Weibull Gamma Research Articles

Fitting Statistical Distributions to Rainfall Data with Different Estimation Techniques: An Empirical Study from Pabna and Dinajpur Districts

Bangladesh is a country of diverse climatic conditions because of its rainfall and other geographical conditions which have a complex impact on economic and social aspects. The statistical distributions are used in many real life data for modeling and predicting. Knowing the real distribution of rainfall rather than depending on basic summary statistics would improve many uses of rainfall data. The aim of this paper is twofold: first, the performance of different statistical distributions such as Normal, Log-Normal, Gamma Weibull, and Gumbel are compared for modeling the monthly rainfall data from Pabna and Dinajpur districts from January 1971 to December 2015; second, the performance of the Maximum Likelihood Estimation (MLE), Quantile Matching Estimation (QME), and the Maximum Spacing Estimation are also compared for fitting these statistical distributions. The empirical study showed that Gamma distribution performs better for fitting the monthly rainfall data for both Pabna and Dinajpur districts of the three methods like Maximum Likelihood Estimation (MLE) method, the Quantile Matching Estimation (QME), and the Maximum Spacing Estimation (MSE) method. The Normal distribution performs worse of these study areas. By the comparison of these three methods we found that Maximum Likelihood Estimation (MLE) gives better results. This study provides the actual distribution of rainfall data of these study areas. International Journal of Statistical Sciences, Vol. 23(2), November, 2023, pp 87-106

On the Comparison between the Reliability of Units Produced by Different Production Lines

AbstractThe paper discusses how to evaluate the reliability of units produced by different production lines. The procedure is based on selecting independent random samples of units produced by different production lines and then evaluating reliability functions for each group of units. The comparison between these reliability functions at a given time allows manufacturing experts to evaluate the effectiveness of production lines. A statistical methodology has been taken based on the assumption that the lifetime of units produced by each product line has a Weibull Gamma distribution. Then, real‐world data are used to illustrate the study's contribution to reliability theory applications.

Survival analysis of women breast cancer patients in Northwest Amhara, Ethiopia.

Breast cancer, the most common cause of cancer death and the most frequently diagnosed cancer among women worldwide, ranks as the second cause of death next to lung cancer. Thus, the main objective was to assess the factors that affect the survival time of breast cancer patients using the shared frailty model. A retrospective study design was used to collect relevant data on the survival time of breast cancer patients from the medical charts of 322 breast cancer patients under follow-up at the Felege Hiwot Comprehensive Specialized Hospital (FHCSH). The data were explored using the Cox proportional hazard model, the accelerated failure time model, and shared frailty models. The model comparison was done using AIC and BIC. As a result, the Weibull gamma shared frailty model had a minimum AIC and BIC value. From a total of 322 patients, about 95 (29.5%) died and 227 (70.5%) were censored. The overall mean and median estimated survival times of breast cancer patients under study were 43.7 and 45 months, respectively. The unobserved heterogeneity in the population of clusters (residence) as estimated by the Weibull-gamma shared frailty model was 0.002 (p-value = 0.000), indicating the presence of residential variation in the survival time of breast cancer patients. The estimated hazard rate of patients who had not had recurrent breast cancer was 0.724 (95% CI: 0.571, 0.917) times the estimated hazard rate of patients who had had recurrent breast cancer. The prevalence of breast cancer was considerably high. Under this investigation, older patients, patients in stages III and IV, anemic and diabetes patients, patients who took only chemotherapy treatment, metastasized patients, patients with an AB blood type, patients with a positive breast cancer family history, and patients whose cancer was recurrent had high death rates. Patient characteristics such as age, stage, complications, treatment, metastasis, blood type, family history, and recurrence were significant factors associated with the survival time of women with breastcancer.

