The Dhaka University Journal of Science, May 29, 2023
This paper aims to evaluate the accuracy of probability calculation using Chebyshev's inequality ... more This paper aims to evaluate the accuracy of probability calculation using Chebyshev's inequality based on simulation study. We consider symmetric Pr k X k µ σ µ σ − ≤ ≤ + calculated using Chebyshev's inequality is underapproximated for all the probability distributions considered in the study.
The Dhaka University Journal of Science, May 29, 2023
This paper aims to find efficient methods for estimating the parameters (shape α = , scale β =) o... more This paper aims to find efficient methods for estimating the parameters (shape α = , scale β =) of Weibull distribution in different α β >> and the WLSM is suitable in such a situation (lowest RMSE) irrespective of all sample sizes. Finally, the utility of simulation results have been illustrated by analyzing two real-life data sets.
Access to and use of the material held within the University of Nottingham's Institutional Reposi... more Access to and use of the material held within the University of Nottingham's Institutional Repository (the Repository) is based on your acceptance of the following terms and conditions:
This paper aims to find efficient methods for estimating the parameters (shape =α , scale = β ) o... more This paper aims to find efficient methods for estimating the parameters (shape =α , scale = β ) of Weibull distribution in different situations. The maximum likelihood estimation method (MLE), the median rank regression method (MRR), the least square method (LSM) and the weighted least square method (WLSM) are considered for the estimation of the parameters. The root mean square error (RMSE) criterion is used to measure the relative efficiency of the estimators experimentally (Monte Carlo simulation). From the simulation study, it is observed that the MLE produces the lowest RMSE, irrespective of all sample sizes, for decreasing hazard function(α << β ) (α is considerably smaller than β ) and roughly linear hazard function with a positive slope (α >1) . When (α >> β ) the WLSM produces the lowest RMSE for small sample sizes (n ≤ 40) but for large sample sizes it is the MLE, irrespective of all types of hazard functions. When (α ,β →1), the WLSM produces the lowest RMS...
This paper aims to evaluate the accuracy of probability calculation using Chebyshev’s inequality ... more This paper aims to evaluate the accuracy of probability calculation using Chebyshev’s inequality based on simulation study. We consider symmetric (Normal (3,1.52 ), Laplace (3, 2 ) Beta (7.7 ) t5) positively skewed, negatively skewed (5 χ2, Beta (3, 8 ) Gamma (5,1)), (Beta (7, 2)), Exponential (5) and Uniform (0, 1 ) distributions, fx(x) in our simulation study to measure the performance of Chebyshev’s inequality. We then calculate Pr (μ − kσ ≤ X ≤ μ + kσ ) for ~ ( ) X X f x , μ = E ( X ) and σ 2 =Var ( X ), and compare this with the approximated probability obtained from Chebyshev’s inequality to measure the accuracy of Chebyshev’s inequality. From our simulation study, it is observed that loss due to using Chebyshev’s inequality for probability calculation is the least and the maximum when fx(x) is the Exponential and the Beta (symmetric) distributions, respectively for k ≥ 2.5. Moreover, the value of Pr (μ − kσ ≤ X ≤ μ + kσ ) calculated using Chebyshev’s inequality is underapp...
This study investigates the relationship between remittance and import for the economy of Banglad... more This study investigates the relationship between remittance and import for the economy of Bangladesh. The study used different econometric techniques of measuring the long and short term relationship between variables. The Johansen Cointegration test is used to determine the existence of a long term relationships between study variables. The normalized Cointegrating coefficients are found statistically significant and show a stable and positive relationship between study variables. Our Granger causality analysis suggests the existence of a unidirectional causality running from import to remittance. This confirms that remittances have no significant impact on the demand for imported goods rather import exerts a positive shock on the remittance of Bangladesh.
International journal of business and social research, 2013
This study investigates the relationship between remittance and import for the economy of Banglad... more This study investigates the relationship between remittance and import for the economy of Bangladesh. The study used different econometric techniques of measuring the long and short term relationship between variables. The Johansen Cointegration test is used to determine the existence of a long term relationships between study variables. The normalized Cointegrating coefficients are found statistically significant and show a stable and positive relationship between study variables. Our Granger causality analysis suggests the existence of a unidirectional causality running from import to remittance. This confirms that remittances have no significant impact on the demand for imported goods rather import exerts a positive shock on the remittance of Bangladesh.
