# An Exponential-Cum-Sine-Type Hybrid Imputation Technique for Missing Data

Authors: D. Bhattacharyya; G.N. Singh; Taghreed M. Jawa; Neveen Sayed-Ahmed; Awadhesh K. Pandey
Journal: Computational Intelligence and Neuroscience
 (2021)

---

## Abstract

In this study, a new exponential-cum-sine-type hybrid imputation technique has been proposed to handle missing data when conducting surveys. The properties of the corresponding point estimator for population mean have been examined in terms of bias and mean square errors. An extensive simulation study using data generated from normal, Poisson, and Gamma distributions has been conducted to evaluate how the proposed estimator performs in comparison to several contemporary estimators. The results have been summarized, and discussion regarding real-life applications of the estimator follows.


---

## Body

## 1. Introduction

The impracticality of measuring the entire population for any realistic project due to budgetary, time, or other constraints makes sampling indispensible for any field of study [1–12]. The widespread applications of acceptance sampling in various industries for manufacturing and other processes have been noted for a considerable period of time. Sampling can also be applied to obtain vital information on the chief characteristics of items ranging from electrical appliances to machine parts such as screws and bolts, automobiles, and computer parts such as chip. In addition, many environmental problems involve physical, geographical, economical, and other characteristics which need to be estimated prior to data analysis, model formulation, and predictions. Studies related to the amount of rainfall received annually in a flood-prone area, the quality of drinking water near an industrial zone, the soil quality of an agricultural land, etc. are some instances where estimation of mean, median, variance, and other statistics is essential. Such information can be collected via sample surveys [4, 6, 7, 9, 13].Missing data is a universal occurrence in sample surveys, leading to a decline in data quality and complications in making inferences. It is pivotal for survey statisticians to factor in the stochastic nature of incomplete data. This brings forth the question of what assumptions have to be made or which techniques have to be employed to handle the problem of ignorability of completeness mechanism. The mechanisms of missing data have been studied in detail in [9, 13], among others. Three missing data mechanisms are mostly of interest in the survey literature, namely, missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). MCAR is said to occur when data is missing randomly or by chance, MAR occurs when the missingness does not depend on the variable under study (which may be unobserved), but on some other variables (which is fully observed), and MNAR occurs when missingness depends on the variable under study.Numerous statistical methods have been devised over the years to overcome the problem of missing data. Subsampling of nonrespondents in surveys via mail questionnaire was pioneered in [8]. Another commonly used method is imputation, in which the missing values are filled in by a suitable function of the available values, to ensure the structural completeness of the sample before analysis begins. Popular imputation techniques include mean imputation, regression imputation, hot deck imputation, cold deck imputation, and nearest neighbor method. Imputation techniques in the survey literature are from [3, 5, 14–21], among others. Some recent works in the area of imputation and estimation of population mean have been done in [22–29] and others.Information from an auxiliary variable can be utilized to provide an improved estimate for population characteristics. Such information may be readily available as secondary data from previous surveys or census or may be collected during the survey procedure at little to no additional cost. Some examples of such auxiliary information include the lifetime of a previous batch of bulbs when studying the life of a current lot of bulbs, the speed of cars when studying the mileage of cars, etc.In this manuscript, a new exponential-cum-sine-type hybrid imputation technique and corresponding point estimator have been proposed for estimation of population mean. Motivation for this estimator, its properties, and its uses have been discussed in the subsequent sections. The manuscript is henceforth divided into the following sections: Section2 introduces the sample structure and notations used in the manuscript. Section 3 discusses some conventional estimators of population mean. Section 4 discusses the proposed estimator, including its existence, consistency, properties, and implementation in R. The simulation study has been presented in Section 5, the results and discussion in Section 6, and the conclusions in Section 7.


## 2. Sample Structure and Notations Used

Let the character of interest be denoted byY. We consider the scenario in which complete information on a correlated auxiliary variable X is available to the survey statisticians and its population mean is known.The sample structure and the notations used henceforth have been introduced in Table1.Table 1
Sample structure and notations.
StructureSizePopulationNSampleNRespondentsRNonrespondentsN − rCharacteristicNotationThe population mean ofYY¯The population mean ofXX¯The sample mean ofY based on the responding part of the sampley¯rThe sample mean ofX based on the responding part of the samplex¯rThe sample means ofX, respectively, based on the entire samplex¯nThe correlation coefficient betweenX and YρThe population mean square ofXSX2The population mean square ofYSY2The coefficient of variation ofXCXThe coefficient of variationYCY


## 3. Some Conventional Estimators

Before the proposed estimator is introduced, it is important to examine some existing estimators for population mean and study their strengths and limitations. A few such estimators have been discussed in this section.The mean estimator is a simple and traditional estimator, which makes use of the average of the responses to provide an estimate of the population mean. The ratio estimator tries to make an improvement over the mean estimator by incorporating auxiliary information into a correlated variable. Various other estimators that make innovative use of auxiliary information have been proposed, for instance, the estimator proposed in [30], regression-type estimators proposed in [10], and exponential type estimators in [31], among others.The structures of some of these estimators have been given in Table2, while the expressions for their respective variances (V) or mean square errors (MSEs) have been given in Table 3.Table 2
Structures of some well-known estimators.
EstimatorNotation usedStructureMean estimatory¯my¯rRatio estimatory¯RATy¯rx¯n/x¯rKadilar and Cingi [10] estimator ATKCAy¯r+bX¯−x¯n/x¯nX¯Kadilar and Cingi [10] estimator BTKCBy¯r+bX¯−x¯r/x¯rX¯Kadilar and Cingi [10] estimator CTKCCy¯r+bx¯n−x¯n/x¯rX¯Toutenberg and Srivastava [30] estimatorTTSSy¯r+r/ny¯r/x¯nx¯n−x¯rSingh et al. [31]TSMKKy¯rx¯n/x¯rexpX¯−x¯r/X¯+x¯rTable 3
MSEs of some well-known estimators.
EstimatorVariance (V) or mean square error (MSE)y¯mVy¯m=θ1SY2y¯RATMSEy¯RAT=θ2SY2+θ3SY2+R12SX2−2R1ρSYSXTKCAMSETKCA=1/r−1/NSY2+1/n−1/NSX2R12−B2TKCBMSETKCB=1/r−1/NSY2−BSYX+R2SX2TKCCMSETKCC=1/r−1/NSY2+1/r−1/NR+B2SX2−2R+BSXYTTSSMSETTSS=1/r−1/NSY2+Y¯21/r−1/nr/nr/nCX2−2ρCYCXTSMKKMTSMKK=Y¯21/r−1/NCY2+9/4CX2−3ρCYCX+21/n−1/NρCYCX−CX2WhereR1=R=Y¯/X¯, B=SXY/SX2It is to be noted that most conventional estimators make use of simple functional forms, such as linear combinations, exponential functions, and chains. Combination of multiple mathematical functions is rarely seen. This can be attributed to computational limitations associated with such functions. However, with the advent of supercomputers and improvement in computational powers, such obstructions have been eliminated. It is worth exploring whether combinations of mathematical functions produce better estimates than traditional estimators. This has been the motivation behind the construction of the proposed estimator.Two such functions have been used, namely, the exponential and sine functions. Such particular functions were selected based on their use in real-life situations. The exponential function is usually used to model growth and decay observed in nature, such as growth and decay of microorganisms like bacteria, human population, spread of pandemics, and compound interests. Sine function is commonly utilized for the purpose of modeling natural phenomena which are periodic in nature, such as sound waves, light waves, tides, sunlight intensity, and average temperature variations through the year, as well as ballistic trajectories, electrical currents, and GPS locations.


## 4. Formulation of the Proposed Estimator

Letyi and xi be the values of Y and X, respectively, for the ith unit in the population. The following imputation method may be suggested to deal with the problem of missing data:(1)y⋅i=yi,ifi∈R,nn−rxiexpsinx¯n−sinx¯r1+sinx¯n+sinx¯r−rn−ry¯r,ifi∈Rc.The point estimator under an imputation method is given in(2)T=1n∑i∈Sy⋅i=1n∑i∈Ry⋅i+∑i∈Rcy⋅i.Using equation (2), under the imputation outlined in equation (1), the expression for the point estimator of Y¯ is obtained as(3)T=y¯rexpsinx¯n−sinx¯r1+sinx¯n+sinx¯r.


### 4.1. Existence and Consistency of the Estimator

It is important to specify the domain of values for which an estimator exists, so that survey statisticians or those working in the field can determine whether an estimator can be reasonably used in a practical scenario.The given estimator consists of two major functions: the trigonometrical functionsin and the exponential function exp. Both sinx and expx exist in ∀x∈ℝ, so y⋅iandT exist in ∀x∈ℝ.Hence, the proposed estimator can be used for all real values of the characters under study. For real-world scenarios, most, if not all, characters of interest take only real values. For example, measurements such as length, breadth, height, weight, diameter, currencies, and number of an item do not take nonreal values. Hence, the proposed estimator can be used in all practical scenarios.It is to be noted that the structure of the estimator is consistent for large sample approximations. Asn⟶∞, y¯r⟶Y¯, x¯r⟶X¯, x¯n⟶X¯, and exp0=1. Hence, T⟶Y¯.


### 4.2. Properties of the Proposed Estimator

The “goodness” of an estimator can be measured in terms of various properties. Two such properties, namely, bias and mean squared error (MSE), have been explored here. The bias gives an idea about the expected deviation from the true value of a parameter, while MSE deals with the degree of spread. The expressions for the same have been derived under large sample assumptions up to the first order of approximations. Some transformations involving error terms have been used for the purpose, indicated as follows:(4)y¯r=Y¯1+η0,x¯r=X¯1+η1,x¯n=X¯1+η2,θ1=1r−1N,θ2=1n−1N,θ3=1r−1n.The error terms have the following expectations:(5)Eη0=Eη1=Eη2=0,Eη02=θ1CY2,Eη12=θ1CX2,Eη22=θ2CX2,Eη0η1=θ1ρCYCX,Eη1η2=θ2CX2,Eη0η2=θ2ρCYCX.To obtain the expressions for bias and MSE, in the first step, algebraic expansion of the expression of the estimator given in equation (3) is done, using the following Taylor’s series:(1)
sinx=x−x3/3!+x5/5!−x7/7!−⋯(2)
expx=1+x+x2/2!+x3/3!+x4/4!+⋯(3)
1+x−1=1−x+x2−x3+⋯The estimator takes the following form:(6)T=y¯r1+x¯n−x¯r+2x¯r2−2x¯nx¯r.In the second step, the transformations in equation (4) are applied to equation (6) to obtain the following form of the estimator:(7)T=Y¯1+X¯η2−η1+η0η2X¯2η1−η2+η12−η1η2+η0η1−η0η2+X¯η0η2−η0η1.Hence,T−Y¯=Y¯X¯η2−η1+η0η2X¯2η1−η2+η12−η1η2+η0η1−η0η2+X¯η0η2−η0η1.Expectations taken on both sides and use of the expected values ofηi,i=0,1,2, yield the expectations for bias B. and MSE M., obtained up to the first order of approximations of the estimators Ti,i=1,2,…,6, as follows:(8)BT=ET−Y¯=Y¯2X¯2−X¯θ3ρCYCX−2X¯2θ2CX2,(9)MT=ET−Y¯2=θ1SY2+Y¯2θ3C2CX2+2CρCYCX,where C=2X¯2−X¯.


### 4.3. Implementation in R

In the current day and age, most computations are carried out using a suitable software environment. The following R [32] code snippet has been developed to carry out the proposed imputation on a data set of interest and calculate the value of the corresponding point estimator:#Import data of respondents from filedfresp < −read.table (file.choose())#Import data of nonrespondents from filedfnonresp < −read.table (file.choose())xrbar = mean (dfresp[, 1])yrbar < −mean (dfresp[, 2])xbarnonresp = mean (dfnonresp[, 1])r = nrow (dfresp) #no. of respondentsnonresp =nrow (dfnonresp) #no. of nonrespondentsn = r + nonresp #sample sizexnbar=(r∗xrbar + nonresp∗xbarnonresp)/nnum = sin(xnbar) − sin(xrbar)den = 1 + sin(xnbar)+sin(xrbar)#imputationt < −c()for (i in 1 : (n − r)){t[i] = n/(n − r)∗x[i]∗exp(num/den) − r/(n − r)∗yrbar}#point estimationest =yrbar∗exp(num/den)


## 4.1. Existence and Consistency of the Estimator

It is important to specify the domain of values for which an estimator exists, so that survey statisticians or those working in the field can determine whether an estimator can be reasonably used in a practical scenario.The given estimator consists of two major functions: the trigonometrical functionsin and the exponential function exp. Both sinx and expx exist in ∀x∈ℝ, so y⋅iandT exist in ∀x∈ℝ.Hence, the proposed estimator can be used for all real values of the characters under study. For real-world scenarios, most, if not all, characters of interest take only real values. For example, measurements such as length, breadth, height, weight, diameter, currencies, and number of an item do not take nonreal values. Hence, the proposed estimator can be used in all practical scenarios.It is to be noted that the structure of the estimator is consistent for large sample approximations. Asn⟶∞, y¯r⟶Y¯, x¯r⟶X¯, x¯n⟶X¯, and exp0=1. Hence, T⟶Y¯.


## 4.2. Properties of the Proposed Estimator

The “goodness” of an estimator can be measured in terms of various properties. Two such properties, namely, bias and mean squared error (MSE), have been explored here. The bias gives an idea about the expected deviation from the true value of a parameter, while MSE deals with the degree of spread. The expressions for the same have been derived under large sample assumptions up to the first order of approximations. Some transformations involving error terms have been used for the purpose, indicated as follows:(4)y¯r=Y¯1+η0,x¯r=X¯1+η1,x¯n=X¯1+η2,θ1=1r−1N,θ2=1n−1N,θ3=1r−1n.The error terms have the following expectations:(5)Eη0=Eη1=Eη2=0,Eη02=θ1CY2,Eη12=θ1CX2,Eη22=θ2CX2,Eη0η1=θ1ρCYCX,Eη1η2=θ2CX2,Eη0η2=θ2ρCYCX.To obtain the expressions for bias and MSE, in the first step, algebraic expansion of the expression of the estimator given in equation (3) is done, using the following Taylor’s series:(1)
sinx=x−x3/3!+x5/5!−x7/7!−⋯(2)
expx=1+x+x2/2!+x3/3!+x4/4!+⋯(3)
1+x−1=1−x+x2−x3+⋯The estimator takes the following form:(6)T=y¯r1+x¯n−x¯r+2x¯r2−2x¯nx¯r.In the second step, the transformations in equation (4) are applied to equation (6) to obtain the following form of the estimator:(7)T=Y¯1+X¯η2−η1+η0η2X¯2η1−η2+η12−η1η2+η0η1−η0η2+X¯η0η2−η0η1.Hence,T−Y¯=Y¯X¯η2−η1+η0η2X¯2η1−η2+η12−η1η2+η0η1−η0η2+X¯η0η2−η0η1.Expectations taken on both sides and use of the expected values ofηi,i=0,1,2, yield the expectations for bias B. and MSE M., obtained up to the first order of approximations of the estimators Ti,i=1,2,…,6, as follows:(8)BT=ET−Y¯=Y¯2X¯2−X¯θ3ρCYCX−2X¯2θ2CX2,(9)MT=ET−Y¯2=θ1SY2+Y¯2θ3C2CX2+2CρCYCX,where C=2X¯2−X¯.


## 4.3. Implementation in R

In the current day and age, most computations are carried out using a suitable software environment. The following R [32] code snippet has been developed to carry out the proposed imputation on a data set of interest and calculate the value of the corresponding point estimator:#Import data of respondents from filedfresp < −read.table (file.choose())#Import data of nonrespondents from filedfnonresp < −read.table (file.choose())xrbar = mean (dfresp[, 1])yrbar < −mean (dfresp[, 2])xbarnonresp = mean (dfnonresp[, 1])r = nrow (dfresp) #no. of respondentsnonresp =nrow (dfnonresp) #no. of nonrespondentsn = r + nonresp #sample sizexnbar=(r∗xrbar + nonresp∗xbarnonresp)/nnum = sin(xnbar) − sin(xrbar)den = 1 + sin(xnbar)+sin(xrbar)#imputationt < −c()for (i in 1 : (n − r)){t[i] = n/(n − r)∗x[i]∗exp(num/den) − r/(n − r)∗yrbar}#point estimationest =yrbar∗exp(num/den)


## 5. Simulation Study

Before an estimator can be used in practical scenarios, its performance must be examined, in terms of its properties. To this end, the bias of the estimator is calculated and the MSE is compared with that of the contemporary estimators given in Table2 in terms of percentage relative efficiencies (PREs).The PREs of the estimator with respect to the contemporary estimators are defined as follows:(10)PRE1=Vy¯mMT×100,PRE2=My¯RATMT×100,PRE3=MTTSSMT×100,PRE4=MTSMKKMT×100,PRE5=MTKC1MT×100,PRE6=MTKC2MT×100,PRE7=MTKC3MT×100,where the expression for the MSE of the proposed estimator T is given in equation (9), while that of the contemporary estimators is given in Table 3.Using R [32], an extensive simulation study has been carried out on sufficiently large fictitious populations to compute the bias and the PREs defined above. Data is generated from three different probability distributions, namely, normal and Gamma distributions (continuous distributions) and Poisson distribution (discrete distribution). Some important properties of the distributions have been summarized in Table 4. Such distributions are chosen based on their occurrence in real-life situations.Table 4
Some properties of normal, Poisson, and Gamma distributions.
DistributionNormalParametersμ,σ2Pdffx=1/σ2πexp−x−μ2/2σ2,−∞<x<∞MeanEXμVarianceVXσ2DistributionPoissonParameterλ>0Pmffx=λxe−λ/x!MeanEXλVarianceVXλDistributionGammaParametersα,λPdffx=λαxα−1e−λx/Γx,ifx>0,0,otherwiseMeanEXα/λVarianceVXα/λ2Data from normal distribution is rampant in nature. It can be used to model heights of individuals, test scores of students, blood pressure, daily returns of any particular stock, weights of items produced by a manufacturing process, etc. Poisson distribution can be used to model the probability that a given number of events occur in a specific time interval, for example, the number of insurance claims filed per month, the number of network failures occurring per week, and the number of bulbs manufactured per minute. It also finds use by medical statisticians, such as for estimating the number of births that may be expected on a particular night, the number of patients with an infectious disease arriving at a clinic within a given hour, the number of mutations on a given strand of DNA per time unit, etc. Gamma distribution can be used for modeling wait time, reliability, service time in queuing theory, etc. For example, it can be used to model the amount of rainfall that accumulates in a given reservoir, the flow of items through manufacturing as well as distribution processes, the size of loan defaults, etc. Thus, these three distributions are chosen based on their importance in practical scenarios.It is seen through trial and error that the estimator performs well whenX and Y take small values and the variation in X is greater than that in Y.The steps of the simulation are as follows:(1)
The sizes of the population, the sample, and the responding part of the sample are defined. For the purpose of the study, sufficiently large values ofN=100000,n=40000, and r=35000 have been chosen.(2)
The parameters of the population are defined.(3)
Simulation is conducted for various values ofρ. For the purpose of the study, ρ in the range 0.1,0.9; i.e., positively correlated variable X is considered.The results of the simulation study related to the PREs have been presented in Tables5–11, while the biases have been presented in Table 12.Table 5
Values ofPRE1 when ρ∈0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 and response rates are 75%,80%,85%,90%,and95% for data generated from normal, Gamma, and Poisson distributions.
