Javid Shabbir - Academia.edu (original) (raw)
Papers by Javid Shabbir
Repeated measurements designs (RMDs) are always economical but with the use of these designs, the... more Repeated measurements designs (RMDs) are always economical but with the use of these designs, there may arise residual effects. Minimal strongly balanced RMDs are well known to estimate the treatment effects and residual effects independently. In the situation, where these designs cannot be constructed, minimal nearly strongly balanced RMDs are used which is an efficient class of RMDs to control the residual effects. In this article, efficient minimal circular nearly strongly balanced RMDs are constructed in periods of three different sizes.
Communications in Statistics, Jan 17, 2020
The randomized response technique (RRT) is an effective method designed to obtain the sensitive i... more The randomized response technique (RRT) is an effective method designed to obtain the sensitive information from respondents while assuring the privacy. Narjis and Shabbir [Narjis, G., and J. Shabbir. 2018. Estimation of population proportion and sensitivity level using optional unrelated question randomized response techniques. Communications in Statistics-Simulation and Computation 0 (0):1-15] proposed three binary optional unrelated question RRT models for estimating the proportion of population that possess a sensitive characteristic ðpÞ and the sensitivity level ðxÞ of the question. In this study, we have developed the Bayes estimators of two parameters ðp, xÞ for optional unrelated question RRT model along with their corresponding minimal Bayes posterior expected losses (BPEL) under squared error loss function (SELF) using beta prior. Relative losses, mean squared error (MSE) and absolute bias are also examined to compare the performances of the Bayes estimates with those of the classical estimates obtained by Narjis and Shabbir (2018). A real survey data are provided for practical utilizations.
Cogent Medicine, 2021
Abstract Abstract: COVID-19 has big health issues which affect worldwide beyond their borders, ra... more Abstract Abstract: COVID-19 has big health issues which affect worldwide beyond their borders, race, and ethnicity. All the countries faced this pandemic challenge but most of the underdeveloped countries are facing more dangerous situations due to limited financial and health infrastructure to respond against it. Overall, more than 100 million people are affected by the Novel Virus which results in 2.15 million people dying within a small interval of time. The current pandemic has brought unpredicted challenges to societies and also threatened humanity and global resilience. According to the National Command Operation Center, Pakistan, more than 0.534 million people are suffering with COVID-19 with more than 11 thousand deaths across the country. The Government of Pakistan has taken different initiatives like complete and smart lockdown to control the pandemic as much as possible. After the removal of the first lockdown, the high peak was observed across the country and created a panic situation among people and the government again closed all the educational and religious institutions with immediate effect to tackle the second wave of pandemic. Further, the interconnected nature of COVID-19 crises demands an integrated approach and coordination between all stakeholders to handle the pandemic in a significant way. Identifying the best set of policies and guidelines to handle COVID-19 challenges, and align them for the sustainable recovery from pandemic. The basic challenge facing the policy makers of underdeveloped countries is how to utilize limited resources to achieve interconnected goals for managing health recovery, economic crises, and creating environmental sustainability. We present a framework for identifying and prioritizing policy action to address COVID-19 and ensure sustainable recovery. The framework outlines principles and criteria, and shared policy goals, identifying smart strategies, accessing policy compatibility, aligning policy instruments and improving sustainability in short and long term policy decisions. This framework can be helpful for policy makers in the short and long run for mapping policy options and accessing cross-sectoral implementation. This framework is also helpful for policy makers to prioritize policy choice and allocate limited resources in such a way that they are directed toward actions and achieve interconnected solutions of health, economy, and environment.
Communications in Statistics, Jun 18, 2019
In this article, we propose a new partial randomized response technique (RRT) model to estimate t... more In this article, we propose a new partial randomized response technique (RRT) model to estimate the mean of the number of persons possessing a rare sensitive attribute using the Poisson distribution. Properties of the proposed partial RRT model have been studied. The utility of proposed partial RRT model under stratification is also explored. Efficiency comparison between proposed partial RRT model is carried out numerically under simple and stratified random sampling.