This study considers the classification problem for binary output attribute when input attributes... more This study considers the classification problem for binary output attribute when input attributes are drawn from multivariate normal distribution, in both clean and contaminated case. Classical metrics are affected by the outliers, while robust metrics are computationally inefficient. In order to achieve robustness and computational efficiency at the same time, we propose a new robust distance metric for K-Nearest Neighbor (KNN) method. We call our proposed metric Alternative Robust Mahalanobis Distance (ARMD) metric. Thus KNN using ARMD is alternative KNN method. The classical metrics use non robust estimate (mean) as the building block. To construct the proposed ARMD metric, we replace non robust estimate (mean) by its robust counterpart median. Thus, we developed ARMD metric for alternative KNN classification technique. Our simulation studies show that the proposed alternative KNN method gives better results in case of contaminated data compared to the classical KNN. The performa...
Normal distribution is one of the most commonly used non-uniform distributions in applications in... more Normal distribution is one of the most commonly used non-uniform distributions in applications involving simulations. Advanced computing facilities make the simulation tasks simple but the challenge is to meet the increasingly stringent requirements on the statistical quality of the generated samples. In this paper, we examine performances of different existing methods available to generate random samples from normal distribution based on statistical quality of the generated samples (randomness and normality) and computational complexities. From the simulation study, it is observed that CDF approximation based method and acceptance-rejection method devised by Rao et al. 12 and Sigman 14 are the fastest and the slowest respectively among all algorithms considered in this paper while generated samples produced by all methods satisfy randomness and normality properties. An application involving simulation from normal distribution is shown by considering a Monte Carlo integration problem.
The Gaussian distribution is often considered to be the underlying distribution of many observed ... more The Gaussian distribution is often considered to be the underlying distribution of many observed samples for modelling purposes, and hence simulation from the Gaussian density is required to verify the fitted model. Several methods, most importantly, Box-Muller method, inverse transformation method and acceptance-rejection method devised by Box and Muller1, Rao et al.7 and Sigman8 respectively, are available in the literature to generate samples from the Gaussian distribution. Among these methods, Box-Muller method is the most popular and widely used because of its easy implementation and high efficiency,which produces exact samples2. However, generalizing this method for generating non-standard multivariate Gaussian variates is not discovered yet. On the other hand, inverse transformation method uses numerical approximation to the CDF of Gaussian density which may not be desirable in some situations while performance of acceptance-rejection method depends on choosing efficient prop...
This thesis considers the development of efficient MCMC sampling methods for Bayesian models used... more This thesis considers the development of efficient MCMC sampling methods for Bayesian models used for the pairwise alignment of two unlabelled configurations. We introduce ideas from differential geometry along with other recent developments in unlabelled shape analysis as a means of creating novel and more efficient MCMC sampling methods for such models. For example, we have improved the performance of the sampler for the model of Green and Mardia (2006) by sampling rotation, A ∈ SO(3), and matching matrix using geodesic Monte Carlo (MCMC defined on manifold) and Forbes and Lauritzen (2014) matching sampler, developed for finger print matching problem, respectively. We also propose a new Bayesian model, together with implementation methods, motivated by the desire for further improvement. The model and its implementation methods proposed exploit the continuous nature of the parameter space of our Bayesian model and thus move around easily in this continuous space, providing highly ...
Simulating random variates from arbitrary non-normalized probability densities, very often they d... more Simulating random variates from arbitrary non-normalized probability densities, very often they do not have familiar forms, is an increasingly important requirement in many different fields, especially in Bayesian statistics. Accept-reject algorithm is one of the commonly used methods to simulate random variates from such densities but restriction on choosing proposal density under this framework (heavier tails than the target density) limits its applicability to a larger extent. On the other hand, Markov Chain Monte Carlo (MCMC) method can choose proposal density arbitrary which makes this method applicable to a larger class of target densities5. In addition to MCMC method, a more general widely used method known as ratio-of-uniforms (RoU) which requires only two uniform variates to simulate one variates from such densities. However, no empirical comparison among these methods for simulating random variates from such densities was seen in the literature. In this paper, we limit our...
