Maximum Likelihood Estimates Of Parameters Research Articles

Folded normal regression models with applications in biomedicine

In biomedical studies, a difference or deviation is usually measured and only the magnitude is recorded but the algebraic sign of the data is irretrievably lost, the resulting observed variable no longer follows a normal distribution, rather it follows a folded normal (FN) distribution. More importantly, the FN distribution could be used to fit data sets with the following two characteristics: (i) The density curve is similar to the normal density but truncated somewhere; (ii) The density curve of the truncated side is significantly higher than that of the other side. There are several issues on the statistical inferences with the FN distribution which are not (well) addressed in the existing literature. In this paper, starting from the stochastic representation, we develop a new expectation–maximization (EM) algorithm to calculate the maximum likelihood estimates of parameters in both FN distribution and the FN regression models. The EM structure can also facilitate the Bayesian inferences about the FN distribution and the FN regression models. Extensions to the generalized FN distribution are provided. Simulation studies are conducted to assess the estimation performances for the FN distribution and the FN regression model. Two real data sets are analyzed to illustrate the proposed methods.

Inferences for two Lindley populations based on joint progressive type-II censored data

The joint censoring scheme is of great importance when the motive of study is to compare the relative merits of products in relation of their service times. In last few years, progressive censoring received considerable attention in order to save cost and time of the experiment. This paper deals with inferences for Lindley populations, when joint progressive type-II censoring scheme is applied on two samples in a joint manner. Here, the maximum likelihood estimators of parameters are derived along with their associated confidence intervals which are dependent on the Fisher’s information matrix. The boot-p and boot-t confidence intervals are also obtained. Bayes estimators of the unknown parameters assuming gamma priors are calculated. The concept of importance sampling technique and Gibbs sampling technique are used as the Bayes estimators cannot be calculated in closed form. HPD credible intervals are also constructed. A Monte Carlo simulation study is performed to measure the efficiency of the estimates. A real data set is given for illustrative purpose. Finally some criteria for an optimum censoring scheme are discussed.

Spatial auto-correlation and auto-regressive models estimation from sample survey data.

Maximum likelihood estimation of the model parameters for a spatial population based on data collected from a survey sample is usually straightforward when sampling and non-response are both non-informative, since the model can then usually be fitted using the available sample data, and no allowance is necessary for the fact that only a part of the population has been observed. Although for many regression models this naive strategy yields consistent estimates, this is not the case for some models, such as spatial auto-regressive models. In this paper, we show that for a broad class of such models, a maximum marginal likelihood approach that uses both sample and population data leads to more efficient estimates since it uses spatial information from sampled as well as non-sampled units. Extensive simulation experiments based on two well-known data sets are used to assess the impact of the spatial sampling design, the auto-correlation parameter and the sample size on the performance of this approach. When compared to some widely used methods that use only sample data, the results from these experiments show that the maximum marginal likelihood approach is much more precise.

Gaussian copula joint models to analysis mixed correlated longitudinal count and continuous responses

Gaussian Copula joint models for analyzing mixed correlated longitudinal continuous and count responses with random effects are presented where the count response is defined by a latent variable approach, and its distribution is a member of the power series family of distributions. A copula-based joint model is proposed that accounts for associations between count and continuous responses. A full likelihood-based inference method for estimation is used by which maximum likelihood estimation of parameters is obtained. To illustrate the utility of the models, some simulation studies are performed. Finally, the proposed models are motivated by analyzing a medical data set where the correlated responses are the number of joint damaged obtained from the severity of osteoporosis (count response) and Body Mass Index (continuous response). Effects of some covariates on responses are investigated simultaneously. Besides, identifiability of parameters, detection of outlines by examining residuals and sensitivity analyses are fully investigated.

The Weibull-G Poisson Family for Analyzing Lifetime Data

We study a new family of distributions defined by the minimum of the Poissonrandom number of independent identically distributed random variables having a general Weibull-G distribution (see Bourguignon et al. (2014)). Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics, reliability and entropies are derived. Maximum likelihood estimation of the model parameters is investigated. Three special models of the new family are discussed. We perform three applications to real data sets to show the potentiality of theproposed family.

The Generalized Inflection S-Shaped Software Reliability Growth Model

In this paper, the new generalized inflection S-shaped software reliability growth model is proposed. It is a very flexible finite failure Poisson process that possesses two distinguishing features: 1) includes as special cases the popular inflection S-shaped model, Goel generalized nonhomogenous Poisson process, and Goel–Okumoto model and 2) differently than these latter models, allows for modeling nonmonotonic failure rate per fault functions. The properties of the generalized inflection S-shaped model are discussed and intuitive arguments are provided to justify its structure. Maximum-likelihood estimators of model parameters are formulated and their properties are summarized. A special attention is devoted to the nonexistence issue of maximum-likelihood estimates. The problem of estimating the optimal release time of a software product is also addressed. Affordability and flexibility of the proposed model are demonstrated via four applicative examples, based on real sets of software reliability data. Attained results show that the generalized inflection S-shaped model provides outputs that may significantly differ from those provided by the models nested within it. As a side result, the developed examples also show that the nonexistence issue of maximum-likelihood estimates, in the case of finite failure Poisson processes, cannot be considered an oddity and that its occurrence is not necessarily related to model complexity.

Parameter Estimation of the Negative Binomial—New Weighted Lindley Distribution by the Method of Maximum Likelihood

In this article we investigate the parameter estimation of the Negative Binomial—New Weighted Lindley distribution. We are interested in the maximum likelihood method because it provides estimators with many superior properties, such as minimum variance and asymptotically unbiased estimators. The simulation study is performed in order to investigate the accuracy of the maximum likelihood estimators of the parameters of the Negative Binomial—New Weighted Lindley distribution.

Generalized AIC and chi-squared statistics for path models consistent with directed acyclic graphs.

We explain how to obtain a generalized maximum-likelihood chi-square statistic, , and a full-model Akaike Information Criterion (AIC) statistic for piecewise structural equation modeling (SEM); that is, structural equations without latent variables whose causal topology can be represented as a directed acyclic graph (DAG). The full piecewise SEM is decomposed into submodels as a Markov network, each of which can have different distributional assumptions or functional links and that can be modeled by any method that produces maximum-likelihood parameter estimates. The generalized is a function of the difference in the maximum likelihoods of the model and its saturated equivalent and the full-model AIC is calculated by summing the AIC statistics of each of the submodels.

Copula models for one-shot device testing data with correlated failure modes

Copula models have become one of the most popular tools, especially in finance and insurance, for modeling multivariate distributions in the past few decades, and they have recently received increasing attention for data analysis in reliability engineering and survival analysis. This paper considers two Archimedean copula models — the Gumbel-Hougaard copula and Frank copula — for analyzing one-shot device data with two correlated failure modes, which are collected from constant-stress accelerated life tests. A one-shot device is a unit that cannot be used again after a test, e.g., munitions, rockets, and automobile airbags. Only either left- or right-censored data are collected instead of the actual lifetimes of the devices under test. With the aid of Kendall’s tau correlation coefficient, initial values of the dependence parameter for the copula models are presented to determine maximum likelihood estimates of model parameters through a numerical approach. Furthermore, the proposed model can be used to examine whether the correlation between times to failure modes changes over stress levels. Real data from a survival experiment are also re-analyzed to illustrate the proposed methods.

Joint Maximum Likelihood Estimation of Motion and T1 Parameters from Magnetic Resonance Images in a Super-resolution Framework: a Simulation Study

Joint Maximum Likelihood Estimation of Motion and T1 Parameters from Magnetic Resonance Images in a Super-resolution Framework: a Simulation Study

Numerical characteristics and parameter estimation of finite mixed generalized normal distribution

In this paper, a univariate finite mixed generalized normal distribution (MixGND) is proposed. First, we derive some probabilistic properties including hazard rate function, characteristic function, kurtosis and skewness, for a mixture of two generalized normal distributions. In particular, we use a geometric analysis and numerical simulation technique to study the monotonicity of skewness and kurtosis from prescribing corresponding parameters. Then moment estimation and maximum likelihood estimation of parameters are also given. To use the maximum likelihood estimation (MLE) method, an expectation conditional maximization (ECM) algorithm is proposed to estimate and numerically simulate seven parameters of a two-component MixGND under the same variance and heteroscedasticity. By using data sets of the S&P 500 and Shanghai Stock Exchange Composite Index (SSEC), we compare goodness-of-fit performance between the mixture of two generalized normal distributions and the mixture of two normal distributions. The empirical analysis results show that the former better describes the heavy-tailed and leptokurtic characteristics of the daily returns.

