Context-specific independence in graphical log-linear models

Log-linear models are the popular workhorses of analyzing contingency tables. A log-linear parameterization of an interaction model can be more expressive than a direct parameterization based on probabilities, leading to a powerful way of defining restrictions derived from marginal, conditional and context-specific independence. However, parameter estimation is often simpler under a direct parameterization, provided that the model enjoys certain decomposability properties. Here we introduce a cyclical projection algorithm for obtaining maximum likelihood estimates of log-linear parameters under an arbitrary context-specific graphical log-linear model, which needs not satisfy criteria of decomposability. We illustrate that lifting the restriction of decomposability makes the models more expressive, such that additional context-specific independencies embedded in real data can be identified. It is also shown how a context-specific graphical model can correspond to a non-hierarchical log-linear parameterization with a concise interpretation. This observation can pave way to further development of non-hierarchical log-linear models, which have been largely neglected due to their believed lack of interpretability.     

Log-linear Poisson autoregression

We consider a log-linear model for time series of counts. This type of model provides a framework where both negative and positive association can be taken into account. In addition time dependent covariates are accommodated in a straightforward way. We study its probabilistic properties and maximum likelihood estimation. It is shown that a perturbed version of the process is geometrically ergodic, and, under some conditions, it approaches the non-perturbed version. In addition, it is proved that the maximum likelihood estimator of the vector of unknown parameters is asymptotically normal with a covariance matrix that can be consistently estimated. The results are based on minimal assumptions and can be extended to the case of log-linear regression with continuous exogenous variables. The theory is applied to aggregated financial transaction time series. In particular, we discover positive association between the number of transactions and the volatility process of a certain stock.     

���k-out-of-nϵͳ����Լ���²����Ĺ��ƺ��㷨

In this paper we focus on the sequential k-out-of-n model with covariates. We assume that the lifetime distribution given covariates belongs to the exponential family, and deal with log-linear model of the scale parameter of the exponential distribution. The maximum likelihood estimators (MLEs) of the model parameters with order restrictions are derived and some properties of the MLEs are discussed, and we give the algorithm of MLES and the result of simulation.     

New families of estimators and test statistics in log-linear models

In this paper we consider categorical data that are distributed according to a multinomial, product-multinomial or Poisson distribution whose expected values follow a log-linear model and we study the inference problem of hypothesis testing in a log-linear model setting. The family of test statistics considered is based on the family of ?-divergence measures. The unknown parameters in the log-linear model under consideration are also estimated using ?-divergence measures: Minimum ?-divergence estimators. A simulation study is included to find test statistics that offer an attractive alternative to the Pearson chi-square and likelihood-ratio test statistics.     

On the Brown–Proschan model when repair effects are unknown

A device is repaired after failure. The Brown–Proschan (BP) model assumes that the repair is perfect with probability p and minimal with probability (1−p). Theoretical results usually suppose that each repair effect (perfect or minimal repair) is known. However, this is not generally the case in practice. In this paper, we study the behavior of the BP model when repair effects are unknown. In this context, the main features of the failure process are derived: distribution functions of times between failures, failure intensity, likelihood function, etc. We propose to estimate the repair efficiency parameter p and the parameters of the first time to failure distribution with the likelihood function or equivalently the EM algorithm. We also propose to combine a moment estimation of the scale parameter and a maximum likelihood estimation of other parameters. Copyright © 2010 John Wiley & Sons, Ltd.     

Log-linear modeling using conditional log-linear structures

Analysis of large dimensional contingency tables is rather difficult. Fienberg and Kim (1999, Journal of American Statistical Association, 94, 229–239) studied the problem of combining conditional (on single variable) log-linear structures for graphical models to obtain partial information about the full graphical log-linear model. In this paper, we consider the general log-linear models and obtain explicit representation for the log-linear parameters of the full model based on that of conditional structures. As a consequence, we give conditions under which a particular log-linear parameter is present or not in the full model. Some of the main results of Fienberg and Kim follow from our results. The explicit relationships between full model and the conditional structures are also presented. The connections between conditional structures and the layer structures are pointed out. We investigate also the hierarchical nature of the full model, based on conditional structures. Kim (2006, Computational Statistics and Data Analysis, 50, 2044–2064) analyzed graphical log-linear models based on conditional log-linear structures, when a set of variables is conditioned. For this case, we employ the Möbius inversion technique to obtain the interaction parameters of the full log-linear model, and discuss their properties. The hierarchical nature of the full model is also studied based on conditional structures. This result could be effectively used for the model selection also. As applications of our results, we have discussed several typical examples, including a real-life example.     