Minimal neighbor balanced designs are economical, therefore, these are preferred by the experimen... more Minimal neighbor balanced designs are economical, therefore, these are preferred by the experimenters to minimize the bias due to neighbor effects. Minimal circular balanced neighbor designs cannot be constructed for almost every case of v even, where v is number of the treatments to be compared. For v even, the circular GN 2-designs in which each treatment appears exactly once as neighbors with all other treatments except the one with which it appears twice, are considered the better alternate to the minimal balanced neighbor designs. In this article, an algorithm is developed to generate the circular GN 2-designs for v even which can be converted directly into minimal circular balanced and strongly balanced neighbor designs. This algorithm is also coded with R.
Mathematical Population Studies, May 6, 2022
Statistics, computing and interdisciplinary research, Dec 31, 2022
Journal of reliability and statistical studies, Oct 5, 2018
Science Progress, Apr 1, 2023
In this article, we suggest an enhanced family of estimators for estimation of population mean em... more In this article, we suggest an enhanced family of estimators for estimation of population mean employing the supplementary variables under probability proportional to size sampling. Up to the first order of approximation, numerical formulations of the bias and mean square error of estimators are obtained. From our suggested improved family of estimators, we give sixteen different members. The recommended family of estimators has specifically been used to derive the characteristics of sixteen estimators based on the known population parameters of the study as well as auxiliary variables. The performances of the suggested estimators have been assessed using three actual data. Furthermore, a simulation investigation is also accompanied to evaluate the effectiveness of estimators. The proposed estimators have a smaller MSE and an advanced PRE when linked to existing estimators, which are based on actual data sets and simulation studies. Theoretically and empirically studies also reveal that the suggested estimators accomplish well than the usual estimators.
Mathematical Problems in Engineering, Aug 23, 2022
In this study, a new partial randomized response model (RRM) has been proposed for estimating the... more In this study, a new partial randomized response model (RRM) has been proposed for estimating the population mean of two quantitative sensitive variables simultaneously. e utility of proposed model under strati cation is also explored. e e ciency comparisons of the proposed model under simple and strati ed random sampling are carried out numerically. A real data set was collected through direct questioning, proposed partial RRM and competitor randomized device from the students of statistics and animal sciences departments of Quaid-I-Azam University Islamabad, Pakistan. e performance of the proposed partial RRM is better than competitor RRM under simple and strati ed random sampling.
International Journal of Mathematics and Computation, Aug 14, 2018
It is well that, utilization of additional auxiliary information in any form, can improve the per... more It is well that, utilization of additional auxiliary information in any form, can improve the performance of the estimation procedure. In present script, we proposed a class of estimators by using the supplementary and truncated auxiliary information for imputing the missing values. Mathematical results for bias and mean squared error are obtained up to first order approximation. For relative comparison of the proposed estimators with existing ones, real life data sets are used. The numerical study reveals that, the proposed class estimators can perform better than (Rao, 1991), (Grover and Kaur, 2014) and (Haq, Khan and Hussain, 2017) estimators.
Communications in Statistics, Jun 1, 2020
In this paper, we propose a new two-stage unrelated question randomized response technique (RRT) ... more In this paper, we propose a new two-stage unrelated question randomized response technique (RRT) model to estimate the mean of the number of persons possessing a rare sensitive attribute using the Poisson distribution. The utility of proposed two-stage unrelated question RRT model under stratification is also explored. Efficiency comparison between proposed two-stage unrelated question and Singh and Suman (2019) RRT model is carried out numerically under simple random sampling, and with Suman and Singh (2019) in stratified random sampling, respectively.
Iranian Journal of Science and Technology Transaction A-science, Dec 27, 2017
This article focuses on the study of a 3-component mixture of the inverse Weibull distributions u... more This article focuses on the study of a 3-component mixture of the inverse Weibull distributions under Bayesian perspective. The censored sampling scheme is used because it is popular in reliability theory and survival analysis. To achieve this objective, the Bayes estimates of the parameter of the mixture model along with their posterior risks using informative and non-informative priors are attained. These estimates have been acquired under two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the case (a), Bayes estimates are gained under three loss functions while for the case (b) only the squared error loss function is used. To study numerically, the performance of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times.