Time to first birth and its predictors among reproductive age women in high fertility countries in Sub-Saharan Africa: Inverse Weibull gamma shared frailty model

BackgroundEarly initiation of childbearing leads to an increase in total fertility rate and population growth. It has been linked with both maternal and child morbidity and mortality. However, there is limited information on the timing of the first birth and its predictors in the area so far. Therefore, determining the time to first birth and its predictors will help to design strategies to improve fertility rate, maternal and child survival.MethodsThe survey used recent (2010 – 2018) Demographic and Health data; a stratified, two-stage cluster sampling technique was used to select the sample. Inverse Weibull gamma shared frailty model was used to model the data at 95% confidence interval. Adjusted hazard ratio (AHR) and median hazard ratio (MHR) were reported as effect size. Statistical significance was declared at p value < 0.05.ResultsThe overall median age at first birth was found to be 19 years (IQR: 16, 21 years). Rural residency (AHR = 1.02, 95%, CI 1.00,1.04), agricultural employee (AHR = 1.14, 95%, CI 1.13, 1.17), and nonagricultural employee (AHR = 1.06, 95%, CI 1.05, 1.08), marriage below 15 years (AHR = 5.47, 95%, CI 5.37, 5.57) and 15–17 years (AHR = 3.27, 95%, CI 3.22, 3.32), had sex below 15 years (AHR = = 1.57, 95%, CI 1.54, 1.61) and 15–17 years (AHR = 1.38, 95%, CI 1.38, 1.43), women who had unmet need for contraceptive (AHR = 1.39, 95%, CI 1.37, 1.42), and met need (AHR = 1.32, 95%, CI 1.30, 1.35), high spousal age gap (AHR = 1.17, 95%, CI 1.15, 1.19), not heard family planning message (AHR = 1.02, 95%, CI 1.01,1.04) were the higher hazard of early childbirth.ConclusionThe median age at first birth was found to be 19 years. This is lower than the optimal age for giving first birth, which is between late 20 s and early 30 s years. Rural residences, occupation, hearing family planning massage in the media, early sexual intercourse, early age at first marriage, high spousal gap, and unmet need for family planning were predictors of first birth at an early age. Thus, governments and non-governmental organizations should strive to implement programs that aim to reduce early age at first birth by considering these factors.

Statistical modelling of the joint probability density function of air density and wind speed for wind resource assessment: A case study from China

This paper proposes a joint probability density function of air density and wind speed, aiming to integrate the air density distribution into statistical wind resource models, thereby helping better understand the correlation between air density and wind speed at different spatiotemporal scales. Hourly air pressure, temperature, and wind speed time series data from 1745 meteorological stations distributed over China for the period 2008–2019 were used to calculate the empirical distributions of air density and wind speed. Four one-component probability density functions and ten two-component mixture probability density functions were fitted to the empirical distributions. It is indicated that the five-parameter Weibull–Gamma distribution is able to achieves more accurate and stable effects in both unimodal and bimodal air density empirical distribution, and the six-parameter Weibull–Burr distribution is the most appropriate distribution for describing wind speed regimes. The parameterised air density and wind speed marginal distributions were linked using five joint copulas. The goodness-of-fit evaluation of the copula models demonstrates that the Gumbel copula most accurately reproduces the bivariate empirical distribution of air density and wind speed. The application of joint distribution has the capability to simplify the evaluation procedures for wind resource assessment while significantly increasing the accuracy of the evaluation results, and also achieve a systematic assessment of available wind resources, considering the temporal patterns of air density. The results of this study will contribute to offering critical information for making better decisions on wind energy projects to realise full development and effective utilisation of wind resources.

Classical and Bayesian inference of the weighted-exponential distribution with an application to insurance data.

This paper addresses asymmetric flexible two-parameter exponential model called the weighted exponential (WDEx) distribution. Some of its basic mathematical features are evaluated. Its hazard rate accommodates upside-down bathtub, decreasing, decreasing-constant, increasing, and increasing-constant shapes. Five actuarial indicators are studied. We utilize nine classical and Bayesian approaches of estimation for estimating the WDEx parameters. We provide a detailed simulation study to explore and assess the asymptotic behaviors of these estimators. Two approximation methods called the Markov chain Mont Carlo and Tierney and Kadane are applied to obtain the Bayesian estimates. The efficiency and applicability of the WDEx distribution are explored by modeling a lifetime data set from insurance field, showing that the WDEx distribution provides a superior fit over its competing exponential models such as the beta-exponential, Harris extend-exponential, Marshall-Olkin exponential, Marshall-Olkin alpha-power exponential, gamma Weibull, and exponentiated-Weibull distributions.

The extended Weibull distribution with its properties, estimation and modeling skewed data

The present article introduces a new distribution called the odd log–logistic Lindley-Weibull (OLLLW) distribution that provides greater flexibility in modeling data in applied areas such as medicine and engineering. The OLLLW model provides left-skewed, symmetric, right-skewed, and reversed-J shaped densities. Its hazard function can be bathtub, unimodal, increasing or decreasing. The OLLLW density was expressed as a linear mixture of Weibull densities. Some distributional properties of the introduced model are derived. Its parameters are estimated using five classical estimators called, the maximum likelihood, Anderson–Darling, least-squares, Cramér-von Mises, and weighted least squares estimators. The performance of the proposed estimators is explored by detailed simulation results. The flexibility of the OLLLW distribution is studied by two real data sets from medicine and engineering sciences, showing that its capability to fit the data effectively than the Weibull, Fréchet Weibull, transmuted Weibull, gamma Weibull, transmuted exponentiated Weibull, and modified Weibull distributions.

Time to first birth and its predictors among reproductive-age women in Ethiopia: inverse Weibull gamma shared frailty model

BackgroundHigh maternal and child death with high fertility rate have been reported in Ethiopia. Extreme age at first birth is linked with both maternal and child morbidity and mortality. However, literatures showed there were limited studies on the timing of the first birth and its predictors in the area so far. Therefore, determining the time to first birth and its predictors will help to design strategies to improve maternal and child survival.MethodsA community-based cross-sectional study was conducted among reproductive-age women in Ethiopia using the Ethiopian demographic health survey, 2016 data. Stratified two-stage cluster sampling technique was used for sampling. The Kaplan–Meier method was used to estimate time to first birth. Inverse Weibull gamma shared frailty model applied to model the data at 95% confidence interval (CI), adjusted hazard ratio (AHR) and median hazard ratio (MHR) were reported as effect size. Proportional hazard assumption checked using Schoenfeld residual test. Information Criteria were applied to select a parsimonious model. Stratified analysis performed for the interaction terms and statistical significance was declared at p value < 0.05.ResultsThe overall median age at first birth was found to be 20 years (IQR, 16–24 years). The independent predictors of time to first birth were: married 15–17 years (AHR = 2.33, 95% CI 2.08–2.63), secondary education level (AHR = 0.84, 95% CI 0.78–0.96), higher education level (AHR = 0.75, 95% CI 0.65–0.85), intercourse before 15 years in the married stratum (AHR = 23.81, 95% CI 22.22–25.64), intercourse 15–17 years in married stratum (AHR = 5.56, 95% CI 5.26–5.88), spousal age difference (AHR = 1.11, 95% CI 1.05–1.16),and use of contraceptives (AHR = 0.91, 95% CI 0.86–0.97). The median increase in the hazard of early childbirth in a cluster with higher early childbirth is 16% (MHR = 1.16, 95% CI 1.13–1.20) than low risk clusters adjusting for other factors.ConclusionIn this study, first birth was found to be at an early age. Early age at first marriage, at first sexual intercourse and their interaction, high spousal age difference, being Muslim were found to increase early motherhood. Conversely, living in the most urban region, secondary and higher women education were identified to delay the first birth. Investing on women education and protecting them from early marriage is required to optimize time to first birth. The contextual differences in time to first birth are an important finding which requires more study and interventions.

Log-Burr XII Gamma–Weibull Regression Model with Random Effects and Censored Data

It may happen in some applications that the assumption of independence of survival times does not hold. Thus, we propose a new log-Burr XII regression model with log-gamma–Weibull distributions for the random effects. The maximum likelihood method is used to estimate the model parameters based on the Gauss–Hermite numerical integration technique. For different parameter settings, sample sizes, censoring percentages and correlated data, various simulations are performed. Some global-influence measurements are also investigated. In order to assess the robustness of the maximum likelihood estimators, we evaluate local influence on the estimates of the parameters under different perturbation schemes. We illustrate the importance of the new model by means of a real data set in analysis of experiments.