Delayed Reporting of Faults in Warranty Claims

In this article, we present a model for delayed reporting of faults: multiple nonfatal faults are accumulated and then simultaneously reported and repaired. The reporting process is modeled as a stochastic process dependent on the underlying stochastic process generating the faults. We derive the joint distribution of the reporting times and numbers of reported faults, giving general results and results specific to faults generated by a Poisson process. We investigate a number of extensions to the basic model, including multiple fault types (including invisible and fatal faults), preventative maintenance, and customer rush. We show how to simulate from the model and implement maximum likelihood parameter estimation in a simulated dataset and a real dataset of warranty claims from a car manufacturer.

A bivariate joint frailty model with mixture framework for survival analysis of recurrent events with dependent censoring and cure fraction.

In the study of multiple failure time data with recurrent clinical endpoints, the classical independent censoring assumption in survival analysis can be violated when the evolution of the recurrent events is correlated with a censoring mechanism such as death. Moreover, in some situations, a cure fraction appears in the data because a tangible proportion of the study population benefits from treatment and becomes recurrence free and insusceptible to death related to the disease. A bivariate joint frailty mixture cure model is proposed to allow for dependent censoring and cure fraction in recurrent event data. The latency part of the model consists of two intensity functions for the hazard rates of recurrent events and death, wherein a bivariate frailty is introduced by means of the generalized linear mixed model methodology to adjust for dependent censoring. The model allows covariates and frailties in both the incidence and the latency parts, and it further accounts for the possibility of cure after each recurrence. It includes the joint frailty model and other related models as special cases. An expectation-maximization (EM)-type algorithm is developed to provide residual maximum likelihood estimation of model parameters. Through simulation studies, the performance of the model is investigated under different magnitudes of dependent censoring and cure rate. The model is applied to data sets from two colorectal cancer studies to illustrate its practicalvalue.

Inference for dependent competing risks from bivariate Kumaraswamy distribution under generalized progressive hybrid censoring

In this paper, competing risks model is considered when causes of failure are dependent. When latent failure times are distributed by the Marshall-Olkin bivariate Kumaraswamy model, inference for the unknown model parameters is studied under a generalized progressive hybrid censoring. Maximum likelihood estimates of unknown parameters are established, and the associated existence and uniqueness are provided. The approximate confidence intervals are constructed via the observed Fisher information matrix. Moreover, Bayes estimates and the credible intervals of the unknown parameters are also presented based a flexible Gamma-Dirichlet prior, and the importance sampling method is used to compute associated estimates. Simulation study and a lifetime example are given for illustration purposes.

Discrete Circular Distributions with Applications to Shared Orthologs of Paired Circular Genomes

For structural comparisons of paired prokaryotic genomes, an important topic in synthetic and evolutionary biology, the locations of shared orthologous genes (henceforth orthologs) are observed as binned data. This and other data, e.g., wind directions recorded at monitoring sites and intensive care unit arrival times on the 24-hour clock, are counted in binned circular arcs, thus modeling them by discrete circular distributions (DCDs) is required. We propose a novel method to construct a DCD from a base continuous circular distribution (CCD). The probability mass function is defined to take the normalized values of the probability density function at some pre-fixed equidistant points on the circle. Five families of constructed DCDs which have normalizing constants in closed form are presented. Simulation studies show that DCDs outperform the corresponding CCDs in modeling grouped (discrete) circular data, and minimum chi-square estimation outperforms maximum likelihood estimation for parameters. We apply the constructed DCDs, invariant wrapped Poisson and wrapped discrete skew Laplace to compare the structures of paired bacterial genomes. Specifically, discrete four-parameter wrapped Cauchy (nonnegative trigonometric sums) distribution models multi-modal shared orthologs in <i>Clostridium</i> (<i>Sulfolobus</i>) better than the others considered, in terms of AIC and Freedman’s goodness-of-fit test. The result that different DCDs fit the shared orthologs is consistent with the fact they belong to two kingdoms. Nevertheless, these prokaryotes have a common favored site around 70° on the unit circle; this finding is important for building synthetic prokaryotic genomes in synthetic biology. These DCDs can also be applied to other binned circular data.

Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring

This paper considers the problem of estimating the parameters of Poisson-Exponential (PE) distribution under progressive type-I interval censoring scheme. PE is a two-parameter lifetime distribution having an increasing hazard function. It has been applied in complementary risks problems in latent risks, that is in scenarios where maximum lifetime values are observed but information concerning factors accounting for component failure is unavailable. Under progressive type-I interval censoring, observations are known within two consecutively pre-arranged times and items would be withdrawn at pre-scheduled time points. This scheme is most suitable in those cases where continuous examination is impossible. Maximum likelihood estimates of Poisson-Exponential parameters are obtained via Expectation-Maximization (EM) algorithm. The EM algorithm is preferred as it has been confirmed to be a more superior tool when dealing with incomplete data sets having missing values, or models having truncated distributions. Asymptotic properties of the estimates are studied through simulation and compared based on bias and the mean squared error under different censoring schemes and parameter values. It is concluded that for an increasing sample size, the estimated values of the parameters tend to the true value. Among the four censoring schemes considered, the third scheme provides the most precise and accurate results followed by fourth scheme, first scheme and finally the second scheme.

