}}_R\left(t_y\right)\right)& -{\mathrm{MSE}}_{min}\left({\widehat{F^{*}}}_{prop1}\left(t_y\right)\right)>0,\mathrm{or}\hfill \\ \hfill {\displaystyle \frac{V_{110}^{*}}{V_{020}^{*}}}& <1;\hfill \end{array}$$

(b)

From (7) and (22),

$$\begin{array}{cc}\hfill \mathrm{MSE}\left({\widehat{F^{*}}}_{S1}\left(t_y\right)\right)& >{\mathrm{MSE}}_{min}\left({\widehat{F^{*}}}_{prop1}\left(t_y\right)\right)\mathrm{if}\hfill \\ \hfill \mathrm{MSE}\left({\widehat{F^{*}}}_{S1}\left(t_y\right)\right)& -{\mathrm{MSE}}_{min}\left({\widehat{F^{*}}}_{prop1}\left(t_y\right)\right)>0,\mathrm{or}\hfill \\ \hfill {\displaystyle \frac{{(V_{110}^{*}+V_{020}^{*})}^2}{2V_{020}^{*}{V}_{110}^{*}}}& >-1;\hfill \end{array}$$

(c)

From (9) and (22), it can be shown iff $\alpha =\frac{V_{020}^{*}-2V_{110}^{*}}{2V_{020}^{*}}$,

$$\begin{array}{cc}\hfill & {\mathrm{MSE}}_{min}\left({\widehat{F}}_{prop1}^{*}\left(t_y\right)\right)={\mathrm{MSE}}_{min}\left({\widehat{F}}_{Reg}^{*}\left(t_y\right)\right).\hfill \end{array}$$

5.2. Efficiency Comparisons under Scenario II

The proposed estimator under Scenario II is more efficient compared to existing modified estimators if we have the following:

(a)

From (11) and (30),

$$\begin{array}{cc}\hfill \mathrm{MSE}\left({\widehat{F^{\prime }}}_R\left(t_y\right)\right)& >{\mathrm{MSE}}_{min}\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)\mathrm{if}\hfill \\ \hfill \mathrm{MSE}\left({\widehat{F^{\prime }}}_R\left(t_y\right)\right)& -{\mathrm{MSE}}_{min}\left({\widehat{F^{*}}}_{prop2}\left(t_y\right)\right)>0,\mathrm{or}\hfill \\ \hfill {\displaystyle \frac{V_{101}^{{*}^{\prime }}}{V_{002}}}& <1;\hfill \end{array}$$

(b)

From (13) and (30),

$$\begin{array}{cc}\hfill & \mathrm{MSE}\left({\widehat{F^{\prime }}}_{S2}\left(t_y\right)\right)>{\mathrm{MSE}}_{min}\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)\mathrm{if}\hfill \\ \hfill & \mathrm{MSE}\left({\widehat{F}}_{S2}^{\prime }\left(t_y\right)\right)-{\mathrm{MSE}}_{min}\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)>0,\mathrm{or}\hfill \\ \hfill & {\displaystyle \frac{{(V_{101}^{{*}^{\prime }}+V_{002})}^2}{2V_{002}{V}_{101}^{{*}^{\prime }}}}>-1;\hfill \end{array}$$

(c)

From (15) and (30), it can be shown iff ${\alpha }_1=\frac{V_{002}-2V_{110}^{{*}^{\prime }}}{2V_{002}}$,

$${\mathrm{MSE}}_{min}\left({\widehat{F}}_{prop2}^{\prime }\left(t_y\right)\right)={\mathrm{MSE}}_{min}\left({\widehat{F}}_{Reg}^{\prime }\left(t_y\right)\right).$$

6. Numerical Study

An empirical evaluation is presented to evaluate the performance of the proposed estimators and some of the existing estimators, by using three different populations. The summary statistics for these populations are shown in Table 1, Table 2 and Table 3 respectively:

Table 1. Summary statistics for Population I.

