Maximum Likelihood Estimation Method Research Articles

Modelling of Earthquake b-and a-Values Using Least Squares and Maximum Likelihood Estimate Methods in Different Tectonic Regions of the World

Aims: This study modelled a- and b-values of earthquakes employing the least squares regression and maximum likelihood estimate methods. Methodology: Data used in the study were obtained from the International Seismological Centre (ISC), an earthquake catalogue of the United Kingdom. The time window was from 1st January 1988 to 31st December 2010 (30 years) with earthquake focal depth of 0-700km and magnitude Mb ≥ 1.3. Ten different locations were selected and a total of 149,965 events were used. The acquired data were processed and analysed using Microsoft Excel and the hypothesis was tested using independent t-test statistics with the aid of Statistical Software for Social Sciences (SPSS) version 23.0. Results: The findings of the study revealed that the b- and a-values calculated using the least squares regression method were higher than the ones obtained using the maximum likelihood estimate method. Also, the hypothesis revealed that there is a significant difference between the use of the least squares regression method and the maximum likelihood estimate method in the determination of b- and a-value of earthquakes in a given region. Conclusion: The maximum likelihood estimate gives a better estimate of b- and -a values than the least squares regression method.

Frequentist and Bayesian tolerance intervals for setting specification limits for left-censored gamma distributed drug quality attributes.

Tolerance intervals from quality attribute measurements are used to establish specification limits for drug products. Some attribute measurements may be below the reporting limits, that is, left-censored data. When data has a long, right-skew tail, a gamma distribution may be applicable. This paper compares maximum likelihood estimation (MLE) and Bayesian methods to estimate shape and scale parameters of censored gamma distributions and to calculate tolerance intervals under varying sample sizes and extents of censoring. The noninformative reference prior and the maximal data information prior (MDIP) are used to compare the impact of prior choice. Metrics used are bias and root mean square error for the parameter estimation and average length and confidence coefficient for the tolerance interval evaluation. It will be shown that Bayesian method using a reference prior overall performs better than MLE for the scenarios evaluated. When sample size is small, the Bayesian method using MDIP yields conservatively too wide tolerance intervals that are unsuitable basis for specification setting. The metrics for all methods worsened with increasing extent of censoring but improved with increasing sample size, as expected. This study demonstrates that although MLE is relatively simple and available in user-friendly statistical software, it falls short in accurately and precisely producing tolerance limits that maintain the stated confidence depending on the scenario. The Bayesian method using noninformative prior, even though computationally intensive and requires considerable statistical programming, produces tolerance limits which are practically useful for specification setting. Real-world examples are provided to illustrate the findings from the simulation study.

Motifs in earthquake networks: Romania, Italy, United States of America, and Japan

We present a detailed description of seismic activity in Romania, Italy, and Japan, as well as the California seismic zone in the United States of America, based on the statistical analysis of the underlying earthquake networks used to model the aforementioned zones. Our results on network connectivity and simple network motifs allow for a complex description of seismic zones, while at the same time reinforcing the current understanding of seismicity as a critical phenomenon. The reported distributions on node connectivity, three-, and four-node motifs are consistent with power-law, i.e., scale-free, distributions over large intervals and are robust across earthquake networks obtained from different discretizations of the seismic zones of interest. In our analysis of the distributions of node connectivity and simple motifs, we distinguish between the global distribution and the power-law part of it with the help of maximum likelihood estimation (MLE) method and complementary cumulative distribution functions (CCDF). The main message is that the distributions reported for the aforementioned seismic zones have large power-law components, extending over some orders of magnitude, independent of discretization. All the results were obtained using publicly available databases and open-source software, as well as a new toolbox available on GitHub, specifically designed to automatically analyze earthquake databases.

