In this paper, we derive the distribution of life length of a parallel system with random number ... more In this paper, we derive the distribution of life length of a parallel system with random number of components when the life distribution of each component follows a Weibull distribution and the number of components follows a Poisson distribution truncated at zero. For two independent such parallel systems, we are interested in the estimation of the reliability parameter R=P(X>Y), where X and Y are the life lengths of the two parallel systems. The point estimate and confidence interval of R, based on maximum likelihood method, are developed. The performance of each of the point estimate and confidence interval of R is studied through extensive simulation study. A numerical example, based on a real data, is presented to illustrate the implementation of the proposed procedure.
International Journal of Apllied Mathematics, 2022
In this paper, we have studied the generalized Sibuya distribution from a reliability point of vi... more In this paper, we have studied the generalized Sibuya distribution from a reliability point of view. It turns out that this distribution has the logconvex property and hence is infinitely divisible. This enables us to study the monotonic properties of various reliability functions including the failure rate, the mean residual life, the variance residual life and their reversed versions. The monotone properties of the classes of discrete distributions are parallel to those of continuous distributions. Procedures are developed to estimate the reliability functions. Application to real data is provided to illustrate the results.
The Marshall-Olkin extended exponential distribution is considered as a probability model for the... more The Marshall-Olkin extended exponential distribution is considered as a probability model for the lifetime of the product. Sampling plans in which items that are put to test, to collect the life of the items in order to decide upon accepting or rejecting a submitted lot, are called reliability test plans. A test plan to determine the termination time of the experiment for a given sample size, producer’s risk and termination number is constructed. The preferability of the present test plan over similar plans exists in the literature is established with respect to time of the experiment. Results are illustrated by an example.
International Journal of Apllied Mathematics, 2020
In the literature, two-parameter distributions which exhibit all three types of decreasing, incre... more In the literature, two-parameter distributions which exhibit all three types of decreasing, increasing and bathtub shape hazard rate functions are very few. In this paper, we propose a new two-parameter distribution, called Gompertz-weighted exponential distribution, having these three types of hazard rate functions. The proposed distribution is obtained by mixing the frailty parameter of the Gompertz distribution by weighted exponential distribution. The parameters are estimated by the maximum likelihood method and their performance is examined by extensive simulation studies. Three real data applications are provided to illustrate the superiority of the proposed distribution over many well known two-parameter distributions.
The Beta distribution is the standard model for quantifying the influence of covariates on the me... more The Beta distribution is the standard model for quantifying the influence of covariates on the mean of a response variable on the unit interval. However, this well-known distribution is no longer useful when we are interested in quantifying the influence of such covariates on the quantiles of the response variable. Unlike Beta, the Kumaraswamy distribution has a closed-form expression for its quantile and can be useful for the modeling of quantiles in the absence/presence of covariates. As an alternative to the Kumaraswamy distribution for the modeling of quantiles, in this paper the unit-Weibull distribution was considered. This distribution was obtained by the transformation of a random variable with Weibull distribution. The same transformation applied to a random variable with Exponentiated Exponential distribution generates the Kumaraswamy distribution. The suitability of our proposal was demonstrated to model quantiles, conditional on covariates, with two simulated examples and three real applications with datasets from health, accounting and social science. For such data sets, the obtained fits of the proposed regression model were compared with those provided by the Beta and Kumaraswamy regression models.
Brazilian Journal of Probability and Statistics, 2019
This paper focuses on Bayesian estimation of the parameters and reliability function of the power... more This paper focuses on Bayesian estimation of the parameters and reliability function of the power Lindley distribution by using various symmetric and asymmetric loss functions. Assuming suitable priors on the parameters, Bayes estimates are derived by using squared error, linear exponential (linex) and general entropy loss functions. Since, under these loss functions, Bayes estimates of the parameters do not have closed forms we use lindley's approximation technique to calculate the Bayes estimates. Moreover, we obtain the Bayes estimates of the parameters using a Markov Chain Monte Carlo (MCMC) method. Simulation studies are conducted in order to evaluate the performances of the proposed estimators under the considered loss functions. Finally, analysis of a real data set is presented for illustrative purposes.
