Maximum Likelihood Estimates Of Parameters Research Articles

A New Family: Σh Distributions

In this study, a new method is proposed for generating families of the sum of the hazard functions for two distributions named the Σh distributions. This new family will help in the application of a wider range of life time data. Many new distributions, which are members of the family, are presented with emphasis on the Σh Exponential-Lomax distribution. Details and various statistical properties have been introduced. The maximum likelihood estimation for parameters of the Σh Exponential-Lomax distribution has also been discussed alongside Monte Carlo simulation study to assess the accuracy and the performance of the estimation procedure. Finally, the Σh Exponential-Lomax distribution has been fitted to a real data set to provide variability of its applicability.

Kumaraswamy Inverted Topp–Leone Distribution with Applications to COVID-19 Data

In this paper, an attempt is made to discover the distribution of COVID-19 spread in different countries such as; Saudi Arabia, Italy, Argentina and Angola by specifying an optimal statistical distribution for analyzing the mortality rate of COVID-19. A new generalization of the recently inverted Topp Leone distribution, called Kumaraswamy inverted Topp–Leone distribution, is proposed by combining the Kumaraswamy-G family and the inverted Topp–Leone distribution. We initially provide a linear representation of its density function. We give some of its structure properties, such as quantile function, median, moments, incomplete moments, Lorenz and Bonferroni curves, entropies measures and stress-strength reliability. Then, Bayesian and maximum likelihood estimators for parameters of the Kumaraswamy inverted Topp–Leone distribution under Type-II censored sample are considered. Bayesian estimator is regarded using symmetric and asymmetric loss functions. As analytical solution is too hard, behaviours of estimates have been done viz Monte Carlo simulation study and some reasonable comparisons have been presented. The outcomes of the simulation study confirmed the efficiencies of obtained estimates as well as yielded the superiority of Bayesian estimate under adequate priors compared to the maximum likelihood estimate. Application to COVID-19 in some countries showed that the new distribution is more appropriate than some other competitive models.

Stability analysis of intrinsic camera calibration using probability distributions

The currently most widely used calibration method in geometric computer vision is based on Zhang’s method. This approach is easy to implement and delivers detailed result evaluations. One drawback, however, might be that the unexperienced user may obtain unstable and unrepeatable calibration results despite small re-projection errors. Therefore, the data must be interpreted with caution and there is definitely a need for further consideration beside re-projections to be taken for better estimations. So far, statistical inference has been used for presenting precision measures only. In our work, we extend the statistical approach quantitatively using large bundles of images and get calibrations from randomly chosen subsets. Then a Maximum Likelihood Estimation of intrinsic parameters is implemented and the statistical behaviour is analyzed. In addition, the recovered expected values of parameters are utilized as ground truth to scrutinize the single influencing factors of the imaging configuration. According to the results we found out that the image block plays a significant role for camera calibration with regard to the orientation of the imaging configuration. Including also suitable manufactured calibration boards as well as other operational issues, finally a well-designed image block provides a repeatable and reliable calibration.

The bounded inverse Weibull distribution: An extreme value alternative for application to environmental maxima?

Abstract There has long been interest in making inferences about future low-probability natural events that have magnitudes greater than any in the past record. Given a stationary time series, the unbounded Type 1 and Type 2 asymptotic extreme value distributions are often invoked as giving theoretical justification for extrapolating to large magnitudes and long return periods for hydrological variables such as rainfall and river discharge. However, there is a problem in that environmental extremes are bounded above by the bounded nature of their causal variables. Extrapolation using unbounded asymptotic models therefore cannot be justified from extreme value theory and at some point there will be over-prediction of future magnitudes. This creates the apparent contradiction, for example, of annual rainfall maxima being well approximated by Type 2 extreme value distributions despite the bounded nature of rainfall magnitudes. An alternative asymptotic extreme value approach is suggested for further investigation, with the model being the asymptotic distribution of minima (Weibull distribution) applied to block maxima reciprocals. Two examples are presented where data that are well matched by Type 1 or Type 2 extreme value distributions give reciprocals suggestive of lower bounds (upper bound γ to the original data). The asymptotic model here is a 3-parameter Weibull distribution for the reciprocals, with positive location parameter γ−1. When this situation is demonstrated from data, parameter estimation can be carried out with respect to the distribution of reciprocals of 3-parameter Weibull random variables. This distribution is referenced here as the bounded inverse Weibull distribution. A maximum likelihood parameter estimation methodology is presented, together with a parametric bootstrap approach for obtaining one-sided upper confidence limits to γ. When data permits estimation of γ, the bounded inverse Weibull distribution is suggested as an improved alternative to Type 1 or Type 2 extreme value distributions because the upper bound reality is recognised. However, extensive application to many data sets is required to evaluate the practical utility of the bounded approach for extrapolating beyond the largest recorded event.

