Empirical likelihood for generalized linear models with fixed and adaptive designs

Empirical likelihood inference for generalized linear models with fixed and adaptive designs is considered. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter and the resulting estimator is shown to be asymptotically normal. Some simulations are conducted to illustrate the proposed method.     

Generalized mixed estimator for nonlinear models: a maximum likelihood approach

This paper considers the problem of estimating a nonlinear statistical model subject to stochastic linear constraints among unknown parameters. These constraints represent prior information which originates from a previous estimation of the same model using an alternative database. One feature of this specification allows for the disign matrix of stochastic linear restrictions to be estimated. The mixed regression technique and the maximum likelihood approach are used to derive the estimator for both the model coefficients and the unknown elements of this design matrix. The proposed estimator whose asymptotic properties are studied, contains as a special case the conventional mixed regression estimator based on a fixed design matrix. A new test of compatibility between prior and sample information is also introduced. Thesuggested estimator is tested empirically with both simulated and actual marketing data.     

Generalized empirical likelihood inference in partial linear regression model for longitudinal data

In this paper, empirical likelihood inference for longitudinal data within the framework of partial linear regression models are investigated. The proposed procedures take into consideration the correlation within groups without involving direct estimation of nuisance parameters in the correlation matrix. The empirical likelihood method is used to estimate the regression coefficients and the baseline function, and to construct confidence intervals. A nonparametric version of Wilk's theorem for the limiting distribution of the empirical likelihood ratio is derived. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. The finite sample behaviour of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial data set.     

Correlated linear models for the analysis of familial correlations

The problems of estimation and hypotheses testing on the parameters of two correlated linear models are discussed. Such models are known to have direct applications in epidemiologic research, particularly in the field of family studies. When the data are unbalanced, the maximum-likelihood estimation of the parameters is achieved by adopting a fairly simple numerical algorithm. The asymptotic variances and covariances of the estimators are derived, and the procedures are illustrated on arterial-blood-pressure data from the literature.     

Penalized maximum likelihood estimation for Gaussian hidden Markov models

ABSTRACT

The likelihood function of a Gaussian hidden Markov model is unbounded, which is why the maximum likelihood estimator (MLE) is not consistent. A penalized MLE is introduced along with a rigorous consistency proof.     

Weighted empirical likelihood for generalized linear models with longitudinal data

In this paper, we introduce the empirical likelihood (EL) method to longitudinal studies. By considering the dependence within subjects in the auxiliary random vectors, we propose a new weighted empirical likelihood (WEL) inference for generalized linear models with longitudinal data. We show that the weighted empirical likelihood ratio always follows an asymptotically standard chi-squared distribution no matter which working weight matrix that we have chosen, but a well chosen working weight matrix can improve the efficiency of statistical inference. Simulations are conducted to demonstrate the accuracy and efficiency of our proposed WEL method, and a real data set is used to illustrate the proposed method.     

Pseudo maximum likelihood estimation for the dirichlet-multinomial distribution

Pseudo maximum likelihood estimation (PML) for the Dirich-let-multinomial distribution is proposed and examined in this pa-per. The procedure is compared to that based on moments (MM) for its asymptotic relative efficiency (ARE) relative to the maximum likelihood estimate (ML). It is found that PML, requiring much less computational effort than ML and possessing considerably higher ARE than MM, constitutes a good compromise between ML and MM. PML is also found to have very high ARE when an estimate for the scale parameter in the Dirichlet-multinomial distribution is all that is needed.     

On maximum likelihood estimation of the semi-parametric Cox model with time-varying covariates

Including time-varying covariates is a popular extension to the Cox model and a suitable approach for dealing with non-proportional hazards. However, partial likelihood (PL) estimation of this model has three shortcomings: (i) estimated regression coefficients can be less accurate in small samples with heavy censoring; (ii) the baseline hazard is not directly estimated and (iii) a covariance matrix for both the regression coefficients and the baseline hazard is not easily produced.We address these by developing a maximum likelihood (ML) approach to jointly estimate regression coefficients and baseline hazard using a constrained optimisation ensuring the latter''s non-negativity. We demonstrate asymptotic properties of these estimates and show via simulation their increased accuracy compared to PL estimates in small samples and show our method produces smoother baseline hazard estimates than the Breslow estimator.Finally, we apply our method to two examples, including an important real-world financial example to estimate time to default for retail home loans. We demonstrate using our ML estimate for the baseline hazard can give much clearer corroboratory evidence of the ‘humped hazard’, whereby the risk of loan default rises to a peak and then later falls.     

Prediction in linear mixed models

Following estimation of effects from a linear mixed model, it is often useful to form predicted values for certain factor/variate combinations. The process has been well defined for linear models, but the introduction of random effects into the model means that a decision has to be made about the inclusion or exclusion of random model terms from the predictions. This paper discusses the interpretation of predictions formed including or excluding random terms. Four datasets are used to illustrate circumstances where different prediction strategies may be appropriate: in an orthogonal design, an unbalanced nested structure, a model with cubic smoothing spline terms and for kriging after spatial analysis. The examples also show the need for different weighting schemes that recognize nesting and aliasing during prediction, and the necessity of being able to detect inestimable predictions.     

The evaluation of exact maximum likelihood estimates for varma models

This paper describes an algorithm for the evaluation of the exact likelihood function in order to obtain estimates of the coefficients of vector autoregressive moving average (VARMA) models. The use of the algorithm is illustrated by a Monte Carlo experiment and an application to the analysis of a set of bivariate animal population data. Fanally it is shown how to extend the algorithm, in a simple manner, to obtain exact maximum likelihood estimates of the coefficients of vector autoregressive moving average models with included exogenous variables.     

Nonparametric maximum likelihood estimation for shifted curves

The analysis of a sample of curves can be done by self-modelling regression methods. Within this framework we follow the ideas of nonparametric maximum likelihood estimation known from event history analysis and the counting process set-up. We derive an infinite dimensional score equation and from there we suggest an algorithm to estimate the shape function for a simple shape invariant model. The nonparametric maximum likelihood estimator that we find turns out to be a Nadaraya–Watson-like estimator, but unlike in the usual kernel smoothing situation we do not need to select a bandwidth or even a kernel function, since the score equation automatically selects the shape and the smoothing parameter for the estimation. We apply the method to a sample of electrophoretic spectra to illustrate how it works.     

A nonparametric estimation procedure for the Hawkes process: comparison with maximum likelihood estimation

In earlier work, Kirchner [An estimation procedure for the Hawkes process. Quant Financ. 2017;17(4):571–595], we introduced a nonparametric estimation method for the Hawkes point process. In this paper, we present a simulation study that compares this specific nonparametric method to maximum-likelihood estimation. We find that the standard deviations of both estimation methods decrease as power-laws in the sample size. Moreover, the standard deviations are proportional. For example, for a specific Hawkes model, the standard deviation of the branching coefficient estimate is roughly 20% larger than for MLE – over all sample sizes considered. This factor becomes smaller when the true underlying branching coefficient becomes larger. In terms of runtime, our method clearly outperforms MLE. The present bias of our method can be well explained and controlled. As an incidental finding, we see that also MLE estimates seem to be significantly biased when the underlying Hawkes model is near criticality. This asks for a more rigorous analysis of the Hawkes likelihood and its optimization.     

Second-order biases of the maximum likelihood estimates in von Mises regression models

This paper discusses issues related to the improvement of maximum likelihood estimates in von Mises regression models. It obtains general matrix expressions for the second-order biases of maximum likelihood estimates of the mean parameters and concentration parameters. The formulae are simple to compute, and give the biases by means of weighted linear regressions. Simulation results are presented assessing the performance of corrected maximum likelihood estimates in these models.     

Asymptotic properties of maximum quasi-likelihood estimator in quasi-likelihood non linear models with stochastic regression

In this paper, we establish the asymptotic properties of maximum quasi-likelihood estimator (MQLE) in quasi-likelihood non linear models (QLNMs) with stochastic regression under some mild regular conditions. We also investigate the existence, strong consistency, and asymptotic normality of MQLE in QLNMs with stochastic regression.     

Empirical likelihood inference for parameters in a partially linear errors-in-variables model

In this paper, we consider the application of the empirical likelihood method to a partially linear model with measurement errors in the non-parametric part. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter by using the empirical log-likelihood ratio function, and the resulting estimator is shown to be asymptotically normal. Some simulations and an application are conducted to illustrate the proposed method.     

A note on a multivariate version of rolle's theorem and uniqueness of maximum likelihood roots

This note gives a multivariate version of Rolle's theorem and shows its usefulness in establishing the uniqueness of the root of the maximum likelihood equations, the so-called maximum likelihood equation estimator. The technique is used to prove uniqueness in two situations from the literature where the original proof of uniqueness was in error.     

Penalized likelihood ratio test for a biomarker threshold effect in clinical trials based on generalized linear models

In a clinical trial, the responses to the new treatment may vary among patient subsets with different characteristics in a biomarker. It is often necessary to examine whether there is a cutpoint for the biomarker that divides the patients into two subsets of those with more favourable and less favourable responses. More generally, we approach this problem as a test of homogeneity in the effects of a set of covariates in generalized linear regression models. The unknown cutpoint results in a model with nonidentifiability and a nonsmooth likelihood function to which the ordinary likelihood methods do not apply. We first use a smooth continuous function to approximate the indicator function defining the patient subsets. We then propose a penalized likelihood ratio test to overcome the model irregularities. Under the null hypothesis, we prove that the asymptotic distribution of the proposed test statistic is a mixture of chi-squared distributions. Our method is based on established asymptotic theory, is simple to use, and works in a general framework that includes logistic, Poisson, and linear regression models. In extensive simulation studies, we find that the proposed test works well in terms of size and power. We further demonstrate the use of the proposed method by applying it to clinical trial data from the Digitalis Investigation Group (DIG) on heart failure.     

Empirical likelihood for the response mean of generalized linear models with missing at random responses

In this article, we present a new empirical likelihood ratio for constructing the confidence interval of the response mean of generalized linear models with missing at random responses. Compared with the existing methods, the proposal can avoid the so-called “curse of dimensionality” problem when the dimension of covariates is high, and is still chi-squared distributed asymptotically, nevertheless. Simulation studies are also provided to illustrate the performance of the developed method.     

Maximum likelihood and bayes prediction of current system lifetime

The Power Law Process is often used to analyse failure data of repairable systems undergoing development testing where the system failure intensity decreases as a result of repeated application of corrective actions. At the end of the development program, the system failure intensity is assumed to remain constant and the current system lifetime is assumed to be exponentially distributed. In this paper, prediction limits on the current system lifetime have been derived both in the maximum likelihood and Bayesian context. Exact values and a closed form approximation of percentage points of the pivotal quantity used in the classical approach are given in the case of failure truncated testing. For both failure and time truncated testing, the Bayesian approach is developed both when no prior knowledge is available and when information on the reliability growth rate can be given. A numerical example is also given.