Communications in Statistics - Theory and Methods, 2018
A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distrib... more A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distribution, is introduced and studied in detail. The proposed distribution shares many properties with other known distributions on the unit interval, such as Beta, Johnson S B , Unit-Gamma, and Kumaraswamy distributions. Estimation of the parameters of the proposed distribution are obtained by transforming the data to the inverse Gaussian distribution. Unlike most distributions on the unit interval, the maximum likelihood or method of moments estimators of the parameters of the proposed distribution are expressed in simple closed forms which do not need iterative methods to compute. Application of the proposed distribution to a real data set shows better fit than many known two-parameter distributions on the unit interval.
Journal of Statistical Computation and Simulation, 2017
ABSTRACT We propose a modification of the moment estimators for the two-parameter weighted Lindle... more ABSTRACT We propose a modification of the moment estimators for the two-parameter weighted Lindley distribution. The modification replaces the second sample moment (or equivalently the sample variance) by a certain sample average which is bounded on the unit interval for all values in the sample space. In this method, the estimates always exist uniquely over the entire parameter space and have consistency and asymptotic normality over the entire parameter space. The bias and mean squared error of the estimators are also examined by means of a Monte Carlo simulation study, and the empirical results show the small-sample superiority in addition to the desirable large sample properties. Monte Carlo simulation study showed that the proposed modified moment estimators have smaller biases and smaller mean-square errors than the existing moment estimators and are compared favourably with the maximum likelihood estimators in terms of bias and mean-square error. Three illustrative examples are finally presented.
The objective of this book is to provide a practical opportunity for health care trainees and sta... more The objective of this book is to provide a practical opportunity for health care trainees and staff not only to acquaint themselves with statistics in medicine and understand statistical analysis in medical literature, but also to be guided in the application of planning and analysis in their own medical studies. The book has 21 chapters with many exercises at the ends of chapters, with three appendices for chapter summaries, references and data sources, and tables of probability distributions. The book is divided into two parts. Part I (Course of Study) serves as a textbook for an introductory course in biostatistics for students in health care sciences. This part includes Chapters 1–9. Part II (Reference Handbook) serves as a reference manual to support practitioners in reading the medical literature or conducting research studies. Chapter 1 covers the basic concepts of data types, samples and populations, randomness and design of experiments. Chapter 2 is a friendly introduction to commonly used probability distributions and their characteristics. The concept of standard error of the mean is also discussed. The chapter ends with a brief introduction to joint distributions of two random variables. Chapters 3–5 cover standard materials for summarizing data, confidence intervals and hypothesis testing. Chapter 6 is mainly about the logical steps in a statistical test. It emphasizes the unique character of each data type and the statistical test used for that type in medical research. Chapter 7 is devoted to sample size determination in medical studies. Chapter 8 covers standard materials about statistical modelling of response to treatment as well as the relation between regression and correlation. Further, assessment of the regression model is presented. Chapter 9 deals with the stages of scientific knowledge in epidemiology and the type of epidemiological studies. It also describes briefly the statistical methods useful in epidemiology such as the construction of life tables and graphing survival information using Kaplan–Meier curves as well as the log-rank test to compare two survival curves. Useful recommendations to improve efficiency in reading medical articles as well as some guidelines for planning a study are presented in Chapter 10. Calculating probabilities from commonly used probability distributions in medicine is discussed in Chapter11.Classicalmaterialsonconfidenceintervals for the mean, variance, proportion and correlation coefficient are presented in Chapter 12. Common tests on categorical and ranked data are treated in Chapters 13 and 14. Common tests on continuous data means, variances and distribution shapes are given in Chapters 15–17, respectively. Various aspects of sample size determination required in a study are explored in Chapter 18. Modelling a response variable in terms of explanatory variable(s) and clinical decision making are discussed in Chapter 19. Regression and correlation methods are well presented in Chapter 20. Brief introductions to survival and time series analysis are given in Chapter 21. Although the book has been written to provide the reader with a thorough understanding of the basic concepts of statistics, it does not make use of any statistical package to ease the computational and graphical burdens on the reader. Overall, this book is well written and is quite comprehensive in the topics addressed. I recommend it to biomedical researchers and health care planners as a text and reference book for analysing their data.
Communications in Statistics - Theory and Methods, 2016
ABSTRACT This paper proposes a bivariate version of the univariate discrete generalized geometric... more ABSTRACT This paper proposes a bivariate version of the univariate discrete generalized geometric distribution considered by Gómez–Déniz (2010). The proposed bivariate distribution can have a positive or negative correlation coefficient which can be used for modeling bivariate-dependent count data. After discussing some of its properties, maximum likelihood estimation is discussed. Two illustrative examples are given for fitting and demonstrating the usefulness of the new bivariate distribution proposed here.