The parameter estimation of logistic regression with maximum likelihood method and score function modification

The maximum likelihood parameter estimation method with Newton Raphson iteration is used in general to estimate the parameters of the logistic regression model. Parameter estimation using the maximum likelihood method cannot be used if the sample size and proportion of successful events are small, since the iteration process will not yield a convergent result. Therefore, the maximum likelihood method cannot be used to estimate the parameters. One way to resolve this un-convergence problem is using the score function modification. This modification is used to obtain the parameters estimate of logistic regression model. An example of parameter estimation, using maximum likelihood method with small sample size and proportion of successful events equals 0.1, showed that the iteration process is not convergent. This non-convergence can be solved with modifications on a score function. Modification on score function is to change a score function, a matrix of the first derivative of the log likelihood function, to the first derivative matrix itself minus multiplication of information matrix and biased vector. The modification of the score function can quickly yield values of parameter estimates, especially when the sample sizes are larger, and convergence was reached before the 10th iteration.

Modelling High Dimensional Paddy Production Data using Copulas

As the climate change is likely to be adversely affecting the yield of paddy production, thence it has brought a limelight of the probable challenges on human particularly regional food security issues. This paper aims to fit multivariate time series of paddy production variables using copula functions and predicts the next year event based on the data of five countries in southeast Asia. In particular, the most appropriate marginal distribution for each univariate time series was first identified using maximum likelihood parameter estimation method. Next, we performed multivariate copula fitting using two types of copula families, namely, elliptical copula family and Archimedean copula family. Elliptical copula family studied are normal and t copula, while Archimedean copula family considered are Joe, Clayton and Gumbel copulas. The performance of marginal distribution and copula fitting was examined using Akaike information criterion (AIC) values. Finally, we used the best fitted copula model to forecast the succeeding event. In order to assess the performance of copula function, we computed the forecast means and estimation errors of copula function with a generalized autoregressive conditional heteroskedasticity model as reference group. Based on the smallest AIC, the majority of the data favoured the Gumbel copula, which belongs to Archimedean copula family as well as extreme value copula family. Likewise, applying the historical data to forecast the future trends may assist all relevant stakeholders, for instance government, NGO agencies, and professional practitioners in making informed decisions without compromising the environmental as well as economical sustainability in the region.

Inference of edge correlations in multilayer networks.

Many recent developments in network analysis have focused on multilayer networks, which one can use to encode time-dependent interactions, multiple types of interactions, and other complications that arise in complex systems. Like their monolayer counterparts, multilayer networks in applications often have mesoscale features, such as community structure. A prominent approach for inferring such structures is the employment of multilayer stochastic block models (SBMs). A common (but potentially inadequate) assumption of these models is the sampling of edges in different layers independently, conditioned on the community labels of the nodes. In this paper, we relax this assumption of independence by incorporating edge correlations into an SBM-like model. We derive maximum-likelihood estimates of the key parameters of our model, and we propose a measure of layer correlation that reflects the similarity between the connectivity patterns in different layers. Finally, we explain how to use correlated models for edge "prediction" (i.e., inference) in multilayer networks. By incorporating edge correlations, we find that prediction accuracy improves both in synthetic networks and in a temporal network of shoppers who are connected to previously purchased grocery products.

Analysis of progressive Type‐II censoring in presence of competing risk data under step stress modeling

In this article we consider the analysis of progressively censored competing risks data obtained from a simple step‐stress experiment. It is assumed that there are only two competing causes of failures at each stress level and the lifetime distribution of each one of them is one parameter exponential distribution. Based on the cumulative exposure model assumption, the conditional maximum likelihood estimators (MLEs) of the unknown parameters can be obtained in explicit forms. Confidence intervals of the unknown parameters based on the exact distributions of the conditional MLEs and percentile bootstrap method, are constructed. Further we obtain Bayes estimates and the associated credible intervals based on a very flexible Beta‐gamma prior on the unknown parameters. A simulation experiment has been performed to observe the performances of the different estimators.

Bayesian optimal design for non-linear model under non-regularity condition

The common classical approach for finding optimal design is minimizing the variance of an unbiased maximum likelihood estimator (MLE) of parameters. However, under regularity condition, the variance of MLE is approximated by the Cramer–Rao lower bound. In this article, optimal designs are obtained under non-regularity condition in non-linear models. In the Bayesian approach, conditional mutual information is used to propose a new optimality criterion; Bayesian optimal design maximizes the mutual information between the observation and the model parameters. A Bayesian compound criterion is also provided to facilitate the performance comparison of the optimal designs. Finally, the equivalence theorem is given for criterion to allow for checking the optimality of the obtained Bayesian design points.