Three Parameter Generalized Gaussian Type Distribution

Three Parameter Generalized Gaussian Type Distribution

Transmuted Modified Weibull Distribution for Modeling Skewed Lifetime Dataset; Properties and Application

A new five parameter model called Generalized Modified Weibull distribution is proposed and studied. The new distribution generalizes the Modified Weibull Distribution introduced by Lai et al. (2003) using transformed – Transformer framework of Alzaatreh et al. (2014) which resulted to a distribution capable of modeling skewed data (positively or negatively skewed data), as well as, symmetric data. Property of a proper probability density function was used to ascertain that the resulting function is a proper probability density function. Statistical properties of the newly generated distribution were studied. Graph of probability density function of the distribution was used to show that it is capable of modeling skewed data. Graph of cumulative density functions of the distribution was plotted using varying parameter values. Monte Carlo simulation approach was used for the test of homogeneity of the distribution and it was observed that the parameters in the distribution approach the true values as sample size increases. Maximum likelihood Estimation method was used for estimation of the model parameters. Real life dataset was used for model comparison as well as demonstration of its application.

Maximum Likelihood Estimation for the Log-Logistic Distribution Using Whale Optimization Algorithm with Applications

The log-logistic distribution has been widely used in several fields, including engineering, survival analysis, and economics. The method of maximum likelihood estimation is used in this study for estimating the shape and scale parameters for the log-logistic distribution, whereas in the case of the log-logistic distribution, likelihood equations lack explicit solutions. Therefore, problems with solving likelihood equations can be solved by using two highly efficient algorithms, which are the whale optimization algorithm and the Nelder-Mead algorithm, as well as by showing the applicability of this distribution by comparing it with other well-known classical distributions. To demonstrate the performance of each algorithm implemented, an extensive Monte Carlo simulation study has been conducted. The performance of maximum likelihood estimators for each algorithm has been evaluated in terms of mean square error and deficiency criteria. It has been seen that the whale optimization algorithm provides the best estimates for the log-logistic distribution parameters according to the simulation data.

Box-Cox Transformation for Exponential Smoothing With Application

This article introduces a novel algorithm for incorporating power transformation into the estimation process of a Holt-Winters Seasonal model. The algorithm outlines a series of steps aimed at selecting the most appropriate power parameter estimate. This selection is achieved using the conventional Maximum Likelihood Estimation method in combination with various criteria for enhancing statistical modeling efficiency. Supplementary decision rules include assessing Mean Square Error, Mean Absolute Error, and conducting a p-value test for the normality of errors. The algorithm's effectiveness is demonstrated through its application to real-world data. Ultimately, the article affirms the feasibility of obtaining viable solutions for selecting the optimal power parameter.

A New Generalization of the Half-Logistic Model using the Transformed-Transformer Approach with Application to Wind Speed Sample from Nigeria

In this paper, a new generalization of the half-logistic distribution was introduced and called the ‘Odd Weibull Exponentiated Half-Logistic Distribution (OWEHLD)’. The OWEHLD was realized by using the Weibull distribution to transform the exponentiated half-logistic distribution using the quantile function of the log-logistic distribution or the odd ratio as link. This approach for generalizing the half-logistic distribution is the so-called ‘transformed-transformer approach’. Several mathematical properties of the OWEHLD were derived and studied and the maximum likelihood estimation method was employed in estimating its parameters. Real life data sets including a wind speed sample from Nigeria were further used to test the applicability of the new distribution.

Applying Generalized Type-II Hybrid Censored Samples on Generalized and q-Generalized Extreme Value Distributions under Linear Normalization

The generalized extreme value (GEV) distributions have wide applications for describing a variety of random events, such as those that occur during specific survival, financial, or reliability investigations. Also, the q-analogues of GEV distributions, called (q-GEVs), are characterized by their ability to provide more flexibility for modeling, which is due to the influence of the q parameter. In this study, we estimated the parameters of generalized and q-generalized extreme value distributions under linear normalization, called GEVL and q-GEVL, respectively. These parameters were estimated using the maximum likelihood estimator method and are based on the generalized type-II hybrid censored sample (G-Type-II HCS). The confidence intervals for these parameters were evaluated. Also, Shannon entropy was estimated for GEVL and q-GEVL distributions. The accuracy of these parameters and the performance of estimators were demonstrated through a real-life example and a simulation study.

Using machine learning to improve multi-qubit state discrimination of trapped ions from uncertain EMCCD measurements.