The A.W. Marshall and I. Olkin [Biometrika 84, No. 3, 641–652 (1997; Zbl 0888.62012)] extended ex... more The A.W. Marshall and I. Olkin [Biometrika 84, No. 3, 641–652 (1997; Zbl 0888.62012)] extended exponential distribution is considered as a probability model for the life time of products. Sampling plans in which items that are put to test, to collect the life of the items in order to decide upon accepting or rejecting a submitted lot, are called reliability test plans. A test plan to determine the termination time of the experiment for a given sample size, producer’s risk and termination number is constructed. The preferability of the present test plan over similar plans existing in the literature is established with respect to the time of the experiment. The results are illustrated by an example.
The aim of this paper is to study the properties of the asymptotic variances of the maximum likel... more The aim of this paper is to study the properties of the asymptotic variances of the maximum likelihood estimators of the parameters of the exponential mixture model with long‐term survivors for randomly censored data. In addition, we study the asymptotic relative efficiency of these estimators versus those which would be obtained with complete follow‐up. It is shown that fixed censoring at time T produces higher precision as well as higher asymptotic relative efficiency than those obtainable under uniform and uniform‐exponential censoring distributions over (0, T). The results are useful in planning the size and duration of survival experiments with long‐term survivors under random censoring schemes.
The aim of this paper is to compare through Monte Carlo simulations the finite sample properties ... more The aim of this paper is to compare through Monte Carlo simulations the finite sample properties of the estimates of the parameters of the weighted Lindley distribution obtained by four estimation methods: maximum likelihood, method of moments, ordinary leastsquares, and weighted least-squares. The bias and mean-squared error are used as the criterion for comparison. The study reveals that the ordinary and weighted least-squares estimation methods are highly competitive with the maximum likelihood method in small and large samples. Statistical analysis of two real data sets are presented to demonstrate the conclusion of the simulation results.
Although the literature on univariate count regression models allowing for overdispersion is huge... more Although the literature on univariate count regression models allowing for overdispersion is huge, there are few multivariate count regression models allowing for correlation and overdiseprsion. The latter models can find applications in several disciplines such as epidemiology, marketing, sports statistics, criminology, just to name a few. In this paper, we propose a general EM algorithm to facilitate maximum likelihood estimation for a class of multivariate mixed Poisson regression models. We give special emphasis to the multivariate negative binomial, Poisson inverse Gaussian and Poisson lognormal regression models. An application to a real dataset is also given to illustrate the use of the proposed EM algorithm to the considered multivariate regression models.
Computational Statistics & Data Analysis, 2013
A new two-parameter power Lindley distribution is introduced and its properties are discussed. Th... more A new two-parameter power Lindley distribution is introduced and its properties are discussed. These include the shapes of the density and hazard rate functions, the moments, skewness and kurtosis measures, the quantile function, and the limiting distributions of order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators of the parameters as well as the coverage probability and the width of the confidence interval for each parameter. An application of the model to a real data set is presented finally and compared with the fit attained by some other well-known two-parameter distributions.
Brazilian Journal of Probability and Statistics, 2018
Gupta and Kundu (Statistics 43 (2009) 621-643) introduced a new class of weighted exponential dis... more Gupta and Kundu (Statistics 43 (2009) 621-643) introduced a new class of weighted exponential distribution and established its several properties. The probability density function of the proposed weighted exponential distribution is unimodal and it has an increasing hazard function. Following the same line Shahbaz, Shahbaz and Butt (Pak. J. Stat. Oper. Res. VI (2010) 53-59) introduced weighted Weibull distribution, and we derive several new properties of this weighted Weibull distribution. The main aim of this paper is to introduce bivariate and multivariate distributions with weighted Weibull marginals and establish their several properties. It is shown that the hazard function of the weighted Weibull distribution can have increasing, decreasing and inverted bathtub shapes. The proposed multivariate model has been obtained as a hidden truncation model similarly as the univariate weighted Weibull model. It is observed that to compute the maximum likelihood estimators of the unknown parameters for the proposed p-variate distribution, one needs to solve (p + 2) non-linear equations. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters. We obtain the observed Fisher information matrix, which can be used for constructing asymptotic confidence intervals. One data analysis has been performed for illustrative purposes, and it is observed that the proposed EM algorithm is very easy to implement, and the performance is quite satisfactory.
