Existence Of Maximum Likelihood Estimates Research Articles

Statistical inference for Gompertz distribution under adaptive type-II progressive hybrid censoring

Gompertz distribution is a significant and commonly used lifetime distribution, which plays an important role in reliability engineering. In this paper, we study the statistical inference of Gompertz distribution based on adaptive Type-II hybrid progressive censored schemes. From the perspective of frequentist, we derive the point estimations through the method of maximum likelihood estimation (MLE) and the existence of MLE is proved. Besides MLE, we propose the stochastic EM algorithm to reduce complexity and simplify computing. We also apply the method of Bootstraps (Bootstrap-p and Bootstrap-t) to construct confidence intervals. From Bayesian aspect, the Bayes estimates of the unknown parameters are evaluated by applying the MCMC method, the average length and coverage rate of credible intervals are also carried out. The Bayes inference is based on the squared error loss function and LINEX loss function. Furthermore, a numerical simulation is conducted to assess the performance of the proposed methods. Finally, a real-life example is considered to illustrate the application and development of the inference methods. In summary, the Bayesian method seems to perform the best among all approaches, while other approaches also present different advantages.

Estimation for Kies distribution with generalized progressive hybrid censoring under partially observed competing risks model

In this paper, estimation of a competing risks model is discussed with partially observed modes of failure. When latent lifetimes of competing risks follow the Kies distribution under a generalized progressive hybrid censoring, estimation of the unknown parameters is explored under non-order and order restrictions via classical and Bayesian approaches, respectively. The uniqueness and existence of maximum likelihood estimators of parameters are established, and subsequently interval estimators are also derived. Bayesian estimates as well as highest posterior density credible intervals are developed for the model parameters. In addition, when extra order information for the competing risks parameters is available, classical and Bayesian estimates are also established. Extensive simulation studies are conducted to investigate the performance of different methods, and a numerical example is also presented for illustrative purposes.

Hierarchical Aitchison–Silvey models for incomplete binary sample spaces

Multivariate sample spaces may be incomplete Cartesian products, when certain combinations of the categories of the variables are not possible. Traditional log-linear models, which generalize independence and conditional independence, do not apply in such cases, as they may associate positive probabilities with the non-existing cells. To describe the association structure in incomplete sample spaces, this paper develops a class of hierarchical multiplicative models which are defined by setting certain non-homogeneous generalized odds ratios equal to one and are named after Aitchison and Silvey who were among the first to consider such ratios. These models are curved exponential families that do not contain an overall effect and, from an algebraic perspective, are non-homogeneous toric ideals. The relationship of this model class with log-linear models and quasi log-linear models is studied in detail in terms of both statistics and algebraic geometry. The existence of maximum likelihood estimates and their properties, as well as the relevant algorithms are also discussed.

On the existence of posterior mean for Bayesian logistic regression

Abstract Existence conditions for posterior mean of Bayesian logistic regression depend on both chosen prior distributions and a likelihood function. In logistic regression, different patterns of data points can lead to finite maximum likelihood estimates (MLE) or infinite MLE of the regression coefficients. Albert and Anderson [On the existence of maximum likelihood estimates in logistic regression models, Biometrika 71 1984, 1, 1–10] gave definitions of different types of data points, which are complete separation, quasicomplete separation and overlap. Conditions for the existence of the MLE for logistic regression models were proposed under different types of data points. Based on these conditions, we propose the necessary and sufficient conditions for the existence of posterior mean under different choices of prior distributions. In this paper, a general wide class of priors, which are informative priors and non-informative priors having proper distributions and improper distributions, are considered for the existence of posterior mean. In addition, necessary and sufficient conditions for the existence of posterior mean for an individual coefficient is also proposed.

A model for analyzing clustered occurrence data.

Spatial or temporal clustering commonly arises in various biological and ecological applications, for example, species or communities may cluster in groups. In this paper, we develop a new clustered occurrence data model where presence-absence data are modeled under a multivariate negative binomial framework. We account for spatial or temporal clustering by introducing a community parameter in the model that controls the strength of dependence between observations thereby enhancing the estimation of the mean and dispersion parameters. We provide conditions to show the existence of maximum likelihood estimates when cluster sizes are homogeneous and equal to 2 or 3 and consider a composite likelihood approach that allows for additional robustness and flexibility in fitting for clustered occurrence data. The proposed method is evaluated in a simulation study and demonstrated using forest plot data from the Center for Tropical Forest Science. Finally, we present several examples using multiple visit occupancy data to illustrate the difference between the proposed model and those of N-mixturemodels.

Maximum Likelihood Estimation for Matrix Normal Models via Quiver Representations

We study the log-likelihood function and maximum likelihood estimate (MLE) for the matrix normal model for both real and complex models. We describe the exact number of samples needed to achieve (almost surely) three conditions, namely a bounded log-likelihood function, existence of MLEs, and uniqueness of MLEs. As a consequence, we observe that almost sure boundedness of the log-likelihood function guarantees almost sure existence of an MLE, thereby proving a conjecture of Drton, Kuriki, and Hoff [Existence and Uniqueness of the Kronecker Covariance MLE, preprint, arXiv:2003.06024, 2020]. The main tools we use are from the theory of quiver representations, in particular, results of Kac, King, and Schofield on canonical decomposition and stability.