The multivariate normal density (MVN) is considered to be the underlying distribution of many obs... more The multivariate normal density (MVN) is considered to be the underlying distribution of many observed samples in statistics for modelling purpose. Therefore, simulating sample from the MVN is required to verify the efficiency of the fitted model. Decomposition based approach is currently being used to simulate sample from MVN whose building block is Cholesky or eigen decomposition. Unfortunately, there is no concrete study in the literature so far regarding the efficient decomposition technique between these two. In this paper, an attempt is made to determine the efficient decomposition technique between these two in the context of MVN generation through an extensive simulation study. From our simulation study, it is observed that in general the Cholesky decomposition is numerically faster than the eigen decomposition.
The Weibull distribution is extensively used in engineering sectors and biostatistical fields. In... more The Weibull distribution is extensively used in engineering sectors and biostatistical fields. In this article, we have proposed the Weibull mixture of some sampling distributions in which the weight functions are assumed to be chi-square, t and F sampling distributions. Different properties of these proposed distributions have been derived in this paper. The estimators of the parameters of these models using method of moments have also been provided.
In a classification problem with binary outcome attribute, if the input attributes are both conti... more In a classification problem with binary outcome attribute, if the input attributes are both continuous and categorical, the
The main objective of this paper was to study the causal relationships of the economic variables ... more The main objective of this paper was to study the causal relationships of the economic variables GDP, labour, capital and population in Sweden during the time period 1870 to 2000. In this paper the theory of unit root tests, vector auto regressive (VAR) model and Granger-Causality test were ased to find the causality of the variables. Augmented Dickey Fuller test was also used as unit root test. By applying all these tests and methods, the causal relationship among the economic variables has been established. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21970 Dhaka Univ. J. Sci. 62(2): 81-86, 2014 (July)
The Dhaka University Journal of Science, May 29, 2023
This paper aims to evaluate the accuracy of probability calculation using Chebyshev's inequality ... more This paper aims to evaluate the accuracy of probability calculation using Chebyshev's inequality based on simulation study. We consider symmetric Pr k X k µ σ µ σ − ≤ ≤ + calculated using Chebyshev's inequality is underapproximated for all the probability distributions considered in the study.
The Dhaka University Journal of Science, May 29, 2023
This paper aims to find efficient methods for estimating the parameters (shape α = , scale β =) o... more This paper aims to find efficient methods for estimating the parameters (shape α = , scale β =) of Weibull distribution in different α β >> and the WLSM is suitable in such a situation (lowest RMSE) irrespective of all sample sizes. Finally, the utility of simulation results have been illustrated by analyzing two real-life data sets.
Access to and use of the material held within the University of Nottingham's Institutional Reposi... more Access to and use of the material held within the University of Nottingham's Institutional Repository (the Repository) is based on your acceptance of the following terms and conditions:
This paper aims to find efficient methods for estimating the parameters (shape =α , scale = β ) o... more This paper aims to find efficient methods for estimating the parameters (shape =α , scale = β ) of Weibull distribution in different situations. The maximum likelihood estimation method (MLE), the median rank regression method (MRR), the least square method (LSM) and the weighted least square method (WLSM) are considered for the estimation of the parameters. The root mean square error (RMSE) criterion is used to measure the relative efficiency of the estimators experimentally (Monte Carlo simulation). From the simulation study, it is observed that the MLE produces the lowest RMSE, irrespective of all sample sizes, for decreasing hazard function(α << β ) (α is considerably smaller than β ) and roughly linear hazard function with a positive slope (α >1) . When (α >> β ) the WLSM produces the lowest RMSE for small sample sizes (n ≤ 40) but for large sample sizes it is the MLE, irrespective of all types of hazard functions. When (α ,β →1), the WLSM produces the lowest RMS...