Порівняння шести функцій щільності розподілу для моделювання таксаційної будо¬ви за діаметром модальних ялицевих деревостанів Українських Карпат

Для дослідження таксаційної будови за діаметром ялицевого елемента лісу у семи підприємствах лісового господарства Українських Карпат та в НПП "Сколівські Бескиди" закладено 14 прямокутних тимчасових пробних площ та 61 кругову пробну площу (у 15 таксаційних виділах). Дані суцільного переліку дерев ялиці зі замірами їх діаметрів використано для визначення особливостей розподілу кількості дерев за діаметром. Моделювання фактичного розподілу кількості дерев здійснено за допомогою шести фунцій щільності розподілу: дво-, три-, та семипараметричного розподілу Вейбулла, гамма- та бета-розподілу, Sb розподілу Джонсона. Виявлено, що ялицеві деревостани характеризуються значною мінливістю діаметра. За результатами аналізу основних усереднених статистичних показників різних функцій розподілу за діаметром ялицевого елемента встановлено, що дво-, трипараметричний розподіл Вейбулла, гамма- та бета-розподіл завищують показники асиметрії та ексцесу порівняно із фактичним розподілом, Sb розподіл Джонсона занижує ці показники порівняно з фактичними, найближчі результати має бімодальний розподіл Вейбулла. Отримані результати непараметричної оцінки за допомогою χ² критерію Пірсона дають змогу зробити висновок, що найпридатнішою функцією для дослідження таксаційної будови модальних ялицевих деревостанів Українських Карпат є семипараметричний розподіл Вейбулла, а найменш точним – гамма-розподіл.

Odd Burr Lindley Distribution with Properties and Applications

In this study, we introduce a new model called the Odd Burr Lindley distribution which extends the Lindley distribution and has increasing, bathtub and upside down shapes for the hazard rate function. It includes the odd Lindley distribution as a special case. Several statistical properties of the distribution are explored, such as the density, hazard rate, survival, quantile functions, and moments. Estimation using the maximum likelihood and inference of a random sample from the distribution are investigated. A simulation study is performed to compare the performance of the different parameter estimates in terms of bias and mean square error. Two real data applications are modelled with the proposed distribution to illustrate the performance of the new distribution. Based on goodness-of-fi t statistics, the new distribution outperforms the generalized gamma, gamma Weibull, gamma exponentiated exponential, generalized Lindley, Kumaraswamy Lindley, and odd log-logistic Lindley distributions.

Efficient Detection for Improving ASEP Performance of MIMO Composite Fading Channel with Generalized Noise

Multiple-input multiple-output (MIMO) system has achieved significant benefits by applying optimum detection techniques. Maximum likelihood (ML) detection technique improves the error rate performance of MIMO system at optimum level but with enhanced complexity. Therefore, we have considered minimum mean square error-ordered successive interference cancellation with candidate detection to approach the performance of ML with reduced complexity. The proposed system performance is analyzed in composite Weibull–gamma fading and additive white generalized Gaussian noise (AWGGN) scenario. Some special cases of WG fading and AWGGN yield the simplified results.

Inference on the Stress-Strength Model from Weibull Gamma Distribution

Inference on the Stress-Strength Model from Weibull Gamma Distribution

Novel expressions for CEP/BEP under GGD noise for nano communication system

ABSTRACTA newly emerging branch of communication system is nano communication (NC). The NC has enabled numerous important applications, such as immune system support, drug delivery systems, nuclear, biological and chemical defence and air pollution control. However, like in any communication system in NC system also, the medium through which communication takes place introduces various impairments such as fading and noise. In this research paper, channel model 1, i.e. CM1 scenario is analysed over suitable fading channels and noise distribution. First and foremost, a new approximation for Weibull-Gamma (WG) fading distribution that is suitable for CM1 scenario is proposed. Further, suitable noise model for NC is proposed as; generalized Gaussian distribution under CM1 scenario. Furthermore, a novel expression for conditional error probability (CEP) is also presented. Finally, following the proposed CEP closed-form expressions for average bit error rate is derived over WG and Mixture Gamma fading channel in terms of Meijer’s G function. To the best of our knowledge results are novel and never reported in the literature of NC system.