In this paper, we derive the distribution of life length of a parallel system with random number ... more In this paper, we derive the distribution of life length of a parallel system with random number of components when the life distribution of each component follows a Weibull distribution and the number of components follows a Poisson distribution truncated at zero. For two independent such parallel systems, we are interested in the estimation of the reliability parameter R=P(X>Y), where X and Y are the life lengths of the two parallel systems. The point estimate and confidence interval of R, based on maximum likelihood method, are developed. The performance of each of the point estimate and confidence interval of R is studied through extensive simulation study. A numerical example, based on a real data, is presented to illustrate the implementation of the proposed procedure.
International Journal of Apllied Mathematics, 2022
In this paper, we have studied the generalized Sibuya distribution from a reliability point of vi... more In this paper, we have studied the generalized Sibuya distribution from a reliability point of view. It turns out that this distribution has the logconvex property and hence is infinitely divisible. This enables us to study the monotonic properties of various reliability functions including the failure rate, the mean residual life, the variance residual life and their reversed versions. The monotone properties of the classes of discrete distributions are parallel to those of continuous distributions. Procedures are developed to estimate the reliability functions. Application to real data is provided to illustrate the results.
The Marshall-Olkin extended exponential distribution is considered as a probability model for the... more The Marshall-Olkin extended exponential distribution is considered as a probability model for the lifetime of the product. Sampling plans in which items that are put to test, to collect the life of the items in order to decide upon accepting or rejecting a submitted lot, are called reliability test plans. A test plan to determine the termination time of the experiment for a given sample size, producer’s risk and termination number is constructed. The preferability of the present test plan over similar plans exists in the literature is established with respect to time of the experiment. Results are illustrated by an example.
International Journal of Apllied Mathematics, 2020
In the literature, two-parameter distributions which exhibit all three types of decreasing, incre... more In the literature, two-parameter distributions which exhibit all three types of decreasing, increasing and bathtub shape hazard rate functions are very few. In this paper, we propose a new two-parameter distribution, called Gompertz-weighted exponential distribution, having these three types of hazard rate functions. The proposed distribution is obtained by mixing the frailty parameter of the Gompertz distribution by weighted exponential distribution. The parameters are estimated by the maximum likelihood method and their performance is examined by extensive simulation studies. Three real data applications are provided to illustrate the superiority of the proposed distribution over many well known two-parameter distributions.
The Beta distribution is the standard model for quantifying the influence of covariates on the me... more The Beta distribution is the standard model for quantifying the influence of covariates on the mean of a response variable on the unit interval. However, this well-known distribution is no longer useful when we are interested in quantifying the influence of such covariates on the quantiles of the response variable. Unlike Beta, the Kumaraswamy distribution has a closed-form expression for its quantile and can be useful for the modeling of quantiles in the absence/presence of covariates. As an alternative to the Kumaraswamy distribution for the modeling of quantiles, in this paper the unit-Weibull distribution was considered. This distribution was obtained by the transformation of a random variable with Weibull distribution. The same transformation applied to a random variable with Exponentiated Exponential distribution generates the Kumaraswamy distribution. The suitability of our proposal was demonstrated to model quantiles, conditional on covariates, with two simulated examples and three real applications with datasets from health, accounting and social science. For such data sets, the obtained fits of the proposed regression model were compared with those provided by the Beta and Kumaraswamy regression models.
Brazilian Journal of Probability and Statistics, 2019
This paper focuses on Bayesian estimation of the parameters and reliability function of the power... more This paper focuses on Bayesian estimation of the parameters and reliability function of the power Lindley distribution by using various symmetric and asymmetric loss functions. Assuming suitable priors on the parameters, Bayes estimates are derived by using squared error, linear exponential (linex) and general entropy loss functions. Since, under these loss functions, Bayes estimates of the parameters do not have closed forms we use lindley's approximation technique to calculate the Bayes estimates. Moreover, we obtain the Bayes estimates of the parameters using a Markov Chain Monte Carlo (MCMC) method. Simulation studies are conducted in order to evaluate the performances of the proposed estimators under the considered loss functions. Finally, analysis of a real data set is presented for illustrative purposes.