This paper aims to evaluate the accuracy of probability calculation using Chebyshev’s inequality ... more This paper aims to evaluate the accuracy of probability calculation using Chebyshev’s inequality based on simulation study. We consider symmetric (Normal (3,1.52 ), Laplace (3, 2 ) Beta (7.7 ) t5) positively skewed, negatively skewed (5 χ2, Beta (3, 8 ) Gamma (5,1)), (Beta (7, 2)), Exponential (5) and Uniform (0, 1 ) distributions, fx(x) in our simulation study to measure the performance of Chebyshev’s inequality. We then calculate Pr (μ − kσ ≤ X ≤ μ + kσ ) for ~ ( ) X X f x , μ = E ( X ) and σ 2 =Var ( X ), and compare this with the approximated probability obtained from Chebyshev’s inequality to measure the accuracy of Chebyshev’s inequality. From our simulation study, it is observed that loss due to using Chebyshev’s inequality for probability calculation is the least and the maximum when fx(x) is the Exponential and the Beta (symmetric) distributions, respectively for k ≥ 2.5. Moreover, the value of Pr (μ − kσ ≤ X ≤ μ + kσ ) calculated using Chebyshev’s inequality is underapp...
This study investigates the relationship between remittance and import for the economy of Banglad... more This study investigates the relationship between remittance and import for the economy of Bangladesh. The study used different econometric techniques of measuring the long and short term relationship between variables. The Johansen Cointegration test is used to determine the existence of a long term relationships between study variables. The normalized Cointegrating coefficients are found statistically significant and show a stable and positive relationship between study variables. Our Granger causality analysis suggests the existence of a unidirectional causality running from import to remittance. This confirms that remittances have no significant impact on the demand for imported goods rather import exerts a positive shock on the remittance of Bangladesh.
International journal of business and social research, 2013
This study investigates the relationship between remittance and import for the economy of Banglad... more This study investigates the relationship between remittance and import for the economy of Bangladesh. The study used different econometric techniques of measuring the long and short term relationship between variables. The Johansen Cointegration test is used to determine the existence of a long term relationships between study variables. The normalized Cointegrating coefficients are found statistically significant and show a stable and positive relationship between study variables. Our Granger causality analysis suggests the existence of a unidirectional causality running from import to remittance. This confirms that remittances have no significant impact on the demand for imported goods rather import exerts a positive shock on the remittance of Bangladesh.
This study considers the classification problem for binary output attribute when input attributes... more This study considers the classification problem for binary output attribute when input attributes are drawn from multivariate normal distribution, in both clean and contaminated case. Classical metrics are affected by the outliers, while robust metrics are computationally inefficient. In order to achieve robustness and computational efficiency at the same time, we propose a new robust distance metric for K-Nearest Neighbor (KNN) method. We call our proposed metric Alternative Robust Mahalanobis Distance (ARMD) metric. Thus KNN using ARMD is alternative KNN method. The classical metrics use non robust estimate (mean) as the building block. To construct the proposed ARMD metric, we replace non robust estimate (mean) by its robust counterpart median. Thus, we developed ARMD metric for alternative KNN classification technique. Our simulation studies show that the proposed alternative KNN method gives better results in case of contaminated data compared to the classical KNN. The performa...
Normal distribution is one of the most commonly used non-uniform distributions in applications in... more Normal distribution is one of the most commonly used non-uniform distributions in applications involving simulations. Advanced computing facilities make the simulation tasks simple but the challenge is to meet the increasingly stringent requirements on the statistical quality of the generated samples. In this paper, we examine performances of different existing methods available to generate random samples from normal distribution based on statistical quality of the generated samples (randomness and normality) and computational complexities. From the simulation study, it is observed that CDF approximation based method and acceptance-rejection method devised by Rao et al. 12 and Sigman 14 are the fastest and the slowest respectively among all algorithms considered in this paper while generated samples produced by all methods satisfy randomness and normality properties. An application involving simulation from normal distribution is shown by considering a Monte Carlo integration problem.