Pair copula construction for longitudinal data with zero-inflated power series marginal distributions

ABSTRACT Assessing the temporal dependency among outcomes under investigation is critical in many fields. One complication in the modeling process of the discrete longitudinal data is the presence of excess zeros. We propose a framework for modeling count repeated measurements using members of power series family of distributions. The framework accommodates count outcomes having extra zeros. The longitudinal observations of response variable is modeled using pair copula constructions with a D-vine structure. The maximum likelihood estimates of parameters are obtained using a two-stage approach. Some simulation studies are performed for illustration of the proposed methods, for comparing its performance with that of a generalized linear mixed effects (GLME) model and for assessing the robustness of D-vine and GLME models with respect to the distribution of random effects. In the empirical analysis, the proposed method is applied for analysing a real data set of a kidney allograft rejection study.

A new mixed first-order integer-valued autoregressive process with Poisson innovations

Integer-valued time series, seen as a collection of observations measured sequentially over time, have been studied with deep notoriety in recent years, with applications and new proposals of autoregressive models that broaden the field of study. This work proposes a new mixed integer-valued first-order autoregressive model with Poisson innovations, denoted POMINAR(1), mixing two operators known as binomial thinning and Poisson thinning. The proposed process presents some advantages in relation to the most common Poisson innovation processes: (1) this new process allows to capture structural changes in the data; (2) if there are no structural changes, the most common processes with Poisson innovations are particular cases of POMINAR(1). Another important contribution of this work is the establishment of the POMINAR(1) theoretical results, such as the marginal expectation, marginal variance, conditional expectation, conditional variance, transition probabilities. Moreover, the Conditional Maximum Likelihood (CML) and Yule-Walker (YW) estimators for the process parameters are studied. We also present three techniques for one-step-ahead forecasting, the nearest integer of the conditional expectation, conditional median and mode. A simulation study of the forecasting procedures, considering the two estimators, CML and YW methods, is performed, and prediction intervals are presented. Finally, we show an application of the proposed process to a real dataset, referred here as larceny data, including a residual analysis.

Random Effects Models for Analyzing Mixed Overdispersed Binomial and Normal Longitudinal Responses With Application to Kidney Function Data of Cancer Patients

Joint models with random effects are proposed for the analysis of mixed overdispersed binomial and normal longitudinal data. A new parametric distribution, called the log Lindley-binomial distribution, is also introduced for analyzing overdispersed binomial data. The new distribution can be considered as an alternative to the classical beta-binomial distribution. Random effects are used to take into account the correlation between longitudinal responses of the same individual. The full likelihood approach is used to obtain maximum likelihood estimates of parameters of the log Lindley-binomial distribution and proposed models. Also, the generalized estimating equations approach, as an alternative nonlikelihood approach, is used to estimate models parameters. Some simulation studies are conducted to estimate parameters of the log Lindley-binomial distribution and the proposed models. These lead to the validity of the log Lindley-binomial distribution for analyzing related data. Finally, the proposed models are applied to analyze kidney function data of cancer patients to find the best-fitted model for analyzing the data.

Sequential Maximum Likelihood Estimation for the Squared Radial Ornstein-Uhlenbeck Process

In this paper, we study the properties of a sequential maximum likelihood estimator of the unknown parameter for the squared radial Ornstein-Uhlenbeck process. The estimator is proved to be closed, unbiased, normally distributed and strongly consistent. Lastly a simulation study is presented to illustrate the efficiency of the estimators.

Inference of progressively type-II censored competing risks data from Chen distribution with an application

In this paper, the estimation of unknown parameters of Chen distribution is considered under progressive Type-II censoring in the presence of competing failure causes. It is assumed that the latent causes of failures have independent Chen distributions with the common shape parameter, but different scale parameters. From a frequentist perspective, the maximum likelihood estimate of parameters via expectation–maximization (EM) algorithm is obtained. Also, the expected Fisher information matrix based on the missing information principle is computed. By using the obtained expected Fisher information matrix of the MLEs, asymptotic 95% confidence intervals for the parameters are constructed. We also apply the bootstrap methods (Bootstrap-p and Bootstrap-t) to construct confidence intervals. From Bayesian aspect, the Bayes estimates of the unknown parameters are computed by applying the Markov chain Monte Carlo (MCMC) procedure, the average length and coverage rate of credible intervals are also carried out. The Bayes inference is based on the squared error, LINEX, and general entropy loss functions. The performance of point estimators and confidence intervals is evaluated by a simulation study. Finally, a real-life example is considered for illustrative purposes.