Communications in Statistics - Theory and Methods, 2018
A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distrib... more A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distribution, is introduced and studied in detail. The proposed distribution shares many properties with other known distributions on the unit interval, such as Beta, Johnson S B , Unit-Gamma, and Kumaraswamy distributions. Estimation of the parameters of the proposed distribution are obtained by transforming the data to the inverse Gaussian distribution. Unlike most distributions on the unit interval, the maximum likelihood or method of moments estimators of the parameters of the proposed distribution are expressed in simple closed forms which do not need iterative methods to compute. Application of the proposed distribution to a real data set shows better fit than many known two-parameter distributions on the unit interval.
Journal of Statistical Computation and Simulation, 2017
ABSTRACT We propose a modification of the moment estimators for the two-parameter weighted Lindle... more ABSTRACT We propose a modification of the moment estimators for the two-parameter weighted Lindley distribution. The modification replaces the second sample moment (or equivalently the sample variance) by a certain sample average which is bounded on the unit interval for all values in the sample space. In this method, the estimates always exist uniquely over the entire parameter space and have consistency and asymptotic normality over the entire parameter space. The bias and mean squared error of the estimators are also examined by means of a Monte Carlo simulation study, and the empirical results show the small-sample superiority in addition to the desirable large sample properties. Monte Carlo simulation study showed that the proposed modified moment estimators have smaller biases and smaller mean-square errors than the existing moment estimators and are compared favourably with the maximum likelihood estimators in terms of bias and mean-square error. Three illustrative examples are finally presented.
The objective of this book is to provide a practical opportunity for health care trainees and sta... more The objective of this book is to provide a practical opportunity for health care trainees and staff not only to acquaint themselves with statistics in medicine and understand statistical analysis in medical literature, but also to be guided in the application of planning and analysis in their own medical studies. The book has 21 chapters with many exercises at the ends of chapters, with three appendices for chapter summaries, references and data sources, and tables of probability distributions. The book is divided into two parts. Part I (Course of Study) serves as a textbook for an introductory course in biostatistics for students in health care sciences. This part includes Chapters 1–9. Part II (Reference Handbook) serves as a reference manual to support practitioners in reading the medical literature or conducting research studies. Chapter 1 covers the basic concepts of data types, samples and populations, randomness and design of experiments. Chapter 2 is a friendly introduction to commonly used probability distributions and their characteristics. The concept of standard error of the mean is also discussed. The chapter ends with a brief introduction to joint distributions of two random variables. Chapters 3–5 cover standard materials for summarizing data, confidence intervals and hypothesis testing. Chapter 6 is mainly about the logical steps in a statistical test. It emphasizes the unique character of each data type and the statistical test used for that type in medical research. Chapter 7 is devoted to sample size determination in medical studies. Chapter 8 covers standard materials about statistical modelling of response to treatment as well as the relation between regression and correlation. Further, assessment of the regression model is presented. Chapter 9 deals with the stages of scientific knowledge in epidemiology and the type of epidemiological studies. It also describes briefly the statistical methods useful in epidemiology such as the construction of life tables and graphing survival information using Kaplan–Meier curves as well as the log-rank test to compare two survival curves. Useful recommendations to improve efficiency in reading medical articles as well as some guidelines for planning a study are presented in Chapter 10. Calculating probabilities from commonly used probability distributions in medicine is discussed in Chapter11.Classicalmaterialsonconfidenceintervals for the mean, variance, proportion and correlation coefficient are presented in Chapter 12. Common tests on categorical and ranked data are treated in Chapters 13 and 14. Common tests on continuous data means, variances and distribution shapes are given in Chapters 15–17, respectively. Various aspects of sample size determination required in a study are explored in Chapter 18. Modelling a response variable in terms of explanatory variable(s) and clinical decision making are discussed in Chapter 19. Regression and correlation methods are well presented in Chapter 20. Brief introductions to survival and time series analysis are given in Chapter 21. Although the book has been written to provide the reader with a thorough understanding of the basic concepts of statistics, it does not make use of any statistical package to ease the computational and graphical burdens on the reader. Overall, this book is well written and is quite comprehensive in the topics addressed. I recommend it to biomedical researchers and health care planners as a text and reference book for analysing their data.
Communications in Statistics - Theory and Methods, 2016
ABSTRACT This paper proposes a bivariate version of the univariate discrete generalized geometric... more ABSTRACT This paper proposes a bivariate version of the univariate discrete generalized geometric distribution considered by Gómez–Déniz (2010). The proposed bivariate distribution can have a positive or negative correlation coefficient which can be used for modeling bivariate-dependent count data. After discussing some of its properties, maximum likelihood estimation is discussed. Two illustrative examples are given for fitting and demonstrating the usefulness of the new bivariate distribution proposed here.