Scientific Reports, Apr 3, 2023
This article aims to suggest a new improved generalized class of estimators for finite population... more 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. Generally it is a well-established notion that when the auxiliary variable is used appropriately in survey sampling, precision of the estimator is increased. Numbers of estimators exist in the literature for estimating different population parameters, such as mean, variance and total etc. but little attention has been paid to study the distribution (DF). The works on survey sampling discusses a diversity of procedures for incorporating the auxiliary variable via ratio, product, and regression methods of estimation. Numerous researchers have suggested various estimators by adequately adapting the auxiliary variable. These research findings can be investigated by Grover and Kaur 1 suggested a generalized class of ratio type exponential estimators of population mean under linear transformation of auxiliary variable. Ahmad et al. 2 discussed use of extreme values to estimate the finite population mean under PPS sampling scheme. Audu et al. 3 suggested on the efficiency of almost unbiased mean imputation when population mean of auxiliary variable is unknown. Singh and Nigam 4 discussed efficient method of estimating the finite population mean based on two auxiliary variables in the presence of nonresponse under stratified sampling. Shahzad et al. 5 discussed estimation of the population mean by successive use of an auxiliary variable in median ranked set sampling. Singh et al. 6 proposed some imputation methods to deal with the problems of missing data in two-occasion successive sampling. Aggarwal et al. 7 discussed estimation of the population mean by developing a new estimator. Singh et al. 6 suggested an exponential approach for estimating population mean using two auxiliary variables in stratified random sampling. Yadav et al. 8 proposed new modified ratio type estimator of the population mean using the known median of the study variable. Pal and Singh 9 discussed about estimation of finite population mean using auxiliary information in presence of non-response. Pal and Singh 10 proposed an efficient new approach for estimating the general parameter using
Mathematical Problems in Engineering, Nov 17, 2022
In this article, we propose an improved estimator for fnite population variance based on stratife... more In this article, we propose an improved estimator for fnite population variance based on stratifed sampling by using the auxiliary variable as well as the rank of the auxiliary variable. Expressions for the bias and the mean square error of the estimators are derived up to the frst order of approximation. Four real data sets are used to measure the performances of estimators. Moreover, a simulation study is also conducted to observe the efciency of the proposed variance estimator. Te theoretical and numerical results show that the proposed estimator under stratifed random sampling is more efcient as compared to the existing estimators.
Journal of Statistical Computation and Simulation, Mar 4, 2019
In the present article, we propose the generalized ratio-type and generalized ratio-exponential-t... more In the present article, we propose the generalized ratio-type and generalized ratio-exponential-type estimators for population mean in adaptive cluster sampling (ACS) under modified Horvitz-Thompson estimator. The proposed estimators utilize the auxiliary information in combination of conventional measures (coefficient of skewness, coefficient of variation, correlation coefficient, covariance, coefficient of kurtosis) and robust measures (tri-mean, Hodges-Lehmann, midrange) to increase the efficiency of the estimators. Properties of the proposed estimators are discussed using the first order of approximation. The simulation study is conducted to evaluate the performances of the estimators. The results reveal that the proposed estimators are more efficient than competing estimators for population mean in ACS under both modified Hansen-Hurwitz and Horvitz-Thompson estimators.
Communications in Statistics, Sep 8, 2017
ABSTRACT In this article, we propose a new difference-type estimator in estimating the finite pop... more ABSTRACT In this article, we propose a new difference-type estimator in estimating the finite population mean in stratified double sampling by using the ranks of two auxiliary variables as an additional information. The proposed estimator performs better than the usual sample mean estimator, ratio estimator, exponential estimator, Choudhury and Singh (2012) estimator, Vishwakarma and Gangele (2014) estimator, Singh and Khalid (2015) estimator, Khan and Al-Hossain (2016) estimator, Khan (2016) estimator, and the usual difference estimator. Two real datasets are used to observe the performances of estimators.