Estimation of parameters of Weibull\u2013Gamma distribution based on progressively censored data

In this paper, the estimation of parameters of a three-parameter Weibull–Gamma distribution based on progressively type-II right censored sample is studied. The maximum likelihood, Bayes, and parametric bootstrap methods are used for estimating the unknown parameters as well as some lifetime parameters reliability function, hazard function and coefficient of variation. Approximate confidence intervals for the unknown parameters as well as reliability function, hazard function and coefficient of variation are constructed based on the s-normal approximation to the asymptotic distribution of maximum likelihood estimators (MLEs), and log-transformed MLEs. In addition, two bootstrap CIs are also proposed. Bayes estimates of the unknown parameters and the corresponding credible intervals are obtained by using the Gibbs within Metropolis–Hasting samplers procedure. Furthermore, the results of Bayes method are obtained under both the balanced squared error loss and balanced linear-exponential loss. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, a Monte Carlo simulation study is carried out to investigate the precision of the Bayes estimates with MLEs and two bootstrap estimates, also to compare the performance of different corresponding CIs considered.

Antenna selection for MIMO systems over Weibull–Gamma fading channel

In this paper, multiple-input multiple-output (MIMO) system performance is evaluated over composite Weibull–Gamma (WG) fading channel using antenna selection (AS) techniques. WG fading channel is considered to model multipath and shadowing effects. The capacity and error rate performance can be improved by exploiting channel state information on the transmitter side (CSIT) while reducing hardware complexity. Therefore, AS techniques are used to exploit CSIT. Although, optimal AS improves the system performance at the cost of enhanced complexity, sub-optimal AS technique is preferred to improve capacity performance with reduced complexity level. Also, AS is used for orthogonal space time block code (OSTBC) to improve error performance of MIMO system over WG fading channel.

On the Gamma Dual Weibull Model

SYNOPTIC ABSTRACTWe propose a new and simple representation for the gamma dual G probability density function, recently introduced in the statistical literature, as an absolutely convergent power series of the survival function of the baseline G distribution. Additionally, the three-parameter gamma dual Weibull model is studied. Some similarities and differences between the gamma dual Weibull model and gamma exponentiated Weibull model are described. Further, we also provide several properties for both models.

ASEP of MIMO System with MMSE-OSIC Detection over Weibull-Gamma Fading Channel Subject to AWGGN

Ordered successive interference cancellation (OSIC) is adopted with minimum mean square error (MMSE) detection to enhance the multiple-input multiple-output (MIMO) system performance. The optimum detection technique improves the error rate performance but increases system complexity. Therefore, MMSE-OSIC detection is used which reduces error rate compared to traditional MMSE with low complexity. The system performance is analyzed in composite fading environment that includes multipath and shadowing effects known as Weibull-Gamma (WG) fading. Along with the composite fading, a generalized noise that is additive white generalized Gaussian noise (AWGGN) is considered to show the impact of wireless scenario. This noise model includes various forms of noise as special cases such as impulsive, Gamma, Laplacian, Gaussian, and uniform. Consequently, generalizedQ-function is used to model noise. The average symbol error probability (ASEP) of MIMO system is computed for 16-quadrature amplitude modulation (16-QAM) using MMSE-OSIC detection in WG fading perturbed by AWGGN. Analytical expressions are given in terms of Fox-H function (FHF). These expressions demonstrate the best fit to simulation results.

Marshall–Olkin gamma–Weibull distribution with applications

A Marshall–Olkin variant of the Provost type gamma–Weibull probability distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model two real data sets. The new distribution provides a better fit than related distributions as measured by the Anderson–Darling and Cramér–von Mises statistics. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology, engineering, etc.

A distribuição Gama Weibull Poisson aplicada a dados de sobrevivência

A familia de distribuicoes univariadas gama-generalizada proposta por Zografos e Balakrishnan [3] foi discutida por Nadarajah et al. [7] que deram um amplo tratamento matematico a esta classe. Neste artigo, estudamos o modelo Gama Weibull Poisson, que tem como casos especiais, varias distribuicoes discutidas na literatura. Deduzimos uma expressao explicita para a entropia de Renyi, e mostramos que a densidade da distribuicao Gama Weibull Poisson e uma mistura de densidades da distribuicao Weibull Poisson. Estudamos algumas propriedades matematicas importantes como momentos, funcao geratriz de momentos e funcao quantilica.