A New Generalized Class of Exponential Factor-Type Estimators for Population Distribution Function Using Two Auxiliary Variables

Department of Statistics, Abdul Wali Khan University, Mardan, Pakistan Department of Statistics, Quaid-i-azam University, Islamabad, Pakistan Department of Statistics, University of Wah, Wah Cantt, Pakistan Department of Applied Mathematics and Statistics, Institute of Space Techonalogy, Islamabad, Pakistan Faculty of Science and Technology, Rajamangala University of Technology Suvarnabhumi, Nonthaburi 11000, ailand Department of Mathematics, Faculty of Science, Phuket Rajabhat University (PKRU), Raddasa, Phuket 83000, ailand


Introduction
In survey sampling, it is well-known fact that the suitable use of auxiliary information improves the precision of estimators by taking advantage of the correlation between the study variable and the auxiliary variable. Several estimators exist in the literature for estimating various population parameters including mean, median, and total, but little attention has been paid to estimate the distribution function (DF). Some important references to the population mean using the auxiliary information include . In their work, several authors have suggested improved ratio, product, and regression type estimators for estimating the nite population mean. e problem of estimating a nite population (DF) arises when we are interested to nd out the proportion of certain values that are less than or equal to the threshold value. ere are situations where estimating the cumulative distribution function is assessed as essential. For example, for a nutritionist, it is interesting to know the proportion of the population that consumed 25 percent or more of the calorie intake from saturated fat. Similarly, a soil scientist may be interested in knowing the proportion of people living in a developing country below the poverty line. In certain situations, the need of a cumulative distribution function is much more important. Some important work in the eld of distribution function (DF) includes [22], which suggested an estimator for estimating the DF that requires information both on the study variable and the auxiliary variable. On similar lines, [23] proposed ratio and di erence-type estimators for estimating the DF using the auxiliary information. Ahmad and Abu-Dayyeh [24] estimated the DF using the information on multiple auxiliary variables. Rueda et al. [25] used a calibration approach for estimating the DF. Singh et al. [26] considered the problem of estimating the DF and quantiles with the use of auxiliary information at the estimation stage, [27] considered a generalized class of estimators for estimating the DF in the presence of nonresponse, [28] suggested finite population distribution function estimation with dual use of auxiliary information under simple and stratified random sampling. Moreover, two new estimators were proposed for estimating the DF in simple and stratified sampling using the auxiliary variable and rank of the auxiliary variable.
e rest of the article is composed as follows: in Section 2, some notations and symbols are given. In Section 3, the existing estimators for estimating the DF are given. In Section 4, we define two new generalized exponential factor type estimators. Section 6 discusses the numerical study of the proposed class of estimators. We also conduct a simulation study for the support of our proposed generalized family of estimators in Section 7. Section 8 gives the concluding remarks.

Notations and Symbols
A finite population Ω � 1, 2, . . . , N { } of N distinct and identified units is considered. To estimate the finite population distribution function (DF), a sample of size n units is drawn from a population using simple random sampling without replacement (SRSWOR). Let Y i , X i and R x be the values of the study variable Y, the auxiliary variable X and R x rank of the auxiliary variable X, respectively. Let I(Y ≤ t y ) and I(X ≤ t x ) be the indicator variables based on Y and X, respectively.
Let F(t y ) � n i�1 I(Y i ≤ t y )/n and F(t x ) � n i�1 I (X i ≤ t x )/n be the sampled distribution functions corresponding to the population distribution function F(t y ) � n i�1 I(X i ≤ t y )/n and F(t x ) � n i�1 I(X i ≤ t x )/n, respectively. Let Y, X, and R x be the sample means corresponding to population means Y, X, and R x , respectively. Let be the population covariances between I(Y ≤ t y ), I(X ≤ t x ), I(Y ≤ t y ), I(X ≤ t x ) and R x , respectively. Let , be the correlation coefficients between I(Y ≤ t y ) and I(X ≤ t x ), I(Y ≤ t y ) and X, I(X ≤ t x ) and X, I(Y ≤ t y ) and R x , I(X ≤ t x ) and R x , respectively. Let ρ 2 1.23 � (ρ 2 12 + ρ 2 13 − 2ρ 12 ρ 13 ρ 23 )/(1 − ρ 2 23 ), ρ 2 1.24 � (ρ 2 12 + ρ 2 14 − 2ρ 12 ρ 14 ρ 24 )/(1 − ρ 2 24 ) be the population coefficients of multiple determination of (Y ≤ t y ) on I(X ≤ t x ) and X, I(Y ≤ t y ) on I(X ≤ t x ) and R x , respectively.
To obtain the biases and mean squared errors (MSEs) of the adapted and proposed estimators of F(t y ), we consider the following relative error terms. Let (3)

Existing Estimators
In this section, we briefly review some existing estimators of F(t y ).
(1) e conventional unbiased mean per unit estimator of F(t y ) is e variance of F(t y ) is given by the following equation: 2 Mathematical Problems in Engineering (2) e traditional ratio estimator of F(t y ) is e bias and MSE of F(t y ) R , to the first order of approximation, respectively, are given by the following equation: e ratio estimator F(t y ) R performs better than F(t y ), in terms of MSE, if ρ 12 > C 2 /(2C 1 ).
(3) Reference [29] suggested the usual product estimator of F(t y ): e bias and MSE of F(t y ) P , to the first order of approximation are given by the following equation: e product estimator F(t y ) P is better than F(t y ), in terms of MSE, if ρ 12 <− C 2 /(2C 1 ). (4) e conventional difference estimator of F(t y ) is where m is an unknown constant. e minimum variance of F(t y ) Reg at the optimum value of m (opt) � ρ 12 (δ 1 /δ 2 ), is given by the following equation: (5) Reference [4] suggested an improved difference-type estimator of F(t y ), given by the following equation: where m 1 and m 2 are unknown constants. e optimum values of m 1 and m 2 are e bias and minimum MSE of F(t y ) R,D , to the first order of approximation, respectively, are given by the following equation: (6) Reference [30] suggested the exponential ratio-type and product-type estimators are given by the following equation: e biases and MSEs of F(t y ) BT,R and F(t y ) BT,P , to the first order of approximation, respectively, are given by the following equation: Bias F t y BT,P � λF t y (7) Reference [14] suggested a generalized class of ratiotype exponential estimators as follows: Mathematical Problems in Engineering where m 3 and m 4 are unknown constants.
e optimum values of m 3 and m 4 , determined by minimizing (26) are given by the following equation: e bias of (F(y) GK ), are given by the following equation: e minimum MSE of F(y) GK at the optimum values of m 3 and m 4 is given by the following equation:

Proposed Estimator
Using the appropriate auxiliary information during the estimation stage or at the design stage improves an estimator's efficiency. e sample distribution function of the auxiliary variable has already been employed to increase the efficiency and quality of estimators. e study of [20] suggested using the rank of the auxiliary variable as an additional auxiliary variable to improve the precision of a population distribution function estimator. In this article, we used two auxiliary variables to estimate the finite distribution function; we need additional auxiliary information on the sample mean and sample distribution function of the auxiliary variable, as well as the sample distribution function of the study variable. In literature, auxiliary information using the distribution function has been rarely attempted, therefore we are motivated towards it. e principal advantage of our proposed generalized class of estimator is that it is more flexible, and efficient than the existing estimators.

First Proposed Estimator.
On the lines of [31], we suggest a generalized class of exponential factor type estimators which contains many stable and efficient estimators. By combining the idea of [30,31], the first estimator is given by the following equation: where Substituting different values of K i (i � 1, 2, 3, 4) in Equation (32), we can generate many more types of estimators from our general proposed class of estimators, given in Table 1.
By solving F(t y ) (k 1 ,k 2 ) Prop1 given in (32) up-to first order of approximation, we have the following equation: where Or F t y Using (39), the bias and MSE of F(t y ) (k 1 ,k 2 ) Prop1 are given by the following equation: Bias F t y Differentiate (40) with respect to ϑ 1 and ϑ 2 , we get the optimum values of ϑ 1 and ϑ 2 i.e., Substituting the optimum values of ϑ 1(opt) and Prop1 and is given by the following equation: where is the coefficient of multiple determination of F(t y ) on F(t x ) and X.

Second Proposed Estimator.
To increase the efficiency of the estimators both at the design stage as well as at the estimation stage, we utilize the auxiliary information. When there exists a correlation between the study variable and the auxiliary variable, then the rank of the auxiliary variable is also correlated with the study variable. e rank of the auxiliary variable can be treated as a new auxiliary variable, and this information may also be used to increase the precision of the estimators. Based on the idea of rank, we propose a second new class of factor type estimators of the finite population distribution function. e estimator is given by the following equation: where Substituting different values of K i (i � 1, 2, 3, 4) in Equation (45), we can generate many more different types of estimators from our general proposed class of estimators, given in Table 2.
Solving F(t y ) (k 1 ,k 2 ) Prop2 given in (45) in terms of errors, we have the following equation: where With first order approximation, we have the following equation: 6 Mathematical Problems in Engineering Using (52) the bias and MSE of F(t y ) (k 1 ,k 2 ) Prop2 are given by the following equation: (53) Differentiate Equation (54) with respect to ϑ 1 and ϑ 2 for minimum MSE, we get the optimum values of ϑ 1 and ϑ 2 , Substituting the optimum values of ϑ 1(opt) and ϑ 2(opt) in Equation ≠ 25, we get minimum MSE of F(t y ) (k 1 ,k 2 ) Prop2 is given by the following equation: where is the coefficient of multiple determination of F(t y ) on F(t x ) and R x .

Theoretical Comparison
In this section, the adapted and proposed estimators of F(t y ) are compared in terms of the minimum mean square error.

Numerical Study
In this section, we conduct a numerical study to investigate the performances of the adapted and proposed DF estimators. For this purpose, three populations are     considered. e summary statistics of these populations are reported in Tables 3-5. e percentage relative efficiency (PRE) of the estimator F i (t y ) with respect to F (t y ) is given by the following equation:: where i � R, P, Reg, (R, D), (BT, R), (BT, P), (G, K), e PREs of the distribution function estimators, computed from three populations, are given in Tables 6-9 Population 1. (Source: [32]) Y: number of teachers and X: number of students.     Table 9: PREs of distribution function estimators for population I, II, and III, when F(t y ) and F(t x ) � Q 3 (y), Q 3 (x). In Tables 6-9 we use Y, X, Y, X, Q 1 (y), Q 1 (x), Q 3 (y) and Q 3 (x) for indicator functions of Y and X, respectively. Here,

Estimators Population 1 Population 2 Population 3
when we used Y as indicator function of Y and X as indicator function of X, we get PRE in Table 6, and when we used Q 1 (y) as indicator function of Y and Q 1 (x) as indicator function of X, we get PRE given in Table 7. And, similarly when we used Y as indicator function of Y and X as indicator function of X, we get PRE in Table 8, and when we used Q 3 (y) as indicator function of Y and Q 3 (x) as indicator function of X, we get PRE given in Table 9.
Here we take three data sets for numerical illustration, respectively.
In Tables 6-9, we observe that the proposed class of estimators are more precise than the existing estimators in terms of PREs.

Simulation Study
A simulation study is conducted to obtain the efficiency of the suggested estimators under simple random sampling when the auxiliary variables and rank of the auxiliary variable are used. We have generated three populations of size 1,000 from a multivariate normal distribution with different covariance matrices. All populations have different correlations, i.e., Population I is negatively correlated, Population II is positively correlated, and Population III has a strong positive correlation between X and Y variables. Population averages and covariance matrices are given as follows.

Population III
In this study, we consider the generated population for summarizing the simulation procedures. e simulation results of PRE are given in Tables 10-13.
In Tables 10-13, it can be seen that the proposed estimators perform better than all existing estimators. e percent relative efficiency shows that the second proposed family of estimators with simple random sampling yields the best result when the variables Y and X have a positive correlation. Overall, we can conclude that the performance of the family of suggested estimators is better than all existing estimators.

Concluding Remarks
In this article, we proposed a generalized class of exponential factor type estimators, utilizing the supplementary information in the form of the mean and rank of the auxiliary variable for estimating the finite population distribution function. e expressions for biases and mean squared errors of the proposed generalized class of estimators are derived up to the first order of approximation. e proposed estimators F(t y ) (k 1 ,k 2 ) Prop1 and F(t y ) (k 1 ,k 2 ) Prop2 are compared to all existing estimators numerically and theoretically. Based on the simulation studies as well as on the real data sets, it is observed that the proposed class of estimators performed better than their existing counterparts and should be preferable over the existing estimators available in the literature.

Data Availability
e data used to support the findings of this study are included within the text. Table 13: PREs of distribution function estimators using simulation for populations I, II, and III, when F(t y ) and F(t x ) � Q 3 (y), Q 3 (x).