ρPRE1 when response rate is75%80%85%90%95%When data is generated from normal distribution0.1100.134 4100.110 6100.085 5100.058 7100.030 30.2101.457 1101.196 9100.922 4100.632 3100.325 40.3102.640 5102.164 5101.664 4101.138 3100.584 30.4103.563 8102.916 5102.238 9101.528 5100.783 10.5104.411 4103.604 8102.762 9101.883 2100.963 20.6106.314 4105.142 8103.928 1102.667 8101.359 40.7108.313 4106.747 4105.135 1103.474 6101.763 70.8107.626 8106.197 5104.722 5103.199 5101.626 20.9109.266 4107.508 4105.704 6103.853 2101.952 4When data is generated from Gamma distribution0.199.298 099.421 199.552 199.691 699.840 60.299.950 099.958 999.968 299.978 199.988 70.3100.587 7100.483 5100.373 2100.256 3100.132 10.4101.385 6101.138 3100.877 3100.601 5100.309 50.5102.425 1101.988 6101.529 7101.046 7100.537 50.6103.616 2102.959 2102.271 4101.550 6100.794 30.7105.340 6104.357 1103.333 8102.268 4101.158 10.8106.962 2105.664 0104.321 1102.931 2101.491 70.9109.279 9107.519 1105.712 6103.858 5101.955 0When data is generated from Poisson distribution0.199.970 199.975 399.980 999.986 999.993 20.2100.248 2100.204 3100.157 8100.108 4100.055 90.3100.602 0100.495 2100.382 2100.262 5100.135 30.4100.937 7100.770 9100.594 7100.408 1100.210 20.5101.536 9101.262 3100.972 6100.666 6100.343 00.6101.946 9101.597 9101.230 2100.842 6100.433 10.7102.653 3102.174 9101.672 3101.143 7100.587 10.8103.225 6102.641 4102.028 9101.386 1100.710 60.9104.123 4103.371 2102.585 2101.763 1100.902 3Table 6
Values ofPRE2 when ρ∈0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 and response rates are 75%,80%,85%,90%,and95% for data generated from normal, Gamma, and Poisson distributions.
ρPRE2 when response rate is75%80%85%90%95%When data is generated from normal distribution0.1769.433 6651.167 6525.795 1392.656 0251.005 30.2299.679 7264.020 4226.399 2186.649 7144.586 00.3274.230 3242.818 2209.819 3175.110 1138.554 20.4496.315 6424.337 8348.974 3269.980 6187.087 80.5365.407 7316.882 7266.229 3213.304 3157.951 60.6161.467 9150.062 8138.238 0125.969 8113.233 00.7194.934 1177.050 6158.639 9139.678 2120.140 50.8142.841 8134.812 9126.527 2117.972 2109.134 60.9157.471 7146.568 1135.380 7123.898 2112.108 8When data is generated from Gamma distribution0.1167.446 9155.613 4143.030 6129.625 0115.313 20.2163.157 7152.016 8140.198 6127.639 3114.266 90.3159.970 7149.336 5138.081 8126.150 7113.480 40.4151.366 5142.198 7132.523 9122.299 1111.475 80.5145.333 2137.174 2128.596 2119.566 4110.047 90.6137.744 9130.887 0123.707 7116.184 2108.290 90.7129.067 1123.714 1118.144 9112.346 0106.302 90.8113.216 1110.751 8108.202 6105.564 2102.831 70.9100.842 1100.682 3100.518 4100.350 1100.177 4When data is generated from Poisson distribution0.1139.707 9132.702 4125.271 4117.375 1108.968 40.2136.595 4130.124 2123.267 0115.988 2108.247 60.3132.307 4126.577 8120.514 3114.086 8107.261 40.4129.075 8123.905 2118.439 9112.653 9106.518 20.5122.248 5118.272 8114.079 5109.650 3104.964 80.6117.655 7114.490 2111.156 5107.640 7103.927 60.7109.955 8108.160 7106.274 9104.291 6102.202 80.8104.363 1103.572 8102.744 4101.874 9100.961 20.993.266 094.494 495.778 097.120 698.526 5Table 7
Values ofPRE3 when ρ∈0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 and response rates are 75%,80%,85%,90%,and95% for data generated from normal, Gamma, and Poisson distributions.
ρPRE3 when response rate is75%80%85%90%95%When data is generated from normal distribution0.1470.654 1448.598 9405.082 4335.811 6235.944 60.2205.929 6200.477 9188.554 3168.842 5139.873 10.3188.801 9184.937 7175.366 8158.912 2134.271 40.4303.819 3294.193 0271.676 3233.719 8177.522 60.5229.191 9224.740 8211.484 9187.620 1151.176 60.6120.640 2122.286 4121.653 5118.156 1111.164 60.7133.264 1135.276 3133.806 8128.030 1117.071 00.8103.726 6108.071 5110.493 9110.391 4107.121 90.9105.568 7111.201 2114.246 5113.939 4109.473 9When data is generated from Gamma distribution0.1137.107 3135.015 2130.737 8123.827 1113.774 40.2134.024 1132.235 6128.394 0122.072 7112.790 10.3131.435 3129.955 5126.514 3120.695 9112.033 50.4125.754 8124.791 5122.129 5117.396 2110.175 10.5121.186 1120.746 3118.779 8114.934 1108.818 80.6115.379 1115.641 5114.583 8111.873 7107.146 30.7108.611 5109.731 4109.758 0108.377 1105.247 90.897.570 099.957 8101.676 9102.455 0102.000 30.988.167 891.827 895.109 097.750 499.477 3When data is generated from Poisson distribution0.1121.737 3120.509 3118.000 0113.948 4108.059 80.2119.469 0118.492 9116.324 8112.714 2107.379 00.3116.387 1115.749 6114.042 9111.031 3106.449 90.4113.897 9113.565 1112.251 2109.728 2105.740 30.5108.956 1109.183 7108.621 6107.062 5104.275 00.6105.457 8106.119 4106.113 7105.242 8103.286 80.799.941 6101.235 9102.075 0102.282 5101.663 10.895.626 797.480 599.022 0100.082 7100.477 00.987.555 790.392 293.207 495.855 898.178 2Table 8
