Finite Population Distribution Function Research Articles

Construction of improved comprehensive classes of estimators for population distribution function

The primary purpose of this article is to examine the issue of estimating the finite population distribution function from auxiliary information, such as population mean and rank of the auxiliary variables, that are already known. In order to better estimate the distribution function (DF) of a finite population, two improved estimators are developed. The bias and mean squared error of the suggested and existing estimators are derived up to the first order of approximation. To improve the efficiency of an estimators, we compare the suggested estimators with existing counterpart. Based on the numerical outcomes, it is to be noted that the suggested classes of estimators perform well using six actual data sets. The strength and generalization of the suggested estimators are also verified using a simulation analysis. Based on the result of actual data sets and a simulation study, we observe that the suggested estimator outperforms as compared to all existing estimators which is compared in this study.

Advancing Survey Sampling Efficiency under Stratified Random Sampling and Post-Stratification: Leveraging Symmetry for Enhanced Estimation Accuracy in the Prediction of Exam Scores

This pioneering investigation introduces two innovative estimators crafted to evaluate the finite population distribution function of a study variable, employing auxiliary variables within the framework of stratified random sampling and post-stratification while emphasizing symmetry in the sampling process. The derivation of mathematical expressions for bias and the mean square error up to the first degree of approximation fortifies the credibility of the proposed estimators. Drawing from three distinct datasets, including real-world data capturing student behaviors and exam performances from 500 students, this research highlights the superior efficiency of the proposed estimators compared to existing methods across both sampling schemes. Employing the proposed estimator, we effectively forecast students’ exam scores based on their study hours, backed by empirical evidence showcasing its precision in terms of mean square error and percentage relative efficiency. This study not only introduces inventive solutions to enduring challenges in survey sampling but also provides practical insights into enhancing predictive accuracy in educational assessments.

Predicting the Finite Population Distribution Function under a Multilevel Model

Chambers and Dunstan proposed a model-based predictor of the population distribution function that makes use of auxiliary population information under a general sampling design. Subsequently, Rao, Kovar, and Mantel proposed design-based ratio and difference predictors of the population distribution function that also use this auxiliary information. Both predictors (CD and RKM) assume a single level model for the target population. In this article we develop predictors of the finite population distribution function for a population that follows a multilevel model. These new predictors use the same smearing approach underpinning the CD predictor. We compare our new predictors with the CD and RKM predictors via design-based simulation, and show that they perform better than these single level predictors when there is significant intra-cluster correlation. The performances of these new two level predictors are also examined via an empirical study based on data from a large-scale UK business survey aimed at estimating the distribution of hourly pay rates. AMS Subject Classification: Primary 62G30, Secondary 62G32

A new improved generalized class of estimators for population distribution function using auxiliary variable under simple random sampling

This article aims to suggest a new improved generalized class of estimators for finite population distribution function of the study and the auxiliary variables as well as mean of the usual auxiliary variable under simple random sampling. The numerical expressions for the bias and mean squared error (MSE) are derived up to first degree of approximation. From our generalized class of estimators, we obtained two improved estimators. The gain in second proposed estimator is more as compared to first estimator. Three real data sets and a simulation are accompanied to measure the performances of our generalized class of estimators. The MSE of our proposed estimators is minimum and consequently percentage relative efficiency is higher as compared to their existing counterparts. From the numerical outcomes it has been shown that the proposed estimators perform well as compared to all considered estimators in this study.

Methods to Counter Self-Selection Bias in Estimations of the Distribution Function and Quantiles

Many surveys are performed using non-probability methods such as web surveys, social networks surveys, or opt-in panels. The estimates made from these data sources are usually biased and must be adjusted to make them representative of the target population. Techniques to mitigate this selection bias in non-probability samples often involve calibration, propensity score adjustment, or statistical matching. In this article, we consider the problem of estimating the finite population distribution function in the context of non-probability surveys and show how some methodologies formulated for linear parameters can be adapted to this functional parameter, both theoretically and empirically, thus enhancing the accuracy and efficiency of the estimates made.