Statistical theory and related fields, Sep 26, 2021
Repeated measurements designs (RMDs) are always economical but with the use of these designs, the... more Repeated measurements designs (RMDs) are always economical but with the use of these designs, there may arise residual effects. Minimal strongly balanced RMDs are well known to estimate the treatment effects and residual effects independently. In the situation, where these designs cannot be constructed, minimal nearly strongly balanced RMDs are used which is an efficient class of RMDs to control the residual effects. In this article, efficient minimal circular nearly strongly balanced RMDs are constructed in periods of three different sizes.
Communications in Statistics, Jan 17, 2020
The randomized response technique (RRT) is an effective method designed to obtain the sensitive i... more The randomized response technique (RRT) is an effective method designed to obtain the sensitive information from respondents while assuring the privacy. Narjis and Shabbir [Narjis, G., and J. Shabbir. 2018. Estimation of population proportion and sensitivity level using optional unrelated question randomized response techniques. Communications in Statistics-Simulation and Computation 0 (0):1-15] proposed three binary optional unrelated question RRT models for estimating the proportion of population that possess a sensitive characteristic ðpÞ and the sensitivity level ðxÞ of the question. In this study, we have developed the Bayes estimators of two parameters ðp, xÞ for optional unrelated question RRT model along with their corresponding minimal Bayes posterior expected losses (BPEL) under squared error loss function (SELF) using beta prior. Relative losses, mean squared error (MSE) and absolute bias are also examined to compare the performances of the Bayes estimates with those of the classical estimates obtained by Narjis and Shabbir (2018). A real survey data are provided for practical utilizations.
Cogent Medicine, 2021
Abstract Abstract: COVID-19 has big health issues which affect worldwide beyond their borders, ra... more Abstract Abstract: COVID-19 has big health issues which affect worldwide beyond their borders, race, and ethnicity. All the countries faced this pandemic challenge but most of the underdeveloped countries are facing more dangerous situations due to limited financial and health infrastructure to respond against it. Overall, more than 100 million people are affected by the Novel Virus which results in 2.15 million people dying within a small interval of time. The current pandemic has brought unpredicted challenges to societies and also threatened humanity and global resilience. According to the National Command Operation Center, Pakistan, more than 0.534 million people are suffering with COVID-19 with more than 11 thousand deaths across the country. The Government of Pakistan has taken different initiatives like complete and smart lockdown to control the pandemic as much as possible. After the removal of the first lockdown, the high peak was observed across the country and created a panic situation among people and the government again closed all the educational and religious institutions with immediate effect to tackle the second wave of pandemic. Further, the interconnected nature of COVID-19 crises demands an integrated approach and coordination between all stakeholders to handle the pandemic in a significant way. Identifying the best set of policies and guidelines to handle COVID-19 challenges, and align them for the sustainable recovery from pandemic. The basic challenge facing the policy makers of underdeveloped countries is how to utilize limited resources to achieve interconnected goals for managing health recovery, economic crises, and creating environmental sustainability. We present a framework for identifying and prioritizing policy action to address COVID-19 and ensure sustainable recovery. The framework outlines principles and criteria, and shared policy goals, identifying smart strategies, accessing policy compatibility, aligning policy instruments and improving sustainability in short and long term policy decisions. This framework can be helpful for policy makers in the short and long run for mapping policy options and accessing cross-sectoral implementation. This framework is also helpful for policy makers to prioritize policy choice and allocate limited resources in such a way that they are directed toward actions and achieve interconnected solutions of health, economy, and environment.
Communications in Statistics, Jun 18, 2019
In this article, we propose a new partial randomized response technique (RRT) model to estimate t... more In this article, we propose a new partial randomized response technique (RRT) model to estimate the mean of the number of persons possessing a rare sensitive attribute using the Poisson distribution. Properties of the proposed partial RRT model have been studied. The utility of proposed partial RRT model under stratification is also explored. Efficiency comparison between proposed partial RRT model is carried out numerically under simple and stratified random sampling.