Joint model for longitudinal mixture of normal and zero-inflated power series correlated responses Abbreviated title:mixture of normal and zero-inflated power series random-effects model

ABSTRACT In this paper, a joint model is presented for analyzing longitudinal continuous and count mixed responses. The frequency distribution of continuous longitudinal response variable for each subject at any time has a skewed and or multi-modal form. Then, a suitable finite mixture of normals is used as its distribution. It seems that the continuous response comes from several distinct sub-populations. The number of zeros of the count response is inflated. Also, a zero-inflated power series (ZIPS) distribution is applied as its distribution in order to model the count response. The correlation of longitudinal responses through time and that of mixed continuous and count responses are modeled by utilizing the random-effects vectors in the finite mixtures of regression (FMR) models. Further, a full likelihood-based approach is used to obtain the maximum likelihood estimates of parameters via the EM algorithm. Then, some simulation studies are performed for assessing the performance of the model. Additionally, an application is illustrated for joint analysis of the number of days during the last month that the individual drank alcohol, as well as the respondents’ weight. Finally, the two first times of the Americans Changing Lives survey are evaluated.

Finite mixtures of skew Laplace normal distributions with random skewness

In this paper, the shape mixtures of the skew Laplace normal (SMSLN) distribution is introduced as a flexible extension of the skew Laplace normal distribution which is also a heavy-tailed distribution. The SMSLN distribution includes an extra shape parameter, which controls skewness and kurtosis. Some distributional properties of this distribution are derived. Besides, we propose finite mixtures of SMSLN distributions to model both skewness and heavy-tailedness in heterogeneous data sets. The maximum likelihood estimators for parameters of interests are obtained via the expectation–maximization algorithm. We also give a simulation study and examine a real data example for the numerical illustration of proposed estimators.

A New Lifetime Exponential-X Family of Distributions with Applications to Reliability Data

Modeling reliability data with nonmonotone hazards is a prominent research topic that is quite rich and still growing rapidly. Many studies have suggested introducing new families of distributions to modify the Weibull distribution to model the nonmonotone hazards. In the present study, we propose a new family of distributions called a new lifetime exponential-X family. A special submodel of the proposed family called a new lifetime exponential-Weibull distribution suitable for modeling reliability data with bathtub-shaped hazard rates is discussed. The maximum-likelihood estimators of the model parameters are obtained. A brief Monte Carlo simulation study is conducted to evaluate the performance of these estimators. For illustrative purposes, two real applications from reliability engineering with bathtub-shaped hazard functions are analyzed. The practical applications show that the proposed model provides better fits than the other nonnested models.

Bayesian hierarchical methods for modeling electrical grid component failures

This paper presents a probabilistic modeling framework for component failures in electric power grids. Failure probabilities for grid components are often estimated using parametric models informed by past observations of failure. This work formulates a Bayesian hierarchical framework designed to integrate data and domain expertise to understand the failure properties of a regional power system, where variability in the expected performance of individual components gives rise to failure processes that are heterogeneous and uncertain. We use Bayesian methods to fit failure models to failure data generated in simulation. We test our algorithm by evaluating differences between the data-generating model, our Bayesian hierarchical model, and maximum likelihood parameter estimates. We evaluate how well each model can approximate the failure properties of individual components, and of the system overall. Finally, we define an upgrade policy for achieving targeted reductions in risk exposure, and compare the magnitude of upgrades recommended by each model.

Context-Aware Learning for Generative Models.

This work studies the class of algorithms for learning with side-information that emerges by extending generative models with embedded context-related variables. Using finite mixture models (FMMs) as the prototypical Bayesian network, we show that maximum-likelihood estimation (MLE) of parameters through expectation-maximization (EM) improves over the regular unsupervised case and can approach the performances of supervised learning, despite the absence of any explicit ground-truth data labeling. By direct application of the missing information principle (MIP), the algorithms' performances are proven to range between the conventional supervised and unsupervised MLE extremities proportionally to the information content of the contextual assistance provided. The acquired benefits regard higher estimation precision, smaller standard errors, faster convergence rates, and improved classification accuracy or regression fitness shown in various scenarios while also highlighting important properties and differences among the outlined situations. Applicability is showcased with three real-world unsupervised classification scenarios employing Gaussian mixture models. Importantly, we exemplify the natural extension of this methodology to any type of generative model by deriving an equivalent context-aware algorithm for variational autoencoders (VAs), thus broadening the spectrum of applicability to unsupervised deep learning with artificial neural networks. The latter is contrasted with a neural-symbolic algorithm exploiting side information.

Semiparametric Estimation of a Censored Regression Model Subject to Nonparametric Sample Selection

This study proposes a semiparametric estimation method for a censored regression model subject to nonparametric sample selection without the exclusion restriction. Consistency and asymptotic normality of the proposed estimator are established under mild regularity conditions. A Monte Carlo simulation study indicates that the estimator performs well in various designs and outperforms parametric maximum likelihood estimators. An empirical application to female smoking is provided to illustrate the usefulness of the estimator.