A Simulation Study: Population Distribution Function Estimation Using Dual Auxiliary Information under Stratified Sampling Scheme

In survey sampling, information on auxiliary variables related to the main variable is often available in many practical problems. Since the mid-twentieth century, researchers have taken a keen interest in the use of auxiliary information, due to its usefulness in estimation methods. In this article, our main objective is to discover the problem associated with estimation of the finite population distribution function, using the known auxiliary variable, which occurs as the sample distribution function and the rank of the auxiliary variable. A new family of the finite population distribution function estimators is proposed in the stratified sampling scheme. The mathematical equations for the bias and mean square error have been obtained for each proposed estimator, along with the efficiency conditions. Besides theoretical efficiency comparison, an empirical study has also been conducted to analyze the performance of estimators. A simulation study is also performed to observe the efficiency of the proposed estimators. The implementation of the proposed sampling scheme is illustrated by a practical example.

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

In this article, we propose a generalized class of exponential factor type estimators for estimation of the finite population distribution function (PDF) using an auxiliary variable in the form of the mean and rank of the auxiliary variable exist. The expressions of the bias and mean square error of the estimators are computed up to the first order approximation. The proposed estimators provide minimum mean square error as compared to all other considered estimators. Three real data sets are used to check the performance of the proposed estimators. Moreover, simulation studies are also carried out to observe the performances of the proposed estimators. The proposed estimators confirmed their superiority numerically as well as theoretically by producing efficient results as compared to all other competing estimators.

Modified estimators of finite population distribution function based on dual use of auxiliary information under stratified random sampling.

In survey sampling, information on auxiliary variables related to the main variable is often available in many practical problems. Since the mid-twentieth century, researchers have taken a keen interest in the use of auxiliary information due to its usefulness in estimation methods. The current study presents two new estimators for the distribution function of a finite population based on dual auxiliary variables. The new estimators can be used in situations where the researchers face some sort of complex data set. The mathematical equations for the bias and mean square error have been obtained for each proposed estimator. Besides, an empirical study simulation study has also been conducted to analyse the performance of estimators. It is found that the new suggested estimators of the distribution function of a finite population are more accurate than some of the existing estimators.

A New Exponential Factor-Type Estimator for Population Distribution Function Using Dual Auxiliary Variables under Stratified Random Sampling

In this paper, we propose a generalized class of exponential-type estimators for estimating the finite population distribution function using dual auxiliary variables under stratified sampling. The biases and mean squared errors (MSEs) of the proposed class of estimators are derived up to the first order of approximation. The empirical and theoretical study of comparisons is discussed. Four populations are taken for the support of the theoretical findings. It is observed that the proposed class of estimators performs better as compared to all other considered estimators in stratified sampling.

On use of subsampling of the non-respondents for estimation of distribution function

In this study, we propose a general class of estimators of the finite population distribution function (DF) using two auxiliary variables under subsampling of non-respondents. We use the Hansen and Hurwitz [1] pioneered model in our subsampling technique. Layout of response and non-response classes are discussed in various tables in detail. Expressions for the biases and mean square errors (MSEs) of the estimators are obtained up to first order of approximation. We also obtain the conditions by comparing the proposed estimator with existing estimators. Three real data sets are used to support the theoretical findings. In our findings, it is observed that the proposed class of estimators is more efficient as compared to all other existing estimators including the usual mean estimator, ratio estimator, exponential-ratio estimator, traditional difference estimator, Rao [2] difference estimator, Kumar et al. [3] estimator and many other recent difference type estimators by using the criterion of MSE.

Finite Population Distribution Function Estimation Using Auxiliary Information Under Simple Random Sampling

In this paper, a new estimator for estimating the finite population distribution function(DF) are propose using supplementary information on the DF of the auxiliary variable under simple random sampling. A comparative study is conducted to compare, theoretically and numerically, the adapted distribution function estimators of Cochran (1940), Murthy (1967), Bahl and Tuteja (1991), Rao (1991), Singh et al. (2009) and Grover and Kaur (2014) with the proposed estimators. It is found that the proposed estimators always perform better than the adapted estimators in terms of MSE and percentage relative efficiency.

Estimation of finite population distribution function of sensitive variable*

The finite population proportion of a sensitive characteristic is estimated indirectly by using Randomized Response (RR) Techniques (RRT’s) pioneered by Warner (1965) followed by several other RRT’s in the literature. The existing literature contains several RRT’s for estimating the finite population mean of the sensitive quantitative variable. However, there might be a situation when the population proportion bearing the value of the stigmatizing variable below a threshold is of more concern than the exact population mean. The problem hence reduces to the estimation of the finite population distribution function of a quantitative sensitive variable. Following Chaudhuri and Saha (2004), a logistic regression approach has been used to estimate the finite population proportion bearing value of the stigmatizing variable below a threshold. As an alternative to this method, this article also attempts to provide suitable modifications for sensitive variables, in the estimation of distribution function proposed by Chaudhuri and Shaw (2020), when the variable of interest is innocuous. Numerical results based on a simulated population present interesting finding on the proposed methodologies.

Estimation of finite population distribution function with dual use of auxiliary information under non-response.

In this paper, we propose two new families of estimators for estimating the finite population distribution function in the presence of non-response under simple random sampling. The proposed estimators require information on the sample distribution functions of the study and auxiliary variables, and additional information on either sample mean or ranks of the auxiliary variable. We considered two situations of non-response (i) non-response on both study and auxiliary variables, (ii) non-response occurs only on the study variable. The performance of the proposed estimators are compared with the existing estimators available in the literature, both theoretically and numerically. It is also observed that proposed estimators are more precise than the adapted distribution function estimators in terms of the percentage relative efficiency.

Treating nonresponse in the estimation of the distribution function

The estimation of a finite population distribution function is considered when there are missing data. Calibration adjustment is used for dealing with nonresponse at the estimation stage. Several procedures are proposed and compared. A numerical study is carried out to evaluate the performances of estimators. Computational problems with the implementation of the proposed calibration estimators are also considered.

Kernel density estimation from complex surveys in the presence of complete auxiliary information

Auxiliary information is widely used in survey sampling to enhance the precision of estimators of finite population parameters, such as the finite population mean, percentiles, and distribution function. In the context of complex surveys, we show how auxiliary information can be used effectively in kernel estimation of the superpopulation density function of a given study variable. We propose two classes of “model-assisted” kernel density estimators that make efficient use of auxiliary information. For one class we assume that the functional relationship between the study variable Y and the auxiliary variable X is known, while for the other class the relationship is assumed unknown and is estimated using kernel smoothing techniques. Under the first class, we show that if the functional relationship can be written as a simple linear regression model with constant error variance, the mean of the proposed density estimator will be identical to the well-known regression estimator of the finite population mean. If we drop the intercept from the linear model and allow the error variance to be proportional to the auxiliary variable, the mean of the proposed density estimator matches the ratio estimator of the finite population mean. The properties of the new density estimators are studied under a combined design-model-based inference framework, which accounts for the underlying superpopulation model as well as the randomization distribution induced by the sampling design. Moreover, the asymptotic normality of each estimator is derived under both design-based and combined inference frameworks when the sampling design is simple random sampling without replacement. For the practical implementation of these estimators, we discuss how data-driven bandwidth estimators can be obtained. The finite sample properties of the proposed estimators are addressed via simulations and an example that mimics a real survey. These simulations show that the new estimators perform very well compared to standard kernel estimators which do not utilize the auxiliary information.