Minimal neighbor balanced designs are economical, therefore, these are preferred by the experimen... more Minimal neighbor balanced designs are economical, therefore, these are preferred by the experimenters to minimize the bias due to neighbor effects. Minimal circular balanced neighbor designs cannot be constructed for almost every case of v even, where v is number of the treatments to be compared. For v even, the circular GN 2-designs in which each treatment appears exactly once as neighbors with all other treatments except the one with which it appears twice, are considered the better alternate to the minimal balanced neighbor designs. In this article, an algorithm is developed to generate the circular GN 2-designs for v even which can be converted directly into minimal circular balanced and strongly balanced neighbor designs. This algorithm is also coded with R.
Mathematical Population Studies, May 6, 2022
Statistics, computing and interdisciplinary research, Dec 31, 2022
Journal of reliability and statistical studies, Oct 5, 2018
Science Progress, Apr 1, 2023
In this article, we suggest an enhanced family of estimators for estimation of population mean em... more In this article, we suggest an enhanced family of estimators for estimation of population mean employing the supplementary variables under probability proportional to size sampling. Up to the first order of approximation, numerical formulations of the bias and mean square error of estimators are obtained. From our suggested improved family of estimators, we give sixteen different members. The recommended family of estimators has specifically been used to derive the characteristics of sixteen estimators based on the known population parameters of the study as well as auxiliary variables. The performances of the suggested estimators have been assessed using three actual data. Furthermore, a simulation investigation is also accompanied to evaluate the effectiveness of estimators. The proposed estimators have a smaller MSE and an advanced PRE when linked to existing estimators, which are based on actual data sets and simulation studies. Theoretically and empirically studies also reveal that the suggested estimators accomplish well than the usual estimators.
Mathematical Problems in Engineering, Aug 23, 2022
In this study, a new partial randomized response model (RRM) has been proposed for estimating the... more In this study, a new partial randomized response model (RRM) has been proposed for estimating the population mean of two quantitative sensitive variables simultaneously. e utility of proposed model under strati cation is also explored. e e ciency comparisons of the proposed model under simple and strati ed random sampling are carried out numerically. A real data set was collected through direct questioning, proposed partial RRM and competitor randomized device from the students of statistics and animal sciences departments of Quaid-I-Azam University Islamabad, Pakistan. e performance of the proposed partial RRM is better than competitor RRM under simple and strati ed random sampling.
International Journal of Mathematics and Computation, Aug 14, 2018
It is well that, utilization of additional auxiliary information in any form, can improve the per... more It is well that, utilization of additional auxiliary information in any form, can improve the performance of the estimation procedure. In present script, we proposed a class of estimators by using the supplementary and truncated auxiliary information for imputing the missing values. Mathematical results for bias and mean squared error are obtained up to first order approximation. For relative comparison of the proposed estimators with existing ones, real life data sets are used. The numerical study reveals that, the proposed class estimators can perform better than (Rao, 1991), (Grover and Kaur, 2014) and (Haq, Khan and Hussain, 2017) estimators.
Communications in Statistics, Jun 1, 2020
In this paper, we propose a new two-stage unrelated question randomized response technique (RRT) ... more In this paper, we propose a new two-stage unrelated question randomized response technique (RRT) model to estimate the mean of the number of persons possessing a rare sensitive attribute using the Poisson distribution. The utility of proposed two-stage unrelated question RRT model under stratification is also explored. Efficiency comparison between proposed two-stage unrelated question and Singh and Suman (2019) RRT model is carried out numerically under simple random sampling, and with Suman and Singh (2019) in stratified random sampling, respectively.
Iranian Journal of Science and Technology Transaction A-science, Dec 27, 2017
This article focuses on the study of a 3-component mixture of the inverse Weibull distributions u... more This article focuses on the study of a 3-component mixture of the inverse Weibull distributions under Bayesian perspective. The censored sampling scheme is used because it is popular in reliability theory and survival analysis. To achieve this objective, the Bayes estimates of the parameter of the mixture model along with their posterior risks using informative and non-informative priors are attained. These estimates have been acquired under two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the case (a), Bayes estimates are gained under three loss functions while for the case (b) only the squared error loss function is used. To study numerically, the performance of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times.