Values ofPRE4 when ρ∈0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 and response rates are 75%,80%,85%,90%,and95% for data generated from normal, Gamma, and Poisson distributions.
ρPRE4 when response rate is75%80%85%90%95%When data is generated from normal distribution0.11916.780 11674.549 31417.763 31145.069 6854.942 70.2647.902 5569.861 9487.527 8400.535 9308.479 50.3582.505 4509.820 1433.462 7353.148 0268.559 90.41197.061 41032.498 1860.194 1679.590 3490.072 00.5844.349 3725.848 4602.149 8472.904 1337.729 50.6281.945 8244.044 6204.748 7163.979 6121.653 00.7380.226 8322.547 7263.168 1202.011 5138.996 90.8240.640 9203.163 7164.487 8124.554 883.302 30.9288.721 7238.802 4187.583 5135.013 881.038 9When data is generated from Gamma distribution0.1284.139 1259.915 0234.157 0206.714 6177.417 10.2272.979 2249.299 4224.179 9197.485 3169.062 60.3265.010 4241.396 3216.404 5189.910 7161.775 40.4242.326 3220.621 7197.717 2173.510 2147.886 60.5226.876 8205.793 4183.627 6160.294 0135.697 80.6207.659 0187.420 3166.233 6144.030 8120.736 80.7185.522 7166.100 6145.893 8124.853 9102.928 10.8144.503 8128.566 4112.080 495.017 077.345 50.9112.683 198.642 684.237 369.452 754.273 7When data is generated from Poisson distribution0.1208.198 7193.633 0178.182 8161.765 0144.286 00.2200.344 9186.282 4171.381 0155.563 5138.742 60.3189.485 1176.179 2162.097 7147.170 8131.320 00.4181.498 6168.589 7154.945 0140.499 7125.181 20.5164.227 9152.522 1140.175 9127.135 1113.339 60.6152.881 1141.837 5130.206 9117.941 2104.987 00.7133.280 9123.706 1113.648 1103.069 491.928 30.8119.479 0110.693 3101.483 191.816 981.660 10.991.544 984.914 077.985 170.737 863.149 4Table 9
Values ofPRE5 when ρ∈0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 and response rates are 75%,80%,85%,90%,and95% for data generated from normal, Gamma, and Poisson distributions.
ρPRE5 when response rate is75%80%85%90%95%When data is generated from normal distribution0.11361.451 11484.757 61615.473 21754.286 61901.974 30.2523.041 9562.926 2605.004 8649.463 8696.511 20.3504.883 7541.794 7580.570 3621.355 7664.311 10.4998.053 71078.963 11163.678 31252.474 21345.653 00.5769.279 0828.016 2889.329 7953.392 71020.394 50.6341.143 5360.152 7379.861 5400.309 2421.538 10.7458.557 7485.768 8513.782 2542.633 9572.362 10.8354.110 3373.251 8393.005 4413.401 2434.470 80.9434.663 4459.058 3484.088 3509.778 4536.155 3When data is generated from Gamma distribution0.1226.951 0239.763 0253.386 3267.900 4283.395 80.2227.539 8240.069 8253.361 5267.486 8282.526 40.3230.863 6243.383 3256.633 4270.679 9285.596 70.4224.737 5236.253 2248.405 5261.248 9274.843 90.5226.091 9237.201 0248.880 3261.175 1274.135 10.6226.830 4237.395 3248.455 0260.045 2272.204 90.7227.996 6237.780 7247.960 1258.559 3269.604 70.8213.751 3221.499 4229.514 2237.809 7246.400 80.9207.722 6213.871 4220.180 0226.654 7233.302 1When data is generated from Poisson distribution0.1177.014 5184.577 7192.600 1201.124 8210.200 70.2176.981 0184.423 1192.309 1200.679 9209.581 70.3176.082 6183.287 9190.913 3198.996 4207.579 90.4176.690 9183.813 5191.342 2199.312 6207.764 70.5174.141 1180.768 9187.759 3195.142 9202.953 80.6173.116 4179.477 2186.176 0193.240 6200.701 80.7167.633 3173.193 0179.033 2185.175 9191.645 00.8165.091 7170.188 2175.530 9181.138 2187.030 10.9153.828 1157.554 6161.448 6165.521 6169.786 3Table 10
Values ofPRE6 when ρ∈0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 and response rates are 75%,80%,85%,90%,and95% for data generated from normal, Gamma, and Poisson distributions.
ρPRE6 when response rate is75%80%85%90%95%When data is generated from normal distribution0.12062.182 72061.693 82061.175 52060.625 12060.039 50.2757.255 6755.313 4753.264 3751.099 3748.808 20.3728.352 1724.973 9721.425 0717.692 2713.760 80.41494.992 51485.649 21475.866 31465.612 31454.852 10.51138.649 91129.854 31120.672 81111.079 61101.046 40.6471.604 1466.406 9461.018 4455.428 0449.623 90.7653.137 9643.694 4633.972 5623.959 7613.642 60.8491.045 6484.524 4477.794 5470.845 9463.667 70.9615.439 5605.537 5595.377 6584.949 8574.243 2When data is generated from Gamma distribution0.1297.869 3298.238 8298.631 7299.050 2299.497 10.2298.423 0298.449 3298.477 2298.506 9298.538 50.3303.239 0302.924 8302.592 3302.239 8301.865 50.4293.266 4292.551 0291.796 2290.998 4290.153 90.5294.795 7293.539 4292.218 7290.828 4289.362 90.6295.282 7293.410 3291.450 2289.396 0287.241 00.7296.138 8293.373 9290.497 3287.502 1284.380 70.8273.078 6269.764 2266.335 7262.787 2259.112 20.9262.412 9258.184 8253.846 9249.394 7244.823 8When data is generated from Poisson distribution0.1219.817 0219.828 6219.841 0219.854 1219.868 00.2219.610 4219.514 2219.412 4219.304 2219.189 20.3218.016 2217.784 9217.540 0217.280 5217.004 90.4218.776 0218.414 6218.032 6217.628 2217.199 30.5214.476 7213.896 6213.284 7212.638 5211.954 80.6212.655 0211.926 9211.160 0210.351 3209.497 30.7203.733 4202.783 9201.786 5200.737 4199.632 60.8199.461 7198.332 7197.149 2195.907 1194.601 90.9181.441 7180.131 0178.761 4177.328 8175.828 7Table 11