Parameter	Value	Parameter	Value
N	69	$S_2$	0.50361
n	20	$S_3$	7058.99
${\delta }_1$	0.03550	$S_{44}^2$	20.0602
$\overline{X}$	4954.44	$R_{12}$	0.94202
$\overline{R}$	35.0000	$R_{13}$	$-0.57090$
$F_Y\left(t_y\right)$	0.50725	$R_{23}$	$-0.57274$
$F_X\left(t_x\right)$	0.50725	$R_{14}$	$-0.85584$
$S_1$	0.50361	$R_{24}$	−0.86603
Non-response
Parameter	Value	Parameter	Value
$N_{\left(2\right)}$	17	$S_{4\left(2\right)}$	5.04975
$W_2$	0.24638	$R_{12\left(2\right)}$	0.88273
${\delta }_2$	0.01231	$R_{13\left(2\right)}$	$-0.62943$
$S_{1\left(2\right)}$	0.51449	$R_{23\left(2\right)}$	$-0.57830$
$S_{3\left(2\right)}$	5920.86	$R_{24\left(2\right)}$	−0.85391

Population I (source: [29])

Table 2. Summary statistics for Population II.

Parameter	Value	Parameter	Value
N	50	$S_2$	0.5051
n	15	$S_3$	21.805
$\lambda$	0.04667	$S_4$	15756
$\overline{R}$	25.5000	$R_{13}$	0.2518
$F_Y\left(t_y\right)$	0.5000	$R_{23}$	−0.7952
$F_X\left(t_x\right)$	0.5000	$R_{14}$	0.2134
$S_1$	0.5051	$R_{24}$	−0.8663

Population II (source: [30])

Table 3. Summary statistics for Population III.

Parameter	Value	Parameter	Value
N	50	$S_2$	0.50508
n	15	$S_3$	21.3175
${\delta }_1$	0.04667	$S_4$	14.5756
$\overline{X}$	78.2900	$R_{12}$	$-0.12000$
$\overline{R}$	25.5000	$R_{13}$	0.22925
$F_Y\left(t_y\right)$	0.50000	$R_{23}$	$-0.78936$
$F_X\left(t_x\right)$	0.50000	$R_{14}$	0.18435
$S_1$	0.50508	$R_{24}$	$-0.86630$
Non-response
Parameter	Value	Parameter	Value
$N_2$	12	$S_{4\left(2\right)}$	3.60555
$w_2$	0.24000	$R_{12\left(2\right)}$	$-0.16903$
${\delta }_2$	0.01600	$R_{13\left(2\right)}$	0.25695
$S_{1\left(2\right)}$	0.51493	$R_{23\left(2\right)}$	$-0.81369$
$S_{2\left(2\right)}$	0.52223	$R_{14\left(2\right)}$	0.22034
$S_{3\left(2\right)}$	18.2593	$R_{24\left(2\right)}$	$-0.86905$

Population III (source: [30])

MSEs of the modified estimators and the proposed estimators are given in Table 4 and Table 5, with respect to Scenario I and Scenario II, respectively.

Table 4. MSEs under Scenario I.

Estimator	Data 1	Data 2	Data 3
$\mathrm{MSE}{\widehat{F}}_y^{*}\left(t_y\right)$	0.02942125	0.03513561	0.04084997
$\mathrm{MSE}{\widehat{F}}_R^{*}\left(t_y\right)$	0.10046990	0.11875580	0.13704170
$\mathrm{MSE}{\widehat{F}}_{S1}^{*}\left(t_y\right)$	0.05427591	0.06176173	0.05427591
$\mathrm{MSE}{\widehat{F}}_{Reg}^{*}\left(t_y\right)$	0.01471663	0.01847541	0.02223419
$\mathrm{MSE}{\widehat{F}}_{prop1}^{*}\left(t_y\right)$	0.01469690	0.01840932	0.02210629

Table 5. MSEs under Scenario II.