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.     

The Log-Linear Birnbaum-Saunders Power Model

In this paper the sinh-power model is developed as a natural follow up to the log-linear Birnbaum-Saunders power model. The class of models resulting, incorporates the sinh-power-normal model, the ordinary sinh-normal model and the log-linear Birnbaum-Saunders model (Rieck and Nedelman, Technometrics 33:51–60, 1991). Maximum likelihood estimation is developed with the Hessian matrix used for standard error estimation. An application is reported for the data set on lung cancer studied in Kalbfleisch and Prentice (2002), where it is shown that the log-linear Birnbaum-Saunders power-normal model presents better fit than the log-linear Birnbaum-Saunders model. Another application is devoted to a fatigue data set previously analyzed in the literature. A nonlinear Birnbaum-Saunders power-normal model is fitted to the data set, with satisfactory performance.     

Using a Markov process model of an association football match to determine the optimal timing of substitution and tactical decisions

A football match is modelled as a four-state Markov process. A log-linear model, fed by real data, is used to estimate transition probabilities by means of the maximum likelihood method. This makes it possible to estimate the probability distributions of goals scored and the expected number of league points gained, from any position in a match, for any given set of transition probabilities and hence in principle for any match. This approach is developed in order to estimate the optimal time to change tactics using dynamic programming, either by making a substitution or by some other conscious change of plan. A simple example of this approach is included as an illustration.     

Maximum likelihood estimation in a partially observed stratified regression model with censored data

The stratified proportional intensity model generalizes Cox’s proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also obtained.     

异方差的线性混合模型的参数估计

本文研究既含有固定效应又含有随机效应的线性混合模型,在随机效应的方差不同即异方差情况下,即考虑方差受外界因素的影响,如温度、湿度等,我们称之为协变量,在有协变量情况下对方差建立对数线性模型,运用最大似然估计讨论了固定效应的估计和随机效应的预测,并且用约束最大似然(REML)方法研究对数线性模型中参数和随机误差中参数(离差参数)的估计,并讨论估计量的性质及离差参数估计量的渐近正态性。     

A Non-Iterative Alternative to Ordinal Log-Linear Models

Log-linear modeling is a popular statistical tool for analysing a contingency table. This presentation focuses on an alternative approach to modeling ordinal categorical data. The technique, based on orthogonal polynomials, provides a much simpler method of model fitting than the conventional approach of maximum likelihood estimation, as it does not require iterative calculations nor the fitting and refitting to search for the best model. Another advantage is that quadratic and higher order effects can readily be included, in contrast to conventional log-linear models which incorporate linear terms only.

The focus of the discussion is the application of the new parameter estimation technique to multi-way contingency tables with at least one ordered variable. This will also be done by considering singly and doubly ordered two-way contingency tables. It will be shown by example that the resulting parameter estimates are numerically similar to corresponding maximum likelihood estimates for ordinal log-linear models.     

Standard errors for the cox proportional hazards cure model

Standard errors for the maximum likelihood estimates of the regression parameters in the logistic-proportional-hazards cure model are proposed using an approximate profile likelihood approach and a nonparametric likelihood. Two methods are given and are compared with the standard errors obtained from the inverse of the joint observed information matrix of the regression parameters and the nuisance hazard parameters. The observed information matrix is derived and is shown to be an approximation of the conditional information matrix of the regression parameters given the hazard parameters. Simulations indicate that the standard errors obtained from the inverse of the observed information matrix based on the profile likelihood and the full likelihood are comparable and appropriate. The coverage rates for the logistic regression parameter are generally good. The proportional hazards regression parameter show reasonable coverage rates under ideal conditions but lower coverage rates when the incidence proportion is low or when censoring is heavy. The three methods are applied to a data set to investigate the effects of radiation therapy on tonsil cancer.     

The log-linear model with a generalized gamma distribution for the error: A Bayesian approach

Considering a log-linear model with one covariate and a generalized gamma distribution for the error, we find the posterior densities for the parameters of interest. Since many standard survival distributions are particular cases of the generalized gamma model, the proposed bayesian method is very useful to discriminate between possible models to be used in the data analysis. The Laplace approximation for integrals (see Tierney and Kadane, 1984) is used to find the posterior distributions of the parameters involved when they cannot be obtained explicitly.     