Bootstrap Variance Estimation for Semiparametric Finite Population Distribution Function Estimator

Estimating finite population distribution function (FPDF) emerges as an important problem to the survey statisticians since the pioneering work of Chambers and Dunstan [1] . It unifies estimation of standard finite population parameters, namely, mean and quantiles. Regarding this, estimating variance of FPDF estimator is an important task for accessing the quality of the estimtor and drawing inferences (e.g., confidence interval estimation) on finite population parameters. Due to non-linearity of FPDF estimator, resampling-based methods are developed earlier for parametric or non-parametric Chambers–Dunstan estimator. Here, we attempt the problem of estimating variance of P-splines-based semiparametric model-based Chambers–Dunstan type estimator of the FPDF. The proposed variance estimator involes bootstrapping. Here, the bootstrap procedure is non-trivial since it does not imitate the full mechanism of two-stage sample generating procedure from an infinite hypothetical population (superpopulation). We have established the weak consistency of the proposed resampling-based variance estimator for specific sampling designs, e.g., simple random sampling. Also, the satisfactory empirical performance of the poposed estimator has been shown through simulation studies and a real life example.

Estimation of the finite population distribution function using a global penalized calibration method

Auxiliary information $${\varvec{x}}$$ is commonly used in survey sampling at the estimation stage. We propose an estimator of the finite population distribution function $$F_{y}(t)$$ when $${\varvec{x}}$$ is available for all units in the population and related to the study variable y by a superpopulation model. The new estimator integrates ideas from model calibration and penalized calibration. Calibration estimates of $$F_{y}(t)$$ with the weights satisfying benchmark constraints on the fitted values distribution function $$\hat{F}_{\hat{y}}=F_{\hat{y}}$$ on a set of fixed values of t can be found in the literature. Alternatively, our proposal $$\hat{F}_{y\omega }$$ seeks an estimator taking into account a global distance $$D(\hat{F}_{\hat{y}\omega },F_{\hat{y}})$$ between $$\hat{F}_{\hat{y}\omega }$$ and $${F}_{\hat{y}},$$ and a penalty parameter $$\alpha $$ that assesses the importance of this term in the objective function. The weights are explicitly obtained for the $$L^2$$ distance and conditions are given so that $$\hat{F}_{y\omega }$$ to be a distribution function. In this case $$\hat{F}_{y\omega }$$ can also be used to estimate the population quantiles. Moreover, results on the asymptotic unbiasedness and the asymptotic variance of $$\hat{F}_{y\omega }$$ , for a fixed $$\alpha $$ , are obtained. The results of a simulation study, designed to compare the proposed estimator to other existing ones, reveal that its performance is quite competitive.

Distribution function estimates from dual frame context

The estimation of a finite population distribution function under a dual frame context is considered when auxiliary population information is available. Several procedures are defined and compared to various methods adapted from the literature. The asymptotic distribution of the proposed estimators is established, a brief simulation is implemented and an application to real data is included.

Optimal dimension and optimal auxiliary vector to construct calibration estimators of the distribution function

The calibration technique (Deville and Särndal, 1992) to estimate the finite distribution function has been studied in several papers. Calibration seeks for new weights close enough to sampling weights according to some distance function and that, at the same time, match benchmark constraints on available auxiliary information. The non smooth character of the finite population distribution function causes certain complexities that are resolved by different authors in different ways. One of these is to have consistency at a number of arbitrarily chosen points. This paper deals with the problem of the optimal selection of the number of points and with the optimal selections of these points, when auxiliary information is used by means of calibration.

Doubly Robust Inference for the Distribution Function in the Presence of Missing Survey Data

AbstractItem non‐response in surveys occurs when some, but not all, variables are missing. Unadjusted estimators tend to exhibit some bias, called the non‐response bias, if the respondents differ from the non‐respondents with respect to the study variables. In this paper, we focus on item non‐response, which is usually treated by some form of single imputation. We examine the properties of doubly robust imputation procedures, which are those that lead to an estimator that remains consistent if either the outcome variable or the non‐response mechanism is adequately modelled. We establish the double robustness property of the imputed estimator of the finite population distribution function under random hot‐deck imputation within classes. We also discuss the links between our approach and that of Chambers and Dunstan. The results of a simulation study support our findings.