Mathematical Problems in Engineering, Apr 17, 2023
In some situations, the population of interest difers signifcantly in size, for example, in a med... more In some situations, the population of interest difers signifcantly in size, for example, in a medical study, the number of patients having a specifc disease and the size of health units may vary. Similarly, in a survey related to the income of a household, the household may have a diferent number of siblings, and then in such situations, we use probability proportional to size sampling. In this article, we have proposed an improved class of estimators for the estimation of population mean on the basis of probability proportional to size (PPS) sampling, using two auxiliary variables. Te mathematical expressions of the bias and mean square error (MSE) are derived up to the frst order of approximation. Four real datasets and a simulation study are conducted to assess the efciency of the improved class of estimators. It is found from the real datasets and a simulation study, that the proposed generalized class of estimators produced better results in terms of minimum MSE and higher PRE, as related to other considered estimators. An empirical study is given to support the theoretical results. Te theoretical study also demonstrates that the proposed generalized class of estimators outperforms the existing estimators.
In this article, we proposed an improved finite population variance estimator based on simple ran... more In this article, we proposed an improved finite population variance estimator based on simple random sampling using dual auxiliary information. Mathematical expressions of the proposed and existing estimators are obtained up to the first order of approximation. Two real data sets are used to examine the performances of a new improved proposed estimator. A simulation study is also recognized to assess the robustness and generalizability of the proposed estimator. From the result of real data sets and simulation study, it is examining that the proposed estimator give minimum mean square error and percentage relative efficiency are higher than all existing counterparts, which shown the importance of new improved estimator. The theoretical and numerical result illustrated that the proposed variance estimator based on simple random sampling using dual auxiliary information has the best among all existing estimators.
<p>Maps of prediction variance by using universal kriging (left) and Bayesian universal kri... more <p>Maps of prediction variance by using universal kriging (left) and Bayesian universal kriging (right),here X-coord = Latitude and Y-coord = Longitude.</p
In survey sampling, information on auxiliary variables related to the main variable is often avai... more 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 efficie...
<p>RMSPE of spatial covariance models subject to different methods of estimation (ML, REML,... more <p>RMSPE of spatial covariance models subject to different methods of estimation (ML, REML, OLS, and WLS) and universal kriging.</p
<p>Level of sodium concentration (mg/L) at observed locations in three divisions of Punjab,... more <p>Level of sodium concentration (mg/L) at observed locations in three divisions of Punjab, Pakistan (left), and observed locations and gridded locations (right).</p
<p>Level of sodium concentration (mg/L) at observed locations in three divisions of Punjab,... more <p>Level of sodium concentration (mg/L) at observed locations in three divisions of Punjab, Pakistan (left), and observed locations and gridded locations (right).</p
<p>The 95% predictive interval plots and actual values of data for universal kriging (left)... more <p>The 95% predictive interval plots and actual values of data for universal kriging (left) and Bayesian Universal Kriging (right), the dots represent to actual interval and bars represent to 95% credible intervals.</p
<p>Distributions of non-transformed and transformed response variable with their respective... more <p>Distributions of non-transformed and transformed response variable with their respective quantile plots.</p
Mathematical Problems in Engineering, Apr 17, 2023
In some situations, the population of interest difers signifcantly in size, for example, in a med... more In some situations, the population of interest difers signifcantly in size, for example, in a medical study, the number of patients having a specifc disease and the size of health units may vary. Similarly, in a survey related to the income of a household, the household may have a diferent number of siblings, and then in such situations, we use probability proportional to size sampling. In this article, we have proposed an improved class of estimators for the estimation of population mean on the basis of probability proportional to size (PPS) sampling, using two auxiliary variables. Te mathematical expressions of the bias and mean square error (MSE) are derived up to the frst order of approximation. Four real datasets and a simulation study are conducted to assess the efciency of the improved class of estimators. It is found from the real datasets and a simulation study, that the proposed generalized class of estimators produced better results in terms of minimum MSE and higher PRE, as related to other considered estimators. An empirical study is given to support the theoretical results. Te theoretical study also demonstrates that the proposed generalized class of estimators outperforms the existing estimators.
In this article, we proposed an improved finite population variance estimator based on simple ran... more In this article, we proposed an improved finite population variance estimator based on simple random sampling using dual auxiliary information. Mathematical expressions of the proposed and existing estimators are obtained up to the first order of approximation. Two real data sets are used to examine the performances of a new improved proposed estimator. A simulation study is also recognized to assess the robustness and generalizability of the proposed estimator. From the result of real data sets and simulation study, it is examining that the proposed estimator give minimum mean square error and percentage relative efficiency are higher than all existing counterparts, which shown the importance of new improved estimator. The theoretical and numerical result illustrated that the proposed variance estimator based on simple random sampling using dual auxiliary information has the best among all existing estimators.
<p>Maps of prediction variance by using universal kriging (left) and Bayesian universal kri... more <p>Maps of prediction variance by using universal kriging (left) and Bayesian universal kriging (right),here X-coord = Latitude and Y-coord = Longitude.</p
In survey sampling, information on auxiliary variables related to the main variable is often avai... more 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 efficie...
<p>RMSPE of spatial covariance models subject to different methods of estimation (ML, REML,... more <p>RMSPE of spatial covariance models subject to different methods of estimation (ML, REML, OLS, and WLS) and universal kriging.</p
<p>Level of sodium concentration (mg/L) at observed locations in three divisions of Punjab,... more <p>Level of sodium concentration (mg/L) at observed locations in three divisions of Punjab, Pakistan (left), and observed locations and gridded locations (right).</p
<p>Level of sodium concentration (mg/L) at observed locations in three divisions of Punjab,... more <p>Level of sodium concentration (mg/L) at observed locations in three divisions of Punjab, Pakistan (left), and observed locations and gridded locations (right).</p
<p>The 95% predictive interval plots and actual values of data for universal kriging (left)... more <p>The 95% predictive interval plots and actual values of data for universal kriging (left) and Bayesian Universal Kriging (right), the dots represent to actual interval and bars represent to 95% credible intervals.</p
<p>Distributions of non-transformed and transformed response variable with their respective... more <p>Distributions of non-transformed and transformed response variable with their respective quantile plots.</p
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