Estimation of failure probability using semi-definite logit model

We will propose a new and practical method for estimating the failure probability of a large number of small to medium scale companies using their balance sheet data. We will use the maximum likelihood method to estimate the best parameters of the logit function, where the failure intensity function in its exponent is represented as a convex quadratic function instead of a commonly used linear function. The reasons for using this type of function are : (i) it can better represent the observed nonlinear dependence of failure probability on financial attributes, (ii) the resulting likelihood function can be maximized using a cutting plane algorithm developed for nonlinear semi-definite programming problems.We will show that we can achieve better prediction performance than the standard logit model, using thousands of sample companies.Revised: December 2002,     

A transformed random effects model with applications

This paper proposes a transformed random effects model for analyzing non-normal panel data where both the response and (some of) the covariates are subject to transformations for inducing flexible functional form, normality, homoscedasticity, and simple model structure. We develop a maximum likelihood procedure for model estimation and inference, along with a computational device which makes the estimation procedure feasible in cases of large panels. We provide model specification tests that take into account the fact that parameter values for error components cannot be negative. We illustrate the model and methods with two applications: state production and wage distribution. The empirical results strongly favor the new model to the standard ones where either linear or log-linear functional form is employed. Monte Carlo simulation shows that maximum likelihood inference is quite robust against mild departure from normality. Copyright © 2009 John Wiley & Sons, Ltd.     

Binomial proportion estimation in longitudinal data with non-ignorable non-response

Non-random missing data poses serious problems in longitudinal studies. The binomial distribution parameter becomes to be unidentifiable without any other auxiliary information or assumption when it suffers from ignorable missing data. Existing methods are mostly based on the log-linear regression model. In this article, a model is proposed for longitudinal data with non-ignorable non-response. It is considered to use the pre-test baseline data to improve the identifiability of the post-test parameter. Furthermore, we derive the identified estimation (IE), the maximum likelihood estimation (MLE) and its associated variance for the post-test parameter. The simulation study based on the model of this paper shows that the proposed approach gives promising results.     

Likelihood ratio tests for goodness-of-fit of a nonlinear regression model

We propose likelihood and restricted likelihood ratio tests for goodness-of-fit of nonlinear regression. The first-order Taylor approximation around the MLE of the regression parameters is used to approximate the null hypothesis and the alternative is modeled nonparametrically using penalized splines. The exact finite sample distribution of the test statistics is obtained for the linear model approximation and can be easily simulated. We recommend using the restricted likelihood instead of the likelihood ratio test because restricted maximum-likelihood estimates are not as severely biased as the maximum-likelihood estimates in the penalized splines framework.     

Empirical likelihood estimation of discretely sampled processes of OU type

This paper presents an empirical likelihood estimation procedure for parameters of the discretely sampled process of Ornstein-Uhlenbeck type. The proposed procedure is based on the condi- tional characteristic function, and the maximum empirical likelihood estimator is proved to be consistent and asymptotically normal. Moreover, this estimator is shown to be asymptotically efficient under some mild conditions. When the background driving Lévy process is of type A or B, we show that the intensity parameter c...     

Explicit moments for a class of micro-models in non-life insurance

This paper considers properties of the micro-model analysed in Antonio and Plat (2014). The main results are analytical expressions for the moments of the outstanding claims payments subdivided into IBNR claims and individual RBNS claims. These moments are possible to compute explicitly using the discretisation scheme for estimation and simulation used in Antonio and Plat (2014) since the expressions then do not involve any integrals that, typically, would require numerical solutions. Other aspects of the model that are investigated are properties of the maximum likelihood estimators of the model parameters, such as bias and consistency, and a way of computing prediction uncertainty in terms of the mean squared error of prediction that does not require simulations. Moreover, a brief discussion is given on how to compute moments or risk-measures of the claims development result (CDR) using simulations, which based on the results of the present paper can be done without any nested simulations. Based on this it is straightforward to compute the one-year Solvency Capital Requirement, which corresponds to the 99.5% Value-at-Risk of the one-year CDR. A brief numerical illustration is used to show the theoretical performance of the maximum likelihood estimators of the parameters in the claims development process under this model using a realistic set-up based on the case-study of Antonio and Plat (2014). Additionally, the paper ends with a short numerical illustration discussing the model’s robustness under violations of an independence assumption.