Maximum Likelihood Estimates Of Parameters Research Articles

Likelihood-based methods for the zero-one-two inflated Poisson model with applications to biomedicine

ABSTRACT To model count data with excess zeros, ones and twos, for the first time we introduce a so-called zero-one-two-inflated Poisson (ZOTIP) distribution, including the zero-inflated Poisson (ZIP) and the zero-and-one-inflated Poisson (ZOIP) distributions as two special cases. We establish three equivalent stochastic representations for the ZOTIP random variable to develop important distributional properties of the ZOTIP distribution. The Fisher scoring and expectation–maximization (EM) algorithms are derived to obtain the maximum likelihood estimates of parameters of interest. Bootstrap confidence intervals are also provided. Testing hypotheses are considered, simulation studies are conducted, and two real data sets are used to illustrate the proposed methods.

A joint model for mixed longitudinal k-category inflation ordinal and continuous responses

In this paper, a joint model is presented to analyse longitudinal continuous and ordinal mixed responses. The ordinal longitudinal response inflates in the kth category, e.g., in the first, middle, or last. A k-category-inflated proportional odds () model with the random effects vectors is applied for modelling the ordinal response. The correlation of longitudinal responses through time as well as two response variables are modelled by utilizing the random effects vectors in the joint model. 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 to assess the performance of the model. Simulations show that the model performs well in coverage rates, estimation bias, and consistency. However, there are minor restrictions on the nature of the covariates for the identifiability of the model. Additionally, an application of our model is illustrated for joint analysis of the child's reading recognition skill (read) and the child's antisocial behaviour (anti) of the Peabody Individual Achievement Test (PIAT) dataset. The four-time points of the PIAT survey are evaluated.

Two-stage sampling in the estimation of growth parameters and percentile norms: sample weights versus auxiliary variable estimation

BackgroundThe use of auxiliary variables with maximum likelihood parameter estimation for surveys that miss data by design is not a widespread approach, despite its documented improved efficiency over traditional approaches that deploy sampling weights. Although efficiency gains from the use of Normally distributed auxiliary variables in a model have been recorded in the literature, little is known about the effects of non-Normal auxiliary variables in the parameter estimation.MethodsWe simulate growth data to mimic SCALES, a two-stage survey of language development with a screening phase (stage one) for which data are observed for the whole sample and an intensive assessments phase (stage two), for which data are observed for a sub-sample, selected using stratified random sampling. In the simulation, we allow a fully observed Poisson distributed stratification criterion to be correlated with the partially observed model responses and develop five generalised structural equation growth models that host the auxiliary information from this criterion. We compare these models with each other and with a weighted growth model in terms of bias, efficiency, and coverage. We finally apply our best performing model to SCALES data and show how to obtain growth parameters and population norms.ResultsParameter estimation from a model that incorporates a non-Normal auxiliary variable is unbiased and more efficient than its weighted counterpart. The auxiliary variable method is capable of producing efficient population percentile norms and velocities.ConclusionsThe deployment of a fully observed variable that dominates the selection of the sample and correlates strongly with the incomplete variable of interest appears beneficial for the estimation process.

Actual error rates in linear discrimination of spatial Gaussian data in terms of semivariograms

The novel approach for classifying spatial Gaussian data based on Bayes discriminant functions in terms of semivariograms has been developed. We derived the closed-form expressions for the Maximum Likelihood estimator of regression parameters and the actual error rate (AER). Results are illustrated through simulation study of spatial Gaussian data sampled on a regular two-dimensional unit spacing lattice endowed with neighborhood structure based on Euclidean distance between sites. The accuracy of the derived AER for various values of spatial dependence parameters, Mahalanobis distances and neighborhood sizes are evaluated and compared.

A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations

Abstract In this paper, we introduce a new two-parameter lifetime distribution which is called extended Half-Logistic (EHL) distribution. Theoretical properties of this model including the hazard function, quantile function, asymptotic, extreme value, moments, conditional moments, mean residual life, mean past lifetime, residual entropy, cumulative residual entropy and order statistics are derived and studied in details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, two real data sets are illustration the purposes.