Scientific Reports, Apr 3, 2023
This article aims to suggest a new improved generalized class of estimators for finite population... more 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. Generally it is a well-established notion that when the auxiliary variable is used appropriately in survey sampling, precision of the estimator is increased. Numbers of estimators exist in the literature for estimating different population parameters, such as mean, variance and total etc. but little attention has been paid to study the distribution (DF). The works on survey sampling discusses a diversity of procedures for incorporating the auxiliary variable via ratio, product, and regression methods of estimation. Numerous researchers have suggested various estimators by adequately adapting the auxiliary variable. These research findings can be investigated by Grover and Kaur 1 suggested a generalized class of ratio type exponential estimators of population mean under linear transformation of auxiliary variable. Ahmad et al. 2 discussed use of extreme values to estimate the finite population mean under PPS sampling scheme. Audu et al. 3 suggested on the efficiency of almost unbiased mean imputation when population mean of auxiliary variable is unknown. Singh and Nigam 4 discussed efficient method of estimating the finite population mean based on two auxiliary variables in the presence of nonresponse under stratified sampling. Shahzad et al. 5 discussed estimation of the population mean by successive use of an auxiliary variable in median ranked set sampling. Singh et al. 6 proposed some imputation methods to deal with the problems of missing data in two-occasion successive sampling. Aggarwal et al. 7 discussed estimation of the population mean by developing a new estimator. Singh et al. 6 suggested an exponential approach for estimating population mean using two auxiliary variables in stratified random sampling. Yadav et al. 8 proposed new modified ratio type estimator of the population mean using the known median of the study variable. Pal and Singh 9 discussed about estimation of finite population mean using auxiliary information in presence of non-response. Pal and Singh 10 proposed an efficient new approach for estimating the general parameter using
Mathematical Problems in Engineering, Nov 17, 2022
In this article, we propose an improved estimator for fnite population variance based on stratife... more In this article, we propose an improved estimator for fnite population variance based on stratifed sampling by using the auxiliary variable as well as the rank of the auxiliary variable. Expressions for the bias and the mean square error of the estimators are derived up to the frst order of approximation. Four real data sets are used to measure the performances of estimators. Moreover, a simulation study is also conducted to observe the efciency of the proposed variance estimator. Te theoretical and numerical results show that the proposed estimator under stratifed random sampling is more efcient as compared to the existing estimators.
Journal of Statistical Computation and Simulation, Mar 4, 2019
In the present article, we propose the generalized ratio-type and generalized ratio-exponential-t... more In the present article, we propose the generalized ratio-type and generalized ratio-exponential-type estimators for population mean in adaptive cluster sampling (ACS) under modified Horvitz-Thompson estimator. The proposed estimators utilize the auxiliary information in combination of conventional measures (coefficient of skewness, coefficient of variation, correlation coefficient, covariance, coefficient of kurtosis) and robust measures (tri-mean, Hodges-Lehmann, midrange) to increase the efficiency of the estimators. Properties of the proposed estimators are discussed using the first order of approximation. The simulation study is conducted to evaluate the performances of the estimators. The results reveal that the proposed estimators are more efficient than competing estimators for population mean in ACS under both modified Hansen-Hurwitz and Horvitz-Thompson estimators.
Communications in Statistics, Sep 8, 2017
ABSTRACT In this article, we propose a new difference-type estimator in estimating the finite pop... more ABSTRACT In this article, we propose a new difference-type estimator in estimating the finite population mean in stratified double sampling by using the ranks of two auxiliary variables as an additional information. The proposed estimator performs better than the usual sample mean estimator, ratio estimator, exponential estimator, Choudhury and Singh (2012) estimator, Vishwakarma and Gangele (2014) estimator, Singh and Khalid (2015) estimator, Khan and Al-Hossain (2016) estimator, Khan (2016) estimator, and the usual difference estimator. Two real datasets are used to observe the performances of estimators.
Statistical theory and related fields, Sep 26, 2021