Values ofPRE7 when ρ∈0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 and response rates are 75%,80%,85%,90%,and95% for data generated from normal, Gamma, and Poisson distributions.
ρPRE7 when response rate is75%80%85%90%95%When data is generated from normal distribution0.12062.182 72061.693 82061.175 52060.625 12060.039 50.2757.255 6755.313 4753.264 3751.099 3748.808 20.3728.352 1724.973 9721.425 0717.692 2713.760 80.41494.992 51485.649 21475.866 31465.612 31454.852 10.51138.649 91129.854 31120.672 81111.079 61101.046 40.6471.604 1466.406 9461.018 4455.428 0449.623 90.7653.137 9643.694 4633.972 5623.959 7613.642 60.8491.045 6484.524 4477.794 5470.845 9463.667 70.9615.439 5605.537 5595.377 6584.949 8574.243 2When data is generated from Gamma distribution0.1297.869 3298.238 8298.631 7299.050 2299.497 10.2298.423 0298.449 3298.477 2298.506 9298.538 50.3303.239 0302.924 8302.592 3302.239 8301.865 50.4293.266 4292.551 0291.796 2290.998 4290.153 90.5294.795 7293.539 4292.218 7290.828 4289.362 90.6295.282 7293.410 3291.450 2289.396 0287.241 00.7296.138 8293.373 9290.497 3287.502 1284.380 70.8273.078 6269.764 2266.335 7262.787 2259.112 20.9262.412 9258.184 8253.846 9249.394 7244.823 8When data is generated from Poisson distribution0.1219.817 0219.828 6219.841 0219.854 1219.868 00.2219.610 4219.514 2219.412 4219.304 2219.189 20.3218.016 2217.784 9217.540 0217.280 5217.004 90.4218.776 0218.414 6218.032 6217.628 2217.199 30.5214.476 7213.896 6213.284 7212.638 5211.954 80.6212.655 0211.926 9211.160 0210.351 3209.497 30.7203.733 4202.783 9201.786 5200.737 4199.632 60.8199.461 7198.332 7197.149 2195.907 1194.601 90.9181.441 7180.131 0178.761 4177.328 8175.828 7Table 12
Values of bias whenρ∈0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 and response rates are 75%,80%,85%,90%,and95% for data generated from normal, Gamma, and Poisson distributions.
ρBias of the proposed estimator when response rate is75%80%85%90%95%When data is generated from normal distribution0.1−0.000 095 000 0−0.000 092 000 0−0.000 090 000 0−0.000 088 000 0−0.000 086 000 00.2−0.000 093 000 0−0.000 088 000 0−0.000 083 000 0−0.000 078 000 0−0.000 075 000 00.3−0.000 121 000 0−0.000 114 000 0−0.000 108 000 0−0.000 102 000 0−0.000 097 000 00.4−0.000 114 000 0−0.000 104 000 0−0.000 096 000 0−0.000 089 000 0−0.000 082 000 00.5−0.000 126 000 0−0.000 113 000 0−0.000 101 000 0−0.000 091 000 0−0.000 082 000 00.6−0.000 135 000 0−0.000 120 000 0−0.000 106 000 0−0.000 094 000 0−0.000 083 000 00.7−0.000 143 000 0−0.000 126 000 0−0.000 111 000 0−0.000 098 000 0−0.000 086 000 00.8−0.000 154 000 0−0.000 133 000 0−0.000 115 000 0−0.000 099 000 0−0.000 084 000 00.9−0.000 159 000 0−0.000 136 000 0−0.000 115 000 0−0.000 097 000 0−0.000 081 000 0When data is generated from Gamma distribution0.1−0.000 030 580 0−0.000 030 510 0−0.000 030 460 0−0.000 030 400 0−0.000 030 360 00.2−0.000 026 300 0−0.000 026 150 0−0.000 026 020 0−0.000 025 900 0−0.000 025 800 00.3−0.000 030 400 0−0.000 030 190 0−0.000 030 010 0−0.000 029 840 0−0.000 029 690 00.4−0.000 027 760 0−0.000 027 420 0−0.000 027 110 0−0.000 026 840 0−0.000 026 600 00.5−0.000 027 550 0−0.000 027 110 0−0.000 026 730 0−0.000 026 390 0−0.000 026 080 00.6−0.000 032 820 0−0.000 032 170 0−0.000 031 590 0−0.000 031 080 0−0.000 030 620 00.7−0.000 029 350 0−0.000 028 530 0−0.000 027 810 0−0.000 027 170 0−0.000 026 590 00.8−0.000 031 190 0−0.000 030 150 0−0.000 029 230 0−0.000 028 420 0−0.000 027 690 00.9−0.000 034 810 0−0.000 033 330 0−0.000 032 030 0−0.000 030 880 0−0.000 029 850 0When data is generated from Poisson distribution0.1−0.000 000 403 0−0.000 000 395 0−0.000 000 387 0−0.000 000 381 0−0.000 000 375 00.2−0.000 000 447 0−0.000 000 430 0−0.000 000 415 0−0.000 000 402 0−0.000 000 390 00.3−0.000 000 455 0−0.000 000 431 0−0.000 000 411 0−0.000 000 392 0−0.000 000 376 00.4−0.000 000 510 0−0.000 000 473 0−0.000 000 440 0−0.000 000 411 0−0.000 000 385 00.5−0.000 000 552 0−0.000 000 503 0−0.000 000 460 0−0.000 000 421 0−0.000 000 387 00.6−0.000 000 632 0−0.000 000 568 0−0.000 000 511 0−0.000 000 461 0−0.000 000 416 00.7−0.000 000 703 0−0.000 000 620 0−0.000 000 547 0−0.000 000 483 0−0.000 000 425 00.8−0.000 000 777 0−0.000 000 675 0−0.000 000 585 0−0.000 000 505 0−0.000 000 434 00.9−0.000 000 871 0−0.000 000 746 0−0.000 000 635 0−0.000 000 536 0−0.000 000 448 0