The Gaussian distribution is often considered to be the underlying distribution of many observed ... more The Gaussian distribution is often considered to be the underlying distribution of many observed samples for modelling purposes, and hence simulation from the Gaussian density is required to verify the fitted model. Several methods, most importantly, Box-Muller method, inverse transformation method and acceptance-rejection method devised by Box and Muller1, Rao et al.7 and Sigman8 respectively, are available in the literature to generate samples from the Gaussian distribution. Among these methods, Box-Muller method is the most popular and widely used because of its easy implementation and high efficiency,which produces exact samples2. However, generalizing this method for generating non-standard multivariate Gaussian variates is not discovered yet. On the other hand, inverse transformation method uses numerical approximation to the CDF of Gaussian density which may not be desirable in some situations while performance of acceptance-rejection method depends on choosing efficient prop...
This thesis considers the development of efficient MCMC sampling methods for Bayesian models used... more This thesis considers the development of efficient MCMC sampling methods for Bayesian models used for the pairwise alignment of two unlabelled configurations. We introduce ideas from differential geometry along with other recent developments in unlabelled shape analysis as a means of creating novel and more efficient MCMC sampling methods for such models. For example, we have improved the performance of the sampler for the model of Green and Mardia (2006) by sampling rotation, A ∈ SO(3), and matching matrix using geodesic Monte Carlo (MCMC defined on manifold) and Forbes and Lauritzen (2014) matching sampler, developed for finger print matching problem, respectively. We also propose a new Bayesian model, together with implementation methods, motivated by the desire for further improvement. The model and its implementation methods proposed exploit the continuous nature of the parameter space of our Bayesian model and thus move around easily in this continuous space, providing highly ...
Simulating random variates from arbitrary non-normalized probability densities, very often they d... more Simulating random variates from arbitrary non-normalized probability densities, very often they do not have familiar forms, is an increasingly important requirement in many different fields, especially in Bayesian statistics. Accept-reject algorithm is one of the commonly used methods to simulate random variates from such densities but restriction on choosing proposal density under this framework (heavier tails than the target density) limits its applicability to a larger extent. On the other hand, Markov Chain Monte Carlo (MCMC) method can choose proposal density arbitrary which makes this method applicable to a larger class of target densities5. In addition to MCMC method, a more general widely used method known as ratio-of-uniforms (RoU) which requires only two uniform variates to simulate one variates from such densities. However, no empirical comparison among these methods for simulating random variates from such densities was seen in the literature. In this paper, we limit our...
The multivariate normal density (MVN) is considered to be the underlying distribution of many obs... more The multivariate normal density (MVN) is considered to be the underlying distribution of many observed samples in statistics for modelling purpose. Therefore, simulating sample from the MVN is required to verify the efficiency of the fitted model. Decomposition based approach is currently being used to simulate sample from MVN whose building block is Cholesky or eigen decomposition. Unfortunately, there is no concrete study in the literature so far regarding the efficient decomposition technique between these two. In this paper, an attempt is made to determine the efficient decomposition technique between these two in the context of MVN generation through an extensive simulation study. From our simulation study, it is observed that in general the Cholesky decomposition is numerically faster than the eigen decomposition.
The Weibull distribution is extensively used in engineering sectors and biostatistical fields. In... more The Weibull distribution is extensively used in engineering sectors and biostatistical fields. In this article, we have proposed the Weibull mixture of some sampling distributions in which the weight functions are assumed to be chi-square, t and F sampling distributions. Different properties of these proposed distributions have been derived in this paper. The estimators of the parameters of these models using method of moments have also been provided.
In a classification problem with binary outcome attribute, if the input attributes are both conti... more In a classification problem with binary outcome attribute, if the input attributes are both continuous and categorical, the
The main objective of this paper was to study the causal relationships of the economic variables ... more The main objective of this paper was to study the causal relationships of the economic variables GDP, labour, capital and population in Sweden during the time period 1870 to 2000. In this paper the theory of unit root tests, vector auto regressive (VAR) model and Granger-Causality test were ased to find the causality of the variables. Augmented Dickey Fuller test was also used as unit root test. By applying all these tests and methods, the causal relationship among the economic variables has been established. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21970 Dhaka Univ. J. Sci. 62(2): 81-86, 2014 (July)
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