This paper proposes a residual network (ResNet)-based convolutional neural network (CNN) model to improve multi-qubit state measurements using an electron-multiplying charge-coupled device (EMCCD). The CNN model is developed to simultaneously use the intensity of pixel values and the shape of ion images in determining the quantum states of ions. In contrast, conventional methods use only the intensity values. In our experiments, the proposed model achieved a 99.53±0.14% mean individual measurement fidelity (MIMF) of 4 trapped ions, reducing the error by 46% when compared to the MIMF of maximum likelihood estimation method of 99.13±0.08%. In addition, it is experimentally shown that the model is also robust against the ion image drift, which was tested by intentionally shifting the ion images.

SURVIVAL FUNCTION AND HAZARD FUNCTION ANALYSIS OF EXPONENTIAL DISTRIBUTION IN TYPE I CENSORED SURVIVAL DATA: A CASE STUDY OF BREAST CANCER PATIENTS

Breast cancer is the most common cancer in women and the leading cause of cancer-related death in Indonesia. Analysis of survival data is important for improving the treatment and care of breast cancer patients. This study aims to estimate the parameters, find the survival function, and hazard function of breast cancer patients using a parametric method with an exponential distribution. Previous studies have shown that the Maximum Likelihood Estimation (MLE) method is suitable for estimating the survival function from exponential survival data by censoring. In this study, the exponential distribution was found to be the best for data on breast cancer patients from Surabaya Ontology Hospital. The estimated parameters are θ = 33.9157, and the survival function is calculated using The estimated hazard function for patient death or failure is 0.0295. The results of this study can contribute to the development of better treatment and care strategies for breast cancer patients. However, further research is needed because this study only used monthly time units.

Extreme Rainfall Modeling through the Lens of Extreme Value Theory: A Case Study of Katsina City, Nigeria

Climate change has conveyed about strange new weather patterns, among others is changes in rainfall extremes. To make adequate inferences about extreme rainfall, it's important to invest in meteorological research. This can include the use of climate models to predict extreme rainfall events with greater accuracy. Additionally, information sharing can help ensure that countries are well-prepared for extreme weather events and can collaborate to address the global challenges posed by climate change. Monthly rainfall record of Katsina city was collected from the Nigerian Meteorological Agency (NiMet), Nigeria. In pursuit of this objective, the study employed Extreme Value Theory to model and forecast extreme rainfall of Katsina city, utilizing rainfall data spanning from 1989 to 2019. The research employed the Maximum Likelihood Estimation method to derive the model parameters. For the Generalized Extreme Value Distribution (GEVD), the block maxima approach was applied, whereas the Generalized Pareto Distribution (GPD) was fitted using the Peak Over Threshold method. The analysis revealed that the optimal model within the GEVD framework was the Frechet distribution, whereas the ordinary Pareto distribution emerged as the optimal model when considering values above the threshold. Additionally, the study included predictions for return periods of 2, 20, and 100 years based on the return level estimates, accompanied by the presentation of their respective confidence intervals. The analysis demonstrated that as the return periods lengthened, there was a proportional increase in the return levels. The study's model diagnostics, involving probability, density, quantile, and return level plots, collectively indicated that the provided models were well-suited for the dataset.

Improving Value-at-Risk forecast using GA-ARMA-GARCH and AI-KDE models

The classical autoregressive moving average (ARMA) and generalized autoregressive conditional heteroskedastic (GARCH) models have been widely adopted to forecast Value-at-Risk (VaR), the most popular risk measure in quantitative risk management. Their parameters are commonly estimated using the maximum likelihood estimation (MLE) method, which may show biasedness. This reason motivated us to require optimization through a genetic algorithm (GA) or implement artificial intelligence (AI) to improve the VaR forecast. In this paper, we aimed to compare the performances of the GA-based ARMA-GARCH model and the AI models (linear regression, support vector regression, multilayer perceptron, and long short-term memory) in forecasting the VaR. More specifically, we examined which model performs best. Using Bitcoin, crude oil, and stock index returns, we showed that the GA-ARMA-GARCH and AI models outperformed the benchmark MLE-ARMA-GARCH model. We also revealed that the AI models (particularly multilayer perceptron) integrated with kernel density estimation (KDE) performed relatively better than the GA-ARMA-GARCH model because the former resulted in a higher p-value of the VaR backtesting with a smaller error value. In addition, the cross combination between our models was found to have a higher potential to produce an improvement in the VaR forecast.

Accelerating Volumetric CT and MRI Imaging by Reference-Free Deep Learning Transformation from Low-Resolution to High-Resolution

High-resolution (HR) images are important in precision radiation oncology. However, acquiring HR volumetric CT and MRI images is often time consuming; also, the resolution in some direction(s) (e.g., z-direction in the case of CT) is often limited by imaging hardware or fundamental imaging principle. Super-resolution (SR) imaging, i.e., the low-resolution (LR) to HR image transformation, is widely used to improve image resolution. Data-driven deep learning (DL) methods have achieved great success in SR imaging, yet they can hardly be applied to medical imaging as they require large amount of LR-HR image pairs to train the model. We therefore propose a reference-free DL method to increase resolutions of volumetric medical images in an efficient way. We propose a maximum likelihood estimation (MLE)-based implicit neural representation (INR) network for SR imaging. The INR network aims to represent an image as a continuous function parameterized by a coordinate-based multi-layer perceptron. The INR network takes image coordinates as input and outputs corresponding pixel intensities. To train the network without using any HR images, we use a MLE framework to model LR observations' statistics and their relation to the latent HR image. The predicted HR image from the INR's output is transformed to LR images based on the MLE, and the network parameters are then optimized by minimizing the distance between the transformed LR images and actual LR observations. We demonstrate the efficacy of the proposed method on CT and MRI images for 2x, 4x, and 8x SR using only one or two LR image(s). The performance is compared with a conventional SR method named plain MLE, in terms of visual quality and numerical qualities of PSNR and SSIM. Our method outperformed the plain MLE method in the experiment. Table 1 reports the numerical improvements of our method over the compared plain MLE method. For 2x SR with a single LR image, our method achieved significant improvements in both PSNR and SSIM. When using two LR images, the better structural restoration capability of our method became more obvious with higher SR magnifications, as indicated by the increased SSIM differences. Better noise suppression capability of our method is observed in all our studies, as indicated by the PSNR values. In visual quality evaluation, we observed sharper image details with less noise in SR images generated by the proposed method, compared with the plain MLE method. The proposed novel reference-free DL method can efficiently provide high-quality HR images with only one or two LR images for CT and MRI imaging. This method can be easily generalized to many other radiation therapy related applications without the requirement for HR reference images.

Penyesuaian distribusi proses keberangkatan sepeda motor dari lahan parkir saat waktu puncak

The study aims to model the distribution of motorcycle departures from the parking lot at peak times with the Poisson process approach. This process involves a discrete number of departures and continuous interdeparture time. The probability distribution candidate was selected to model the data according to the data nature, the stochastic process, and the empirical observation of the departure process. Parameters are estimated using the Maximum Likelihood Estimation (MLE) method and the bootstrapping procedure to construct confidence intervals for the parameter. The goodness-of-fit test is applied to select the best probability distribution that matches empirical data. Inferences to the distribution parameters suggest that Weibull's distribution is more appropriate for describing the motorcycle's inter-departure time. The number of motorcycle departures fits significantly into a negative binomial distribution. The results of the study concluded that the Poisson process applied was a case of overdispersion, with the motorcycle departure rate decreasing over time.Keywords: Bootstrapping, departure, distribution, goodness-of-fit, PoissonMSC2020: 60E05

Semiparametric transformation model:A hierarchical Bayesian approach

The use of semiparametric or transformation models has been considered by many authors in the analysis of lifetime data in the presence of censoring and covariates as an alternative and generalization of the usual proportional hazards, the proportional odds models, and the accelerated failure time models, extensively used in lifetime data analysis. The inferences for the proportional hazards model introduced by Cox (1972) are usually obtained by maximum likelihood estimation methods assuming the partial likelihood function introduced by Cox (Cox, 1975). In this study, we consider a hierarchical Bayesian analysis of the proportional hazards model assuming the complete likelihood function obtained from a transformation model considering the unknown hazard function as a latent unknown variable under a Bayesian approach. Some applications with real time medical data illustrate the proposed methodology.

Bayesian Estimation Using Product of Spacing for Modified Kies Exponential Progressively Censored Data

In life testing and reliability studies, most researchers have used the maximum likelihood estimation method to estimate unknown parameters, even though it has been proven that the maximum product of spacing method has properties as good as the maximum likelihood estimation method and sometimes even better. In this study, we aim to estimate the unknown parameters of the modified Kies exponential distribution along with the reliability and hazard rate functions under progressive type-II censoring scheme. The maximum likelihood and maximum product of spacing methods are considered in order to find the point estimates and approximate confidence intervals of the various parameters. Moreover, Bayesian estimations based on the likelihood function and the product of the spacing function of the unknown parameters are obtained using the squared error loss function with independent gamma priors. It is observed that the joint posterior distributions have complicated forms. Because of this, Lindley’s approximation and the Markov chain Monte Carlo technique are used to obtain the Bayesian estimates and highest posterior credible intervals. Monte Carlo simulations are performed in order to evaluate the performance of the proposed estimation methods. Two real datasets are studied to demonstrate the efficacy of the offered methodologies and highlight how simple and applicable it might be to apply them in practical fields.

Nonparametric Bayesian modeling on infinite mixture Student t copulas

This article introduces an infinite mixture Student t copula model using a nonparametric Bayesian approach. We establish a corresponding Markov chain Monte Carlo sampler for this model. In contrast to the normal mixture model, our proposed model is more suitable for data exhibiting tail dependence, which is frequently encountered in financial risk management. We evaluated the proposed algorithm through theoretical simulations and real data analysis. Parameter estimation results from the simulations demonstrate that our approach is competitive when compared to the standard maximum likelihood estimation method. The analysis of real financial data supports the validity of our approach and highlights the importance of applying a t copula in the presence of heavy tails.

The Odd Ramos-Louzada Generator of distributions with applications to failure and waiting times

In this study, we have proposed a new generator called the Odd Ramos-Louzada Generator (ORL-G), which can extend many well-known distributions by applying the T-X method. Some of its basic statistical properties such as the quantile function, the rthnon-central moments and Renyi entropy are also investigated. The ORL-G has the ability to model symmetric and asymmetric data as well as non-monotonic and monotonic failure rate datasets. Three new distributions of the ORL-G family are the ORL-Burr XII (ORLBXII), ORL-Log Logistic (ORLLoL), and ORL-Weibull (ORLWei) which are respectively obtained using the Burr XII, the Log-Logistic, and Weibull as baseline models. The RL and Generalized RL distributions are sub-models of the ORLBXII and ORLLoL distributions. The ORL-G model parameters have been estimated using the maximum likelihood (ML) method. Simulation analysis carried out indicates that the ML estimation method and its asymptotic properties performed quite well. The flexibility of the ORL-G family is demonstrated through the application of three real, complete datasets, including failure and waiting times and it is evident that ORLBXII outperformed its sub-models and other competing models.

Solving Fuzzy Rayleigh Model with Application

في هذا البحث اشتققنا وقدرنا معلمتي توزيع رالي باستخدام طريقة تعظيم دالة الامكان المقدرة وطريقة المربعات الصغرى المقدرة وطريقة العزوم المقدرة . تم استخدام اسلوب المحاكاة للعديد من أحجام العينات وللعديد من القيم الابتدائية وتم التوصل بأن طريقة العزوم المقدرة هي الأفضل من بقية الطرق الأخرى باستخدام اسلوب متوسط مربعات الخطأ. ومن ثم تم استخدام تقدير الفترات لإيجاد المقدر لمعلمة الموقع ومعلمة القياس لتوزيع رالي. بعد ذلك استخدمت دالة العضوية المثلثية لإيجاد المعلمات الضبابية المقدرة لطريقة العزوم لتوزيع رالي. وفي النهاية استخدمنا الدالة الزمنية الترتيبية لتحويل المعلمات الضبابية المقدرة الى معلمات تقليدية وتم التوصل بأن المعلمات الضبابية المقدرة هي الافضل.