The A.W. Marshall and I. Olkin [Biometrika 84, No. 3, 641–652 (1997; Zbl 0888.62012)] extended ex... more The A.W. Marshall and I. Olkin [Biometrika 84, No. 3, 641–652 (1997; Zbl 0888.62012)] extended exponential distribution is considered as a probability model for the life time of products. Sampling plans in which items that are put to test, to collect the life of the items in order to decide upon accepting or rejecting a submitted lot, are called reliability test plans. A test plan to determine the termination time of the experiment for a given sample size, producer’s risk and termination number is constructed. The preferability of the present test plan over similar plans existing in the literature is established with respect to the time of the experiment. The results are illustrated by an example.
The aim of this paper is to study the properties of the asymptotic variances of the maximum likel... more The aim of this paper is to study the properties of the asymptotic variances of the maximum likelihood estimators of the parameters of the exponential mixture model with long‐term survivors for randomly censored data. In addition, we study the asymptotic relative efficiency of these estimators versus those which would be obtained with complete follow‐up. It is shown that fixed censoring at time T produces higher precision as well as higher asymptotic relative efficiency than those obtainable under uniform and uniform‐exponential censoring distributions over (0, T). The results are useful in planning the size and duration of survival experiments with long‐term survivors under random censoring schemes.
The aim of this paper is to compare through Monte Carlo simulations the finite sample properties ... more The aim of this paper is to compare through Monte Carlo simulations the finite sample properties of the estimates of the parameters of the weighted Lindley distribution obtained by four estimation methods: maximum likelihood, method of moments, ordinary leastsquares, and weighted least-squares. The bias and mean-squared error are used as the criterion for comparison. The study reveals that the ordinary and weighted least-squares estimation methods are highly competitive with the maximum likelihood method in small and large samples. Statistical analysis of two real data sets are presented to demonstrate the conclusion of the simulation results.
Although the literature on univariate count regression models allowing for overdispersion is huge... more Although the literature on univariate count regression models allowing for overdispersion is huge, there are few multivariate count regression models allowing for correlation and overdiseprsion. The latter models can find applications in several disciplines such as epidemiology, marketing, sports statistics, criminology, just to name a few. In this paper, we propose a general EM algorithm to facilitate maximum likelihood estimation for a class of multivariate mixed Poisson regression models. We give special emphasis to the multivariate negative binomial, Poisson inverse Gaussian and Poisson lognormal regression models. An application to a real dataset is also given to illustrate the use of the proposed EM algorithm to the considered multivariate regression models.
Computational Statistics & Data Analysis, 2013
A new two-parameter power Lindley distribution is introduced and its properties are discussed. Th... more A new two-parameter power Lindley distribution is introduced and its properties are discussed. These include the shapes of the density and hazard rate functions, the moments, skewness and kurtosis measures, the quantile function, and the limiting distributions of order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators of the parameters as well as the coverage probability and the width of the confidence interval for each parameter. An application of the model to a real data set is presented finally and compared with the fit attained by some other well-known two-parameter distributions.
Brazilian Journal of Probability and Statistics, 2018
Gupta and Kundu (Statistics 43 (2009) 621-643) introduced a new class of weighted exponential dis... more Gupta and Kundu (Statistics 43 (2009) 621-643) introduced a new class of weighted exponential distribution and established its several properties. The probability density function of the proposed weighted exponential distribution is unimodal and it has an increasing hazard function. Following the same line Shahbaz, Shahbaz and Butt (Pak. J. Stat. Oper. Res. VI (2010) 53-59) introduced weighted Weibull distribution, and we derive several new properties of this weighted Weibull distribution. The main aim of this paper is to introduce bivariate and multivariate distributions with weighted Weibull marginals and establish their several properties. It is shown that the hazard function of the weighted Weibull distribution can have increasing, decreasing and inverted bathtub shapes. The proposed multivariate model has been obtained as a hidden truncation model similarly as the univariate weighted Weibull model. It is observed that to compute the maximum likelihood estimators of the unknown parameters for the proposed p-variate distribution, one needs to solve (p + 2) non-linear equations. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters. We obtain the observed Fisher information matrix, which can be used for constructing asymptotic confidence intervals. One data analysis has been performed for illustrative purposes, and it is observed that the proposed EM algorithm is very easy to implement, and the performance is quite satisfactory.
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