## 6. Results and Discussion

The simulation study enables us to study the behavior of the proposed estimator under various scenarios involving various values of parameters. The chief conclusions are as follows:(1)
From the values ofPRE1 in Table 5, it is seen that the proposed estimator is more efficient than y¯m for all values of ρ for normal data and for ρ∈0.2,0.9 for Gamma and Poisson data for the various values of response rates.(2)
It is seen that the proposed estimator performs better thany¯RAT for all values of ρ for normal and Gamma data and for ρ∈0.1,0.8 for Poisson data for the various values of response rates from the values of PRE2 in Table 6.(3)
From the values ofPRE3 in Table 7, it is seen that the proposed estimator dominates TTSS for all values of ρ for normal data and for ρ∈0.1,0.7 for Gamma and Poisson data for the various values of response rates.(4)
The values ofPRE4 in Table 8 show that the proposed estimator is more efficient than TSMKK for all values of ρ for normal data and for ρ∈0.1,0.7 for Gamma and Poisson data for the various values of response rates.(5)
In Table9, the values of PRE5 show that the proposed estimator performs better than TKCA for all values of ρ and for the various values of response rates for normal, Gamma, and Poisson data.(6)
From the values ofPRE6 in Table 10, it is seen that the proposed estimator dominates TKCB for all values of ρ and for the various values of response rates for normal, Gamma, and Poisson data.(7)
It is seen that the proposed estimator is more efficient thanTKCC for all values of ρ and for the various values of response rates for normal, Gamma, and Poisson data from the values of PRE7 in Table 11.(8)
From Table12, it is seen that the estimator is negatively biased. The bias is negligible, being of the order 10−5 and 10−7 for various values of the parameter ρ and for various response rates, and hence, bias correction is not needed.


## 7. Conclusion

The following trend in the PREs is noticed from the tables:PRE1 increases with the increase in value of ρ, while PRE2,PRE3,PRE4,PRE5,PRE6,andPRE7 decrease with the increase in value of ρ.The proposed estimator is seen to be consistent, exists for all real values of parameters, has negligible bias, and is more efficient than 7 other contemporary estimators. Hence, the proposed estimator may be recommended for use in field work.



---

*Source: 4845569-2021-12-03.xml*