Power-linear models for incomplete contingency tables with nonignorable non-responses

For categorical data exhibiting nonignorable non-responses, it is well known that maximum likelihood (ML) estimates with a boundary solution are implausible and do not provide a perfect fit to the observed data even for saturated models. We provide the conditions under which ML estimates for the generalized linear model (GLM) with the usual log/logit link function have a boundary solution. These conditions introduce a new GLM with appropriately defined power link functions where its ML estimates resolve the problems arising from a boundary solution and offer useful statistics for identifying the non-response mechanism. This model is applied to a real dataset and compared with Bayesian models.     

Linear regression model with new symmetric distributed errors

Regression models play a dominant role in analyzing several data sets arising from areas like agricultural experiment, space experiment, biological experiment, financial modeling, etc. One of the major strings in developing the regression models is the assumption of the distribution of the error terms. It is customary to consider that the error terms follow the Gaussian distribution. However, there are some drawbacks of Gaussian errors such as the distribution being mesokurtic having kurtosis three. In many practical situations the variables under study may not be having mesokurtic but they are platykurtic. Hence, to analyze these sorts of platykurtic variables, a two-variable regression model with new symmetric distributed errors is developed and analyzed. The maximum likelihood (ML) estimators of the model parameters are derived. The properties of the ML estimators with respect to the new symmetrically distributed errors are also discussed. A simulation study is carried out to compare the proposed model with that of Gaussian errors and found that the proposed model performs better when the variables are platykurtic. Some applications of the developed model are also pointed out.     

Considerations for optimal nonparametric regression under a generalizederror structure

Nonparametric smoothing, such as kernel or spline estimation, has been examined extensively under the assumption of uncorrelated errors. This paper addresses the effects of potential correlation on consistency and other asymptotic properties in a repeated-measures model, using directly optimized linear smoothers of the replicate means. Unrestricted optimal weights, with respect to squared error loss, are used to confirm a lack of consistency for all linear estimators in an autocorrelated errors model. The results indicate kernel methods that work well for an uncorrelated errors model may not have the ability to perform satisfactorily when correlation is introduced, due to an asymmetry in the optimal weights, which disappears for an uncorrelated errors model. These would include data-driven bandwidth selection methods, adjustments of the bandwidth to accommodate correlation, higher-order kernels, and related bias reduction techniques. The analytic results suggest alternative approaches, not considered here in detail, which have shown merit.     

A general maximum likelihood analysis of overdispersion in generalized linear models

This paper presents an EM algorithm for maximum likelihood estimation in generalized linear models with overdispersion. The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully non-parametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters may be sensitive to the specification of a parametric form for the mixing distribution. A listing of a GLIM4 algorithm for fitting the overdispersed binomial logit model is given in an appendix.A simple method is given for obtaining correct standard errors for parameter estimates when using the EM algorithm.Several examples are discussed.     

Small area estimation of expenditure means and ratios under a unit-level bivariate linear mixed model

Under a unit-level bivariate linear mixed model, this paper introduces small area predictors of expenditure means and ratios, and derives approximations and estimators of the corresponding mean squared errors. For the considered model, the REML estimation method is implemented. Several simulation experiments, designed to analyze the behavior of the introduced fitting algorithm, predictors and mean squared error estimators, are carried out. An application to real data from the Spanish household budget survey illustrates the behavior of the proposed statistical methodology. The target is the estimation of means of food and non-food household annual expenditures and of ratios of food household expenditures by Spanish provinces.     

Robust skew-<Emphasis Type="Italic">t</Emphasis> factor analysis models for handling missing data

This paper presents a novel framework for maximum likelihood (ML) estimation in skew-t factor analysis (STFA) models in the presence of missing values or nonresponses. As a robust extension of the ordinary factor analysis model, the STFA model assumes a restricted version of the multivariate skew-t distribution for the latent factors and the unobservable errors to accommodate non-normal features such as asymmetry and heavy tails or outliers. An EM-type algorithm is developed to carry out ML estimation and imputation of missing values under a missing at random mechanism. The practical utility of the proposed methodology is illustrated through real and synthetic data examples.     