Estimator	Data 1	Data 2	Data 3
$\mathrm{MSE}{\widehat{F}}_y^{\prime }\left(t_y\right)$	0.02942125	0.03513561	0.04084997
$\mathrm{MSE}{\widehat{F}}_R^{\prime }\left(t_y\right)$	0.08789827	0.09361262	0.09932698
$\mathrm{MSE}{\widehat{F}}_{S2}\left(t_y\right)$	0.05273304	0.05844739	0.06416175
$\mathrm{MSE}{\widehat{F}}_{Reg}^{\prime }\left(t_y\right)$	0.01667221	0.02238657	0.02810093
$\mathrm{MSE}{\widehat{F}}_{prop2}^{\prime }\left(t_y\right)$	0.01667221	0.02238657	0.02810093

The proposed estimators and the modified estimators of a population CDF were compared to the variance of ${\widehat{F}}_y\left(t_y\right)$ under both scenarios in terms of their percent-relative efficiencies (PREs) by using the following formula:

$$\mathrm{PRE}(·)={\displaystyle \frac{\mathrm{MSE}(·)}{\mathrm{MSE}{\widehat{F}}_y\left(t_y\right)}}\times 100\%.$$

PREs of the proposed estimators, and the modified estimators, are given in Table 6 and Table 7, with respect to Scenario I and Scenario II.

Table 6. PREs of estimators under Scenario I.

Estimator	Data 1	Data 2	Data 3
$\mathrm{PRE}{\widehat{F}}_y^{*}\left(t_y\right)$	100	100	100
$\mathrm{PRE}{\widehat{F}}_R^{*}\left(t_y\right)$	29.28	29.59	29.81
$\mathrm{PRE}{\widehat{F}}_{S1}^{*}\left(t_y\right)$	54.21	56.89	75.26
$\mathrm{PRE}{\widehat{F}}_{Reg}^{*}\left(t_y\right)$	199.99	190.18	184.73
$\mathrm{PRE}{\widehat{F}}_{prop1}^{*}\left(t_y\right)$	200.19	190.86	184.79

Table 7. PREs of the estimators under Scenario II.

Estimator	Data 1	Data 2	Data 3
$\mathrm{PRE}{\widehat{F}}_y^{\prime }\left(t_y\right)$	100	100	100
$\mathrm{PRE}{\widehat{F}}_R^{\prime }\left(t_y\right)$	33.47	37.53	41.13
$\mathrm{PRE}{\widehat{F}}_{S2}^{\prime }\left(t_y\right)$	55.79	60.11	63.67
$\mathrm{PRE}{\widehat{F}}_{Reg}\left(t_y\right)$	176.47	156.95	145.37
$\mathrm{PRE}{\widehat{F}}_{prop2}^{\prime }\left(t_y\right)$	176.47	156.95	145.37

From Table 6 and Table 7 we have the following:

•

It was observed that the PREs corresponding of the estimators ${\widehat{F}}_R^{*}\left(t_y\right)$, ${\widehat{F}}_{S1}^{*}\left(t_y\right)$, ${\widehat{F}}_R^{\prime }\left(t_y\right)$, and ${\widehat{F}}_{S2}^{\prime }\left(t_y\right)$ declined.

•

The PREs corresponding to the proposed estimators, ${\widehat{F}}_{prop1}^{*}\left(t_y\right)$ and ${\widehat{F}}_{prop2}^{\prime }\left(t_y\right)$, and the modified regression estimators, ${\widehat{F}}_{Reg}^{*}\left(t_y\right)$ and ${\widehat{F}}_{Reg}^{\prime }\left(t_y\right)$, showed that the proposed estimators were efficient estimators under both scenarios of non-response.

7. Conclusions

This study proposed an improved class of estimators for the estimation of a finite population CDF under two different scenarios of non-response using SRS. From theoretical and empirical comparisons, the proposed estimators were found to be perform better, based on large PRE and smaller MSE criteria. Therefore, our study suggests the use of the proposed estimators for estimating the CDF in the presence of non-response. Limitations of this study are provided in the Appendix A.