Prediction of dry-cured ham weight loss and prospects of use in a pig breeding program

Large ham weight losses (WL) in dry-curing are undesired as they lead to a loss of marketable product and penalise the quality of the dry-cured ham. The availability of early predictions of WL may ease the adaptation of the dry-curing process to the characteristics of the thighs and increase the effectiveness of selective breeding in enhancing WL. Aims of this study were (i) to develop Bayesian and Random Forests (RFs) regression models for the prediction of ham WL during dry-curing using on-site infrared spectra of raw ham subcutaneous fat, carcass and raw ham traits as predictors and (ii) to estimate genetic parameters for WL and their predictions (P-WL). Visible-near infrared spectra were collected on the transversal section of the subcutaneous fat of raw hams. Carcass traits were carcass weight, carcass backfat depth, lean meat content and weight of raw hams. Raw ham traits included measures of ham subcutaneous fat depth and linear scores for round shape, subcutaneous fat thickness and marbling of the visible muscles of the thigh. Measures of WL were available for 1672 hams. The best prediction accuracies were those of a Bayesian regression model including the average spectrum, carcass and raw ham traits, with R2 values in validation of 0.46, 0.55 and 0.62, for WL at end of salting (23 days), resting (90 days) and curing (12 months), respectively. When WL at salting was used as an additional predictor of total WL, the R2 in validation was 0.67. Bayesian regressions were more accurate than RFs models in predicting all the investigated traits. Restricted maximum likelihood (REML) estimates of genetic parameters for WL and P-WL at the end of curing were estimated through a bivariate animal model including 1672 measures of WL and 8819 P-WL records. Results evidenced that the traits are heritable (h2 ± SE was 0.27 ± 0.04 for WL and 0.39 ± 0.04 for P-WL), and the additive genetic correlation is positive and high (ra = 0.88 ± 0.03). Prediction accuracy of ham WL is high enough to envisage a future use of prediction models in identifying batches of hams requiring an adaptation of the processing conditions to optimise results of the manufacturing process. The positive and high genetic correlation detected between WL and P-WL at the end of dry-curing, as well as the estimated heritability for P-WL, suggests that P-WL can be successfully used as an indicator trait of the measured WL in pig breeding programs.

Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach Part I: Methodology and Experiments

Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularizing the estimation problem to make it we...

New methods to define heavy-tailed distributions with applications to insurance data

Heavy-tailed distributions play an important role in modelling data in actuarial and financial sciences. In this article, nine new methods are suggested to define new distributions suitable for modelling data with an heavy right tail. For illustrative purposes, a special sub-model is considered in detail. Maximum likelihood estimators of the model parameters are obtained and a Monte Carlo simulation study is carried out to assess the behaviour of the estimators. Furthermore, some actuarial measures are calculated. A simulation study based on these actuarial measures is done. The usefulness of the proposed model is proved empirically by means of two real data sets. Finally, Bayesian analysis and performance of Gibbs sampling for the data sets are also carried out.

On the correspondence of deviances and maximum-likelihood and interval estimates from log-linear to logistic regression modelling.

Consider a set of categorical variables where at least one, denoted by Y, is binary. The log-linear model that describes the contingency table counts implies a logistic regression model, with outcome Y. Extending results from Christensen (1997, Log-linear models and logistic regression, 2nd edn. New York, NY, Springer), we prove that the maximum-likelihood estimates (MLE) of the logistic regression parameters equals the MLE for the corresponding log-linear model parameters, also considering the case where contingency table factors are not present in the corresponding logistic regression and some of the contingency table cells are collapsed together. We prove that, asymptotically, standard errors are also equal. These results demonstrate the extent to which inferences from the log-linear framework translate to inferences within the logistic regression framework, on the magnitude of main effects and interactions. Finally, we prove that the deviance of the log-linear model is equal to the deviance of the corresponding logistic regression, provided that no cell observations are collapsed together when one or more factors in become obsolete. We illustrate the derived results with the analysis of a real dataset.