Common Methodological Challenges Encountered With Multiple Systems Estimation Studies

Multiple systems estimation refers to a class of inference procedures that are commonly used to estimate the size of hidden populations based on administrative lists. In this paper we discuss some of the common challenges encountered in such studies. In particular, we summarize theoretical issues relating to the existence of maximum likelihood estimators, model identifiability, and parameter redundancy when there is sparse overlap among the lists. We also discuss techniques for matching records when there are no unique identifiers, exploiting covariate information to improve estimation, and addressing missing data. We offer suggestions for remedial actions when these issues/challenges manifest. The corresponding R coding packages that can assist with the analyses of multiple systems estimation data sets are also discussed.

The existence of maximum likelihood estimate in high-dimensional binary response generalized linear models

Motivated by recent works on the high-dimensional logistic regression, we establish that the existence of the maximum likelihood estimate exhibits a phase transition for a wide range of generalized linear models with binary outcome and elliptical covariates. This extends a previous result of Candès and Sur who proved the phase transition for the logistic regression with Gaussian covariates. Our result reveals a rich structure in the phase transition phenomenon, which is simply overlooked by Gaussianity. The main tools for deriving the result are data separation, convex geometry and stochastic approximation. We also conduct simulation studies to corroborate our theoretical findings, and explore other features of the problem.

On the existence of maximum likelihood estimates for weighted logistic regression

ABSTRACTThe problems of existence and uniqueness of maximum likelihood estimates for logistic regression were completely solved by Silvapulle in 1981 and Albert and Anderson in 1984. In this paper, we extend the well-known results by Silvapulle and by Albert and Anderson to weighted logistic regression. We analytically prove the equivalence between the overlap condition used by Albert and Anderson and that used by Silvapulle. We show that the maximum likelihood estimate of weighted logistic regression does not exist if there is a complete separation or a quasicomplete separation of the data points, and exists and is unique if there is an overlap of data points. Our proofs and results for weighted logistic apply to unweighted logistic regression.

Ranking in the generalized Bradley–Terry models when the strong connection condition fails

ABSTRACTFor non balanced paired comparisons, a wide variety of ranking methods have been proposed. One of the best popular methods is the Bradley–Terry model in which the ranking of a set of objects is decided by the maximum likelihood estimates (MLEs) of merits parameters. However, the existence of MLE for the Bradley–Terry model and its generalized models to allow for tied observation or home-field advantage or both to occur, crucially depends on the strong connection condition on the directed graph constructed by a win–loss matrix. When this condition fails, the MLE does not exist and hence there is no solution of ranking. In this paper, we propose an improved version of the ϵ singular perturbation proposed by Conner and Grant (2000), to address this problem and to extend it to the generalized Bradley–Terry models. Some necessary and sufficient conditions for the existence and uniqueness of the penalized MLEs for these generalized Bradley–Terry-ϵ models are derived. Numerical studies show that the ranking is robust to the different ϵ. We apply the proposed methods to the data of the 2008 NFL regular season.

Separation Issues and Possible Solutions: Part I – Systematic Literature Review on Logistic Models ‐ Part II – Comparison of different methods for separation in logistic regression

Separation Issues and Possible Solutions: Part I – Systematic Literature Review on Logistic Models ‐ Part II – Comparison of different methods for separation in logistic regression

A study of moments and likelihood estimators of the odd Weibull distribution

The odd Weibull distribution is a three-parameter generalization of the Weibull and the inverse Weibull distributions having rich density and hazard shapes for modeling lifetime data. This paper explored the odd Weibull parameter regions having finite moments and examined the relation to some well-known distributions based on skewness and kurtosis functions. The existence of maximum likelihood estimators have shown with complete data for any sample size. The proof for the uniqueness of these estimators is given only when the absolute value of the second shape parameter is between zero and one. Furthermore, elements of the Fisher information matrix are obtained based on complete data using a single integral representation which have shown to exist for any parameter values. The performance of the odd Weibull distribution over various density and hazard shapes is compared with generalized gamma distribution using two different test statistics. Finally, analysis of two data sets has been performed for illustrative purposes.

On the existence of maximum likelihood estimates for presence‐only data

Summary Presence‐only data can be used to determine resource selection and estimate a species’ distribution. Maximum likelihood is a common parameter estimation method used for species distribution models. Maximum likelihood estimates, however, do not always exist for a commonly used species distribution model – the Poisson point process. We demonstrate the issue with conventional maximum likelihood mathematically, using a data example, and a simulation experiment and show alternative estimation methods. We found that when habitat preferences are strong or the number of presence‐only locations is small, by chance, maximum likelihood coefficient estimates for the Poisson point process model may not exist. We found that several alternative estimation methods can produce reliable estimates, but results will depend on the chosen method. It is important to identify conditions for which maximum likelihood estimates are unlikely to be identifiable from presence‐only data. In data sets where the maximum likelihood estimates do not exist, penalized likelihood and Bayesian methods will produce coefficient estimates, but these are sensitive to the choice of estimation procedure and prior or penalty term. When sample size is small or it is thought that habitat preferences are strong, we propose a suite of estimation procedures researchers can consider using.

Meta-analysis of general step-stress experiments under repeated Type-II censoring

A multi-sample model for general step-stress experiments based on sequential order statistics is proposed and analyzed, where the numbers of observations under each stress level are pre-specified. Contrary to common step-stress approaches, an arbitrary absolutely continuous lifetime distribution can be chosen, and, due to the experimental design of repeated Type-II censoring, the existence of maximum likelihood estimators for the parameters of interest is always guaranteed. Estimation and testing of the model parameters associated with the stress levels is examined, and inference under a log-linear link function is considered. Simulation studies illustrate some of the statistical procedures.

Generalized exponential records: existence of maximum likelihood estimates and its comparison with transforming based estimates

In this paper, and based on records of a sequence of iid random variables from the generalized exponential distribution, we consider the problem of the existence of the maximum likelihood estimates of the shape and scale parameters. Existence and uniqueness of the MLE’s are proved. Different transforming based estimates and confidence intervals of these parameters are then derived. The performances of the so obtained estimates and confidence intervals are compared through an extensive numerical simulation study. Analysis of a real data set has also been presented for illustrative purposes.

Existence of MLE and posteriors for a recognition-memory model

Necessary and sufficient conditions are developed for the existence of the maximum likelihood estimate (MLE) for a recognition-memory model. The propriety of posteriors is shown for a class of bounded priors. Under a constant prior, an easy-to-implement Gibbs sampler is developed and illustrated via a real data set.

Relational models for contingency tables

The paper considers general multiplicative models for complete and incomplete contingency tables that generalize log-linear and several other models and are entirely coordinate free. Sufficient conditions for the existence of maximum likelihood estimates under these models are given, and it is shown that the usual equivalence between multinomial and Poisson likelihoods holds if and only if an overall effect is present in the model. If such an effect is not assumed, the model becomes a curved exponential family and a related mixed parameterization is given that relies on non-homogeneous odds ratios. Several examples are presented to illustrate the properties and use of such models.

On the existence of maximum likelihood estimates in random effects models for clustered multivariate binary data

We consider a class of random effects models for clustered multivariate binary data based on the threshold crossing technique of a latent random vector. Components of this latent vector are assumed to have a Laird–Ware structure. However, in place of their Gaussian assumptions, any specified class of multivariate distribution is allowed for the random effects, and the error vector is allowed to have any strictly positive pdf. A well known member of this class of models is the multivariate probit model with random effects. We investigate sufficient and necessary conditions for the existence of maximum likelihood estimates for the location and the association parameters. Implications of our results are illustrated through some hypothetical examples.

Maximum likelihood inference for the Cox regression model with applications to missing covariates

In this paper, we carry out an in-depth theoretical investigation for existence of maximum likelihood estimates for the Cox model [D.R. Cox, Regression models and life tables (with discussion), Journal of the Royal Statistical Society, Series B 34 (1972) 187–220; D.R. Cox, Partial likelihood, Biometrika 62 (1975) 269–276] both in the full data setting as well as in the presence of missing covariate data. The main motivation for this work arises from missing data problems, where models can easily become difficult to estimate with certain missing data configurations or large missing data fractions. We establish necessary and sufficient conditions for existence of the maximum partial likelihood estimate (MPLE) for completely observed data (i.e., no missing data) settings as well as sufficient conditions for existence of the maximum likelihood estimate (MLE) for survival data with missing covariates via a profile likelihood method. Several theorems are given to establish these conditions. A real dataset from a cancer clinical trial is presented to further illustrate the proposed methodology.

Estimation in polytomous logistic model: Comparison of methods

The logistic regression model is a powerful method for modeling the relationship between a categorical variable and a set of explanatory variables. In practice, however, the existence of maximum likelihood estimates is known to be dependent on the data configuration. In fact, the Maximum Likelihood Estimators (MLE) of unknown parameters exists if, and only if, there is data overlapping. The Hidden Logistic Regression (HLR) is an alternative model under which the observed response is related to the unobservable response. The Maximum Estimated Likelihood (MEL) method is also proposed, once it is immune to the complete or quasi-complete separation of data. The Principal Component Logistic Regression (PCLR) model is useful to reduce the number of dimensions of a logistic regression model with continuous covariates avoiding multicollinearity. In this paper we present an extension of the HLR and PCLR models as means for the solution of problems with polytomous responses. The main purpose is to compare the classificatory performance obtained by the models mentioned above with those of the Classical Logistic Regression (CLR) and Individualized Logistic regression (ILR) models, in the case of polytomous responses. The purpose is to propose an alternative approach for the parameter estimation problem in polytomous logistic models when the data groups are completely separated. Simulations results resulting from databases taken from the literature show that the proposed approach is feasible.