Estimating the variance for heterogeneity in arm‐based network meta‐analysis

Network meta‐analysis can be implemented by using arm‐based or contrast‐based models. Here we focus on arm‐based models and fit them using generalized linear mixed model procedures. Full maximum likelihood (ML) estimation leads to biased trial‐by‐treatment interaction variance estimates for heterogeneity. Thus, our objective is to investigate alternative approaches to variance estimation that reduce bias compared with full ML. Specifically, we use penalized quasi‐likelihood/pseudo‐likelihood and hierarchical (h) likelihood approaches. In addition, we consider a novel model modification that yields estimators akin to the residual maximum likelihood estimator for linear mixed models. The proposed methods are compared by simulation, and 2 real datasets are used for illustration. Simulations show that penalized quasi‐likelihood/pseudo‐likelihood and h‐likelihood reduce bias and yield satisfactory coverage rates. Sum‐to‐zero restriction and baseline contrasts for random trial‐by‐treatment interaction effects, as well as a residual ML‐like adjustment, also reduce bias compared with an unconstrained model when ML is used, but coverage rates are not quite as good. Penalized quasi‐likelihood/pseudo‐likelihood and h‐likelihood are therefore recommended.     

Holt-Winters Forecasting: An Alternative Formulation Applied to UK Air Passenger Data

This paper provides a formulation for the additive Holt-Winters forecasting procedure that simplifies both obtaining maximum likelihood estimates of all unknowns, smoothing parameters and initial conditions, and the computation of point forecasts and reliable predictive intervals. The stochastic component of the model is introduced by means of additive, uncorrelated, homoscedastic and Normal errors, and then the joint distribution of the data vector, a multivariate Normal distribution, is obtained. In the case where a data transformation was used to improve the fit of the model, cumulative forecasts are obtained here using a Monte-Carlo approximation. This paper describes the method by applying it to the series of monthly total UK air passengers collected by the Civil Aviation Authority, a long time series from 1949 to the present day, and compares the resulting forecasts with those obtained in previous studies.     

Diagnostics on nonlinear model with scale mixtures of skew-normal and first-order autoregressive errors

In this work, we develop some diagnostics for nonlinear regression model with scale mixtures of skew-normal (SMSN) and first-order autoregressive errors. The SMSN distribution class covers symmetric as well as asymmetric and heavy-tailed distributions, which offers a more flexible framework for modelling. Maximum-likelihood (ML) estimates are computed via an expectation–maximization-type algorithm. Local influence diagnostics and score test for the correlation are also derived. The performances of the ML estimates and the test statistic are investigated through Monte Carlo simulations. Finally, a real data set is used to illustrate our diagnostic methods.     

Developing a Liu estimator for the negative binomial regression model: method and application

This paper introduces a new shrinkage estimator for the negative binomial regression model that is a generalization of the estimator proposed for the linear regression model by Liu [A new class of biased estimate in linear regression, Comm. Stat. Theor. Meth. 22 (1993), pp. 393–402]. This shrinkage estimator is proposed in order to solve the problem of an inflated mean squared error of the classical maximum likelihood (ML) method in the presence of multicollinearity. Furthermore, the paper presents some methods of estimating the shrinkage parameter. By means of Monte Carlo simulations, it is shown that if the Liu estimator is applied with these shrinkage parameters, it always outperforms ML. The benefit of the new estimation method is also illustrated in an empirical application. Finally, based on the results from the simulation study and the empirical application, a recommendation regarding which estimator of the shrinkage parameter that should be used is given.     

Two-sample prediction for progressively Type-II censored Weibull lifetimes

Prediction on the basis of censored data has an important role in many fields. This article develops a non-Bayesian two-sample prediction based on a progressive Type-II right censoring scheme. We obtain the maximum likelihood (ML) prediction in a general form for lifetime models including the Weibull distribution. The Weibull distribution is considered to obtain the ML predictor (MLP), the ML prediction estimate (MLPE), the asymptotic ML prediction interval (AMLPI), and the asymptotic predictive ML intervals of the sth-order statistic in a future random sample (Y_s) drawn independently from the parent population, for an arbitrary progressive censoring scheme. To reach this aim, we present three ML prediction methods namely the numerical solution, the EM algorithm, and the approximate ML prediction. We compare the performances of the different methods of ML prediction under asymptotic normality and bootstrap methods by Monte Carlo simulation with respect to biases and mean square prediction errors (MSPEs) of the MLPs of Y_s as well as coverage probabilities (CP) and average lengths (AL) of the AMLPIs. Finally, we give a numerical example and a real data sample to assess the computational comparison of these methods of the ML prediction.     

Distribution of the biased hypothesis sum of squares in linear models with missing observations

In a linear model with missing observations, one can substitute algebraic quantities and then minimize the error sum of squares for the augmented model. This gives the correct error sum of squares. But this method does not produce the correct hypothesis sum of squares for testing a linear hypothesis about the parameters. The sum of squares obtained is biased but practitioners still use it. The distribution of this biased sum of squares is derived in this paper and the consequences of using this biased sum of squares on the type I and II errors is examined.     

Weak consistency of M-estimator in linear regression model with asymptotically almost negatively associated errors

Abstract

This paper studies a linear regression model with asymptotically almost negatively associated (AANA, in short) random errors. Under some mild conditions, the weak consistency of M-estimator of the unknown parameter is investigated, which extend the corresponding results for independent random errors and negatively associated (NA, in short) random errors. At last, two simulation examples are presented to verify the weak consistency of M-estimator in the model.     

Some theoretical results regarding the polygonal distribution

Polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. We demonstrate that the densities of polygonal distributions are dense in the class of continuous and concave densities with bounded second derivatives. Furthermore, we prove that polygonal density functions provide O(g^{? 2}) approximations (where g is the number of triangular distribution components), in the supremum distance, to any density function from the hypothesized class. Parametric consistency and Hellinger consistency results for the maximum likelihood (ML) estimator are obtained. A result regarding model selection via penalized ML estimation is proved.     

Estimating and testing group effects from type i censored normal samples in experimental design

The maximum likelihood (ML) equations calculated from censored normal samples do not admit explicit solutions. A principle of modification is given and modified maximum likelihood (MML) equations, which admit explicit solutions, are defined. This approach makes it possible to tackle the hitherto unresolved problem of estimating and testing hypotheses about group-effects in one-way classification experimental designs based on Type I censored normal samples. The MML estimators of group-effects are obtained as explicit functions of sample observations and shown to be asymptotically identical with the ML estimators and hence BAN (best asymptotic normal) estimators. A statistic t is defined to test a linear contrast of group-effects and shown to be asymptotically normally distributed. A numerical example is presented which illustrates the procedure.     

Reliability estimation for the exponential distribution

This paper is concerned with classical statistical estimation of the reliability function for the exponential density with unknown mean failure time θ, and with a known and fixed mission time τ. The minimum variance unbiased (MVU) estimator and the maximum likelihood (ML) estimator are reviewed and their mean square errors compared for different sample sizes. These comparisons serve also to extend previous work, and reinforce further the nonexistence of a uniformly best estimator. A class of shrunken estimators is then defined, and it produces a shrunken quasi-estimator and a shrunken estimator. The mean square errors for both these estimators are compared to the mean square errors of the MVU and ML estimators, and the new estimators are found to perform very well. Unfortunately, these estimators are difficult to compute for practical applications. A second class of estimators, which is easy to compute is also developed. Its mean square error properties are compared to the other estimators, and it outperforms all the contending estimators over the high and low reliability parameter space. Since, for all the estimators, analytical mean square error comparisons are not tractable, extensive numerical analyses are done in obtaining both the exact small sample and large sample results.     

Comparative Study of Blu and ML Estimators of Location and Scale Parameters of an Extreme Value Distribution for Small Samples and Type I Censorship

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators.     

Flexible Tweedie regression models for continuous data

Tweedie regression models (TRMs) provide a flexible family of distributions to deal with non-negative right-skewed data and can handle continuous data with probability mass at zero. Estimation and inference of TRMs based on the maximum likelihood (ML) method are challenged by the presence of an infinity sum in the probability function and non-trivial restrictions on the power parameter space. In this paper, we propose two approaches for fitting TRMs, namely quasi-likelihood (QML) and pseudo-likelihood (PML). We discuss their asymptotic properties and perform simulation studies to compare our methods with the ML method. We show that the QML method provides asymptotically efficient estimation for regression parameters. Simulation studies showed that the QML and PML approaches present estimates, standard errors and coverage rates similar to the ML method. Furthermore, the second-moment assumptions required by the QML and PML methods enable us to extend the TRMs to the class of quasi-TRMs in Wedderburn's style. It allows to eliminate the non-trivial restriction on the power parameter space, and thus provides a flexible regression model to deal with continuous data. We provide an R implementation and illustrate the application of TRMs using three data sets.     

Generalized Estimation of the BