Likelihood Equations Research Articles

Maximum likelihood estimation for left-truncated log-logistic distributions with a given truncation point

For a sample X1,X2,…XN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_1, X_2,\\ldots X_N$$\\end{document} of independent identically distributed copies of a log-logistically distributed random variable X the maximum likelihood estimation is analysed in detail if a left-truncation point xL>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_L>0$$\\end{document} is introduced. Due to scaling properties it is sufficient to investigate the case xL=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_L=1$$\\end{document}. Here the corresponding maximum likelihood equations for a normalised sample (i.e. a sample divided by xL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_L$$\\end{document}) do not always possess a solution. A simple criterion guarantees the existence of a solution: Let E(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {E}(\\cdot )$$\\end{document} denote the expectation induced by the normalised sample and denote by β0=E(lnX)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _0=\\mathbb {E}(\\ln {X})^{-1}$$\\end{document}, the inverse value of expectation of the logarithm of the sampled random variable X (which is greater than xL=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_L=1$$\\end{document}). If this value β0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _0$$\\end{document} is bigger than a certain positive number βC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _C$$\\end{document} then a solution of the maximum likelihood equation exists. Here the number βC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _C$$\\end{document} is the unique solution of a moment equation,E(X-βC)=12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {E}(X^{-\\beta _C})=\\frac{1}{2}$$\\end{document}. In the case of existence a profile likelihood function can be constructed and the optimisation problem is reduced to one dimension leading to a robust numerical algorithm. When the maximum likelihood equations do not admit a solution for certain data samples, it is shown that the Pareto distribution is the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-limit of the degenerated left-truncated log-logistic distribution, where L1(R+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\mathbb {R}^+)$$\\end{document} is the usual Banach space of functions whose absolute value is Lebesgue-integrable. A large sample analysis showing consistency and asymptotic normality complements our analysis. Finally, two applications to real world data are presented.

Multivariate normality tests with two-step monotone missing data: a critical review with emphasis on the different methods of handling missing values

Two-step monotone missing data is a special case of missing data that attracted the attention of researchers since it not only appears frequently in applications but also since under the hypothesis of multivariate normality a unique and explicit solution to the likelihood equations exists. This importance prompted Yamada et al. [Kurtosis tests for multivariate normality with monotone incomplete data. Test. 2015;24:532–557. doi: 10.1007/s11749-014-0423-1] and Kurita and Seo [Multivariate normality test based on kurtosis with two-step monotone missing data. J Multivar Anal. 2022;188:104–824. doi: 10.1016/j.jmva.2021.104824] to propose four procedures in total for testing multivariate normality with two-step monotone missing data. The aim of this paper is twofold. On the one, to compare, for the first time, the performance of the four aforementioned multivariate normality tests via a Monte Carlo study. The second aim of the paper is to compare the performance of the four existing testing procedures with the respective procedures derived as a combination of 12 popular methods for handling incomplete data and 6 multivariate normality tests for complete data. We show that in several cases the tests proposed by Yamada et al. [Kurtosis tests for multivariate normality with monotone incomplete data. Test. 2015;24:532–557. doi: 10.1007/s11749-014-0423-1] and Kurita and Seo [Multivariate normality test based on kurtosis with two-step monotone missing data. J Multivar Anal. 2022;188:104–824. doi: 10.1016/j.jmva.2021.104824] are unable to achieve the nominal significance level. At the same time, no single procedure was found to be the most powerful in all situations of multivariate alternatives considered. As a consequence, the use of specific combinations of methods of handling missing data and testing multivariate normality with complete data is recommended.

Novel Closed‐Form Point Estimators for a Weighted Exponential Family Derived From Likelihood Equations

ABSTRACTIn this paper, we propose and investigate closed‐form point estimators for a weighted exponential family. We also develop a bias‐reduced version of these proposed closed‐form estimators through bootstrap methods. Estimators are assessed using a Monte Carlo simulation, revealing favourable results for the proposed bootstrap bias‐reduced estimators. We illustrate the proposed methodology with the use of two real data sets.

A Generalized Finite Mixture Model for Asymmetric Probability Distribution Observations

Finite mixture model is a useful probabilistic model for representing the probability distributions of observations. Gaussian mixture model (GMM) is a widely used one whose parameters are always estimated by the famous EM algorithm. But when probability distributions contain asymmetric modes, GMM requires much more constituent components to achieve a satisfied accuracy. Therefore, a generalized mixture finite model is proposed. First, it adopts the derivative [Formula: see text]-PDF which can well represent the asymmetric probability distributions as the probability density. However, the differential operation and solving the likelihood equations are scarcely possible in the M step of EM algorithm for the complex expression of the derivative [Formula: see text]-PDF. Second, a pseudo EM method is proposed to avoid the abovementioned difficulty which finds the pseudo maximum likelihood estimates of the parameters by utilizing the moment matching principle. It more easily estimates the parameters by solving a series of moment matching equations. Finally, four examples are presented to verify that the proposed generalized finite mixture model has advantage on representing the asymmetric probability distributions observations.

Statistical inference of a series reliability system using shock models with Weibull distribution

AbstractIn this study, we define a series system with non‐independent and non‐identical components using a shock model with sources of fatal shocks. Here, it is assumed that the shocks happen randomly and independently, following a Weibull distribution with various scale and shape parameters. A dependability model with unknown parameters is produced by this process. Making statistical conclusions about the model parameters is the main objective of this research. We apply the maximum likelihood and Bayes approaches to determine the model parameters' point and interval estimates. We shall demonstrate that no analytical solutions to the likelihood equations must be solved to obtain the parameters' maximum likelihood estimates. As a result, we will use the R program to approximate the parameter point and interval estimates. Additionally, we will use the bootstrap‐t and bootstrap‐p methods to approximate the confidence intervals. About the Bayesian approach, we presume that each model parameter is independent and follows a gamma prior distribution with a range of attached hyperparameter values. The model parameters' posterior distribution does not take a practical form. We are unable to derive the Bayes estimates in closed forms as a result. To solve this issue, we use the Gibbs sampler from the Metropolis‐Hasting algorithm based on the Markov chain Monte Carlo method to condense the posterior distribution. To demonstrate the relevance of this research, a real data set application is detailed.

On some algorithms for estimation in Gaussian graphical models

Abstract In Gaussian graphical models, the likelihood equations must typically be solved iteratively. This paper investigates two algorithms: a version of iterative proportional scaling, which avoids inversion of large matrices, and an algorithm based on convex duality and operating on the covariance matrix by neighbourhood coordinate descent, which corresponds to the graphical lasso with zero penalty. For large, sparse graphs, the iterative proportional scaling algorithm appears feasible and has simple convergence properties. The algorithm based on neighbourhood coordinate descent is extremely fast and less dependent on sparsity, but needs a positive-definite starting value to converge. We provide an algorithm for finding such a starting value for graphs with low colouring number. As a consequence, we also obtain a simplified proof of existence of the maximum likelihood estimator in such cases.

Banking sector and economic growth in the digital transformation era: insights from maximum likelihood and Bayesian structural equation modeling

PurposeIn the digital era, the banking sector has transformed into a powerful intermediary, effectively connecting surplus and deficit units. This dynamic landscape empowers savers to secure their finances and generate returns, while simultaneously enabling businesses and individuals to access capital for investment and promoting economic growth. This study explores the relationships among banking development dimensions – represented by primary assets and liabilities, bank capital (core capital and required reserves) and economic growth as measured by components of gross domestic product (GDP).Design/methodology/approachThe study consolidated monthly balance sheets from digital banks over a 20-year period, resulting in an aggregate monthly balance sheet that reflects the financial position of all digital banks in the Palestinian economy. The research employs both maximum likelihood and Bayesian structural equation modeling to measure the causal pathways of the consolidated balance sheet with the individual components of GDP.FindingsThe results revealed that bank main assets (investments and loans) and liabilities (deposits) collectively explain for 97% of bank capital. Investments and loans demonstrate significant negative correlations with bank capital, while deposits exhibit a positive impact. This leads to a fundamental conclusion that a substantial proportion of retained earnings within the banking sector is reinvested, fueling expansion and growth. Additionally, the results showed a significant relationship between bank capital and various GDP components, including private consumption, gross investment and net exports (p = 0.000). However, while the relationship between bank capital and government spending was insignificant in the maximum likelihood estimation, Bayesian estimation revealed a slight yet positive impact of bank capital on government spending.Originality/valueThis research stands out due to its unique exploration of the intricate relationship between bank sector development dimensions, primary assets and liabilities and their impact on bank capital in the digital era. It offers fresh insights by dividing this connection into specific dimensions and constructs, utilizing a comprehensive two-decade dataset covering the digital banks records.

Causal relationship between household consumption transition and CO2 emission in China: a dynamic panel model.

The mitigation of carbon dioxide (CO2) generated from household consumption, accounting for 52% of China's total greenhouse gas emissions, plays a pivotal role in China's pursuit of reaching a carbon peak by 2030. The study used three waves of nationally representative longitudinal data, energy statistics data, and input-output table to estimate household CO2 emissions (HCEs) in China at the micro-scale. The dynamic relationship between household consumption pattern transition and HCEs per capita was explored by applying maximum likelihood and structural equation modeling (ML-SEM) with panel data. The results indicate that per capita HCE level in a given year appears to be positively associated with HCE level for the same household in the previous year. A U-shaped relationship between consumption pattern transition and HCEs per capita was confirmed, as well as the reinforcement effect of income on the impacts of consumption pattern transition. The increase in consumption propensity, household income, share of wage-income, household asset values, and house space results in higher HCEs per capita. The family size and dependency ratio have a negative relationship with HCEs, whereas households that are female-oriented and more Internet-dependent tend to produce more CO2. Exploring the consumption transition of households is crucial for reducing CO2 emissions at the household level in China.

Associations of Anxiety, Insomnia, and Physical Activity during the COVID-19 Pandemic.

Anxiety, insomnia, and physical activity (PA) are interrelated, but the bi-directional relationships between these three variables are not well understood. Less is known of these relationships in settings of disrupted daily activities and acute stress. This study aimed to characterize and examine relationships between insomnia, anxiety, and PA throughout the first year of the COVID-19 pandemic, when many lifestyle behaviors were disrupted. Participants comprised a convenience sample of 204 adults (55.4% female; 43.85 ± 15.85 years old) who completed the Generalized Anxiety Disorder Questionnaire (GAD-7), Insomnia Severity Index (ISI), and the International Physical Activity Questionnaire (IPAQ) at three time points through the first year of the COVID-19 pandemic. A cross-lagged panel model was used to evaluate these variables' concurrent, autoregressive, and cross-lagged relationships across time. Follow-up dynamic panel modeling using maximum likelihood and structural equation modeling was employed. Approximately 64% of participants reported their work/occupation as affected by the pandemic. At baseline, associations between anxiety and insomnia were observed (β-coefficient: 15.87; p < 0.001). Insomnia was a positive future predictor of anxiety (ISI time point 2: 7.9 ± 5.6 points; GAD-7 at time point 3: 4.1 ± 4.2 points; β-coefficient: 0.16; p < 0.01). No associations were observed between PA and anxiety or insomnia (all p > 0.05). Insomnia and anxiety were interrelated, and effects were cross-lagged. These data can inform future work focused on improving anxiety in settings of acute stress and disruptions to daily life, such as changes in occupational structure and stability. Specifically, targeting sleep parameters may be of interest to elicit downstream positive health behaviors.

The household resource efficiency and its economic determinants in China: A DEA and dynamic panel model

Identifying the level of household resource efficiency and its determinants is paramount, given its significant implications for resource governance and national-scale domestic resource planning. This study assesses household resource efficiency in China from 2016 to 2020, employing a super-SBM model alongside three waves of nationally representative longitudinal data. The investigation further delves into the intricate dynamics connecting household income, consumption propensity, and resource efficiency, utilizing maximum likelihood and structural equation modeling (ML-SEM) with dynamic panel data. Findings reveal an overall decrease in household resource efficiency levels, accompanied by diminished variation among surveyed households. Notably, a noteworthy non-monotonic relationship between household income and resource efficiency emerges, confirming a U-shaped pattern. Additionally, consumption propensity exhibits a negative influence, a trend that is further accentuated by increasing income, on household resource efficiency. These insights shed light on the complex interplay between household income, consumption, and resource efficiency, advocating for targeted policies to enhance resource management within households and on a national scale.

Dentist: Quantifying uncertainty by sampling points around maximum likelihood estimates

AbstractIt is standard statistical practice to provide measures of uncertainty around parameter estimates. Unfortunately, this very basic and necessary enterprise is often absent in macroevolutionary studies using maximum likelihood estimates (MLEs).dentist is an R package that allows an approximation of confidence intervals (CI) around parameter estimates without an analytic solution to likelihood equations. This package works by ‘denting’ the likelihood surface by sampling points a specified distance around the MLE following what is essentially a Metropolis‐Hastings walk.We describe the importance of estimating uncertainty around parameter estimates, as well as demonstrate the ability of dentist to accurately approximate CI.We introduce several plotting tools to visualize the results of a dentist analysis. dentist is freely available fromhttps://github.com/bomeara/dentist, written in the R language, and can be used for any given likelihood function.

Comparison of the Meta-Heuristic Algorithms for Maximum Likelihood Estimation of the Exponentially Modified Logistic Distribution

Generalized distributions have been studied a lot recently because of their flexibility and reliability in modeling lifetime data. The two-parameter Exponentially-Modified Logistic distribution is a flexible modified distribution that was introduced in 2018. It is regarded as a strong competitor for widely used classical symmetrical and non-symmetrical distributions such as normal, logistic, lognormal, log-logistic, and others. In this study, the unknown parameters of the Exponentially-Modified Logistic distribution are estimated using the maximum likelihood method. Five meta-heuristic algorithms, including the genetic algorithm, particle swarm optimization algorithm, grey wolf optimization algorithm, whale optimization algorithm, and sine cosine algorithm, are applied in order to solve the nonlinear likelihood equations of the study model. The efficiencies of all maximum likelihood estimates for these algorithms are compared via an extensive Monte Carlo simulation study. The performance of the maximum likelihood estimates for the location and scale parameters of the Exponentially-Modified Logistic distribution developed with the genetic algorithm and grey wolf optimization algorithms is the most efficient among others, according to simulation findings. However, the genetic algorithm is two times faster than grey wolf optimization and can be considered better than grey wolf optimization considering the computation time criterion. Six real datasets are analyzed to show the flexibility of this distribution.

Business Cycle Downturn Likelihood Estimation for Ciudad Juarez

A monthly frequency metropolitan business cycle downturn likelihood equation is estimated for Ciudad Juarez. The binary index of economic conditions is based upon monthly IMMEX export oriented manufacturing employment. A dynamic probit methodology is used for parameter estimation. Continuous explanatory variables include a 1-year minus 1-month Mexico interest rate spread, a 2015 = 100 weighted real exchange rate index, and a 10-year minus 3-month USA interest rate spread. Parameter estimation results confirm the various hypotheses examined. However, model simulation outcomes are less favorable with the results indicating that accurate forecasting of the post-2010 business cycles may require additional refinement to the index.

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.

An EM algorithm for estimating the parameters of the skew generalized t-normal distribution with application to robust finite mixture modeling

The present article describes an EM-type algorithm for estimation of the skew generalized t-normal (SGTN) distribution. The family of SGTN distributions can provide certain types of flexibility such as heavy tails and high kurtosis. The complexity of the SGTN distribution is traced to the ratio of the t density and distribution function of a normal distribution in the likelihood equations. To cope with this problem, we develop a feasible ECME algorithm for computing maximum likelihood estimates of model parameters via a selection mechanism. The proposed approach provides a robust parameter estimation method for the finite mixture model. Standard errors for the parameter estimates can be obtained via a general information-based method. Experimental results on simulated data and one real data example demonstrate the efficacy and usefulness of the proposed methodology.

A stochastic log-logistic diffusion process: Statistical computational aspects and application to real data

This article introduces a new stochastic diffusion process based on the theory of diffusion processes whose mean function is proportional to the log-logistic growth curve. The main characteristics of the process are analyzed, including the transition probability density function, the mean functions and in particular, the auto-correlation function between two times of the process. The parameters of the process are estimated by maximum likelihood method using discrete sampling. The simulated annealing algorithm is applied after bounding the parametric space by a strategy procedure to solve the likelihood equations. The behavior of the diffusion process here derived is finally applied to study an example for the growth data of a microorganism culture.

Reliability Estimation Using EM Algorithm with Censored Data: A Case Study on Centrifugal Pumps in an Oil Refinery

Centrifugal pumps are widely employed in the oil refinery industry due to their efficiency and effectiveness in fluid transfer applications. The reliability of pumps plays a pivotal role in ensuring uninterrupted plant productivity and safe operations. Analysis of failure history data shows that bearings have been identified as critical components in oil refinery pump groups. Analyzing historical failure data for such systems is a complex task due to censored data and missing information. This paper addresses the complexity of estimating the Weibull distribution parameters using the maximum likelihood method under these conditions. The likelihood equation lacks an explicit analytical solution, necessitating numerical methods for resolution. The proposed approach presented in this article leverages the expectation maximization (EM) algorithm for estimating the Weibull distribution parameters in a real-world case study of a complex engineering system. The results demonstrate the superior performance of the EM algorithm with censored data, showcasing its ability to overcome the limitations of traditional methods and provide more accurate estimates for reliability metrics. This highlights the importance of obtaining results through these methodologies in the analysis of reliability and in facilitating more informed decision making in complex systems.

Bidirectional associations of physical activity, sleep, and self-reported mental health in young adults participating in an online wellness intervention during the COVID-19 pandemic.

The purpose of this study was to examine the bidirectional associations of physical activity (PA), sleep, and mental health in young adults participating in an online wellness intervention from October 2021 to April 2022. Participants were a sample of undergraduate students from one US university (N = 89; 28.0% freshman; 73.0% female). The intervention was a 1-h health coaching session that was delivered either once or twice by peer health coaches on Zoom during COVID-19. The number of coaching sessions was determined by random allocation of participants to experimental groups. Lifestyle and mental health assessments were collected at two separate assessment timepoints after each session. PA was assessed using the International Physical Activity Questionnaire-Short Form. Weekday and weekend sleep were assessed by two one-item questionnaires and mental health was calculated from five items. Cross-lagged panel models (CLPMs) examined the crude bidirectional associations of PA, sleep, and mental health across four-time waves (i.e., T1 through T4). To control for individual unit effects and time-invariant covariates, linear dynamic panel-data estimation using maximum likelihood and structural equation modeling (ML-SEM) was also employed. ML-SEMs showed that mental health predicted future weekday sleep (β = 0.46, p < 0.001) and weekend sleep predicted future mental health (β = 0.11, p = 0.028). Although CLPMs showed significant associations between T2 PA and T3 mental health (β = 0.27, p = 0.002), no associations were observed when unit effects and time-invariant covariates were accounted for. Self-reported mental health was a positive predictor of weekday sleep and weekend sleep positively predicted mental health during the online wellness intervention.

Bayesian Estimations of Shannon Entropy and Rényi Entropy of Inverse Weibull Distribution

In this paper, under the symmetric entropy and the scale squared error loss functions, we consider the maximum likelihood (ML) estimation and Bayesian estimation of the Shannon entropy and Rényi entropy of the two-parameter inverse Weibull distribution. In the ML estimation, the dichotomy is used to solve the likelihood equation. In addition, the approximation confidence interval is given by the Delta method. Because the form of estimation results is more complex in the Bayesian estimation, the Lindley approximation method is used to achieve the numerical calculation. Finally, Monte Carlo simulations and a real dataset are used to illustrate the results derived. By comparing the mean square error between the estimated value and the real value, it can be found that the performance of ML estimation of Shannon entropy is better than that of Bayesian estimation, and there is no significant difference between the performance of ML estimation of Rényi entropy and that of Bayesian estimation.

New closed‐form efficient estimator for multivariate gamma distribution

Maximum likelihood estimation is used widely in classical statistics. However, except in a few cases, it does not have a closed form. Furthermore, it takes time to derive the maximum likelihood estimator (MLE) owing to the use of iterative methods such as Newton–Raphson. Nonetheless, this estimation method has several advantages, chief among them being the invariance property and asymptotic normality. Based on the first approximation to the solution of the likelihood equation, we obtain an estimator that has the same asymptotic behavior as the MLE for multivariate gamma distribution. The newly proposed estimator, denoted as , is also in closed form as long as the ‐consistent initial estimator is in the closed form. Hence, we develop some closed‐form ‐consistent estimators for multivariate gamma distribution to improve the small‐sample property. is an alternative to MLE and performs better compared to MLE in terms of computation time, especially for large datasets, and stability. For the bivariate gamma distribution, the is over 130 times faster than the MLE, and as the sample size increasing, the is over 200 times faster than the MLE. Owing to the instant calculation of the proposed estimator, it can be used in state–space modeling or real‐time processing models.