Estimating basis functions in massive fields under the spatial mixed effects model

AbstractSpatial prediction is commonly achieved under the assumption of a Gaussian random field by obtaining maximum likelihood estimates of parameters, and then using the kriging equations to arrive at predicted values. For massive datasets, fixed rank kriging using the expectation–maximization algorithm for estimation has been proposed as an alternative to the usual but computationally prohibitive kriging method. The method reduces computation cost of estimation by redefining the spatial process as a linear combination of basis functions and spatial random effects. A disadvantage of this method is that it imposes constraints on the relationship between the observed locations and the knots. We develop an alternative method that utilizes the spatial mixed effects model, but allows for additional flexibility by estimating the range of the spatial dependence between the observations and the knots via an alternating expectation conditional maximization algorithm. Experiments show that our methodology improves estimation without sacrificing prediction accuracy while also minimizing the additional computational burden of extra parameter estimation. The methodology is applied to a temperature dataset archived by the United States National Climate Data Center, with improved results over previous methodology.

On Progressive Censored Competing Risks Data: Real Data Application and Simulation Study

Competing risks are frequently overlooked, and the event of interest is analyzed with conventional statistical techniques. In this article, we consider the analysis of bi-causes of failure in the context of competing risk models using the extension of the exponential distribution under progressive Type-II censoring. Maximum likelihood estimates for the unknown parameters via the expectation-maximization algorithm are obtained. Moreover, the Bayes estimates of the unknown parameters are approximated using Tierney-Kadane and MCMC techniques. Interval estimates using Bayesian and classical techniques are also considered. Two real data sets are investigated to illustrate the different estimation methods, and to compare the suggested model with Weibull distribution. Furthermore, the estimation methods are compared through a comprehensive simulation study.

A New Weighted Skew Normal Model

Weighted sampling is a useful method for constructing flexible models and analyzing data sets. In this paper, a new weighted distribution of skew normal is introduced with four parameters. The proposed model is a generalized version of several distributions, such as normal, bimodal normal, skew normal and skew bimodal normal-normal. This weighted model is form-invariant under proposed weight function. The basic characteristics of the model are expressed. A method has been used to generate data from the model. The maximum likelihood estimations of parameters are given and evaluated using simulation study. The model is fitted to the three real data sets. The advantage of the proposed model has been shown on the rival distributions using appropriate criteria.

A new approach using the genetic algorithm for parameter estimation in multiple linear regression with long-tailed symmetric distributed error terms: An application to the Covid-19 data

Maximum likelihood (ML) estimators of the model parameters in multiple linear regression are obtained using genetic algorithm (GA) when the distribution of the error terms is long-tailed symmetric. We compare the efficiencies of the ML estimators obtained using GA with the corresponding ML estimators obtained using other iterative techniques via an extensive Monte Carlo simulation study. Robust confidence intervals based on modified ML estimators are used as the search space in GA. Our simulation study shows that GA outperforms traditional algorithms in most cases. Therefore, we suggest using GA to obtain the ML estimates of the multiple linear regression model parameters when the distribution of the error terms is LTS. Finally, real data of the Covid-19 pandemic, a global health crisis in early 2020, is presented for illustrative purposes.

Classical and Bayesian inference approaches for the exponentiated discrete Weibull model with censored data and a cure fraction

In this paper, we introduce maximum likelihood and Bayesian parameter estimation for the exponentiated discrete Weibull (EDW) distribution in presence of randomly right censored data. We also consider the inclusion of a cure fraction in the model. The performance of the maximum likelihood estimation approach is assessed by conducting an extensive simulation study with different sample sizes and different values for the parameters of the EDW distribution. The usefuness of the proposed model is illustrated with two examples considering real data sets.

Matrix Normal Cluster-Weighted Models

Finite mixtures of regressions with fixed covariates are a commonly used model-based clustering methodology to deal with regression data. However, they assume assignment independence, i.e., the allocation of data points to the clusters is made independently of the distribution of the covariates. To take into account the latter aspect, finite mixtures of regressions with random covariates, also known as cluster-weighted models (CWMs), have been proposed in the univariate and multivariate literature. In this paper, the CWM is extended to matrix data, e.g., those data where a set of variables are simultaneously observed at different time points or locations. Specifically, the cluster-specific marginal distribution of the covariates and the cluster-specific conditional distribution of the responses given the covariates are assumed to be matrix normal. Maximum likelihood parameter estimates are derived using an expectation-conditional maximization algorithm. Parameter recovery, classification assessment, and the capability of the Bayesian information criterion to detect the underlying groups are investigated using simulated data. Finally, two real data applications concerning educational indicators and the Italian non-life insurance market are presented.

Semiparametric inference for the scale-mixture of normal partial linear regression model with censored data

In the censored data exploration, the classical linear regression model which assumes normally distributed random errors is perhaps one of the commonly used frameworks. However, practical studies have often criticized the classical linear regression model because of its sensitivity to departure from the normality and partial nonlinearity. This paper proposes to solve these potential issues simultaneously in the context of the partial linear regression model by assuming that the random errors follow a scale-mixture of normal (SMN) family of distributions. The postulated method allows us to model data with great flexibility, accommodating heavy tails and outliers. By implementing the B-spline approximation and using the convenient hierarchical representation of the SMN distributions, a computationally analytical EM-type algorithm is developed for obtaining maximum likelihood (ML) parameter estimates. Various simulation studies are conducted to investigate the finite sample properties, as well as the robustness of the model in dealing with the heavy tails distributed datasets. Real-world data examples are finally analyzed for illustrating the usefulness of the proposed methodology.

Modeling learning in knowledge space theory through bivariate Markov processes

Bivariate Markov processes (BMPs) described by Ephraim and Mark (2012) consist of a pair of stochastic processes in the continuous time, one observable and the other latent, that are jointly Markov. In the present article, the navigation behavior and the learning process of a user of a web-based tutoring system are jointly modeled as BMPs constrained by assumptions that are coherent with the concepts of competence-based knowledge space theory. Such constraints are expressed as formal assumptions about the web-based system and about the nature of the learning process. Scenarios are considered where the observed process is the navigation of an individual through the pages of an intelligent tutoring system, whereas the latent learning process consists of transitions among states in a competence structure. The approach seems to be rather general and flexible in modeling learning scenarios with different assumptions. As an example, BMP models are developed for some exemplary scenarios. Maximum likelihood parameter estimation via expectation–maximization algorithm is presented. The results of a simulation study showed that the parameter values are well-recovered by the estimation algorithm. The results of the application of a bivariate Markov model to the real data of students navigating the intelligent tutoring system Stat-Knowlab showed that the proposed approach provides useful insight into students’ learning processes.

Quantile modeling through multivariate log-normal/independent linear regression models with application to newborn data.

In this article, we propose and study the class of multivariate log-normal/independent distributions and linear regression models based on this class. The class of multivariate log-normal/independent distributions is very attractive for robust statistical modeling because it includes several heavy-tailed distributions suitable for modeling correlated multivariate positive data that are skewed and possibly heavy-tailed. Besides, expectation-maximization (EM)-type algorithms can be easily implemented for maximum likelihood estimation. We model the relationship between quantiles of the response variables and a set of explanatory variables, compute the maximum likelihood estimates of parameters through EM-type algorithms, and evaluate the model fitting based on Mahalanobis-type distances. The satisfactory performance of the quantile estimation is verified by simulation studies. An application to newborn data is presented anddiscussed.

Modeling of Extreme Crude Oil Price using the Generalized Pareto Distribution: Brent and West Texas Benchmark Price

Background and objective: Crude oil is an essential commodity in many countries of the world. This work studies the risk involved in the extreme crude oil price, using the daily crude oil price of the Brent and the West Texas benchmark from year 1990 to 2019. Materials and methods: The Peak Over Threshold (POT) approach of the Generalized Pareto Distribution (GPD) was used to model the extreme crude oil price while the value at risk and the expected shortfall was used to quantify the risk involved in extreme price of crude oil. The GPD, using the Q-Q plot was found to be a good model for the extreme values of the crude oil price. Results: The Value at Risk (VaR) and the Expected Shortfall (ES) calculated at 90%, 95% and 99% with the Maximum Likelihood estimators of GPD parameters and the threshold values were found to decrease with increase in quantile for both benchmark. This shows that risk involved in extreme crude oil price will be borne only by the investors and public. Conclusion: It was also found that the VaR and ES of the Brent are higher than that of West Texas. This implies that it is safer to invest in West Texas crude oil.

Statistical image watermarking using local RHFMs magnitudes and beta exponential distribution

The imperceptibility, robustness and data payload are widely considered as the three main properties vital for any image watermarking systems. They are complimentary to each other and hence challenging to attain the right balance between them. The statistical model-based multiplicative watermarking is an effective way to achieve the tradeoff among imperceptibility, robustness and data payload. Radial harmonic Fourier moments (RHFMs) is a strong tool in image processing, which has many advantages, such as lower noise sensitivity, powerful image description ability and geometric invariance feature. In this paper, we propose a new statistical image watermarking scheme using local RHFMs magnitudes and Beta exponential distribution model. Our image watermarking scheme consists of two parts, namely, embedding and extraction. In the embedding process, we divide the host image into no-overlapping blocks and compute the local RHFMs of image blocks, then insert the watermark signal into the robust local RHFMs magnitudes through multiplicative approach. In the extraction phase, robust local RHFMs magnitudes are firstly modeled by employing the Beta exponential distribution, where the statistical properties of local RHFMs magnitudes are captured accurately. Then the modified maximum likelihood parameter estimation (MMLE) approach is introduced to estimate the statistical parameters of Beta exponential distribution model. And finally an image watermark decoder for multiplicative watermarking is developed using Beta exponential distribution and maximum likelihood decision criterion. Experimental results on some test images and comparison with well-known existing methods demonstrate the efficacy and superiority of the proposed statistical image watermarking.

An Analytical EM Algorithm for Sub-Gaussian Vectors

The area in which a multivariate α-stable distribution could be applied is vast; however, a lack of parameter estimation methods and theoretical limitations diminish its potential. Traditionally, the maximum likelihood estimation of parameters has been considered using a representation of the multivariate stable vector through a multivariate normal vector and an α-stable subordinator. This paper introduces an analytical expectation maximization (EM) algorithm for the estimation of parameters of symmetric multivariate α-stable random variables. Our numerical results show that the convergence of the proposed algorithm is much faster than that of existing algorithms. Moreover, the likelihood ratio (goodness-of-fit) test for a multivariate α-stable distribution was implemented. Empirical examples with simulated and real world (stocks, AIS and cryptocurrencies) data showed that the likelihood ratio test can be useful for assessing goodness-of-fit.

A Latent-Class Model for Clustering Incomplete Linear and Circular Data in Marine Studies

Identification of representative regimes of wave height and direction under different wind conditions is complicated by issues that relate to the specification of the joint distribution of variables that are defined on linear and circular supports and the occurrence of missing values. We take a latent-class approach and jointly model wave and wind data by a finite mixture of conditionally independent Gamma and von Mises distributions. Maximum-likelihood estimates of parameters are obtained by exploiting a suitable EM algorithm that allows for missing data. The proposed model is validated on hourly marine data obtained from a buoy and two tide gauges in the Adriatic Sea.

A plotless density estimator with a Norton-Rice distribution for ordered distances

A Norton-Rice distribution (NRD) is a versatile, flexible distribution for k ordered distances from a random location to the k nearest objects. In a context of plotless density estimation (PDE) with n randomly chosen sample locations, and distances measured to the k = 6 nearest objects, the NRD provided a good fit to distance data from seven populations with a census of forest tree stem locations. More importantly, the three parameters of a NRD followed a simple trend with the order (1, …, 6) of observed distances. The trend is quantified and exploited in a proposed new PDE through a joint maximum likelihood estimation of the NRD parameters expressed as a functions of distance order. In simulated probability sampling from the seven populations, the proposed PDE had the lowest overall bias with a good performance potential when compared to three alternative PDEs. However, absolute bias increased by 0.8 percentage points when sample size decreased from 20 to 10. In terms of root mean squared error (RMSE), the new proposed estimator was at par with an estimator published in Ecology when this study was wrapping up, but otherwise superior to the remaining two investigated PDEs. Coverage of nominal 95% confidence intervals averaged 0.94 for the new proposed estimators and 0.90, 0.96, and 0.90 for the comparison PDEs. Despite tangible improvements in PDEs over the last decades, a globally least biased PDE remains elusive.

Likelihood analysis and stochastic EM algorithm for left truncated right censored data and associated model selection from the Lehmann family of life distributions

The Lehmann family of distributions includes Weibull, Gompertz, and Lomax models as special cases, all of which are quite useful for modeling lifetime data. Analyses of left truncated right censored data from the Lehmann family of distributions are carried out in this article. In particular, the special cases of Weibull, Gompertz, and Lomax distributions are considered. Maximum likelihood estimates of the model parameters are obtained. The steps of the stochastic expectation maximization algorithm are developed in this context. Asymptotic confidence intervals for the model parameters are constructed using the missing information principle and two parametric bootstrap approaches. Through extensive Monte Carlo simulations, performances of the inferential methods are examined. The effect of model misspecification is also assessed within this family of distributions through simulations. As it is quite important to select a suitable model for a given data, a study of model selection based on left truncated right censored data is carried out through extensive Monte Carlo simulations. Two examples based on real datasets are provided for illustrating the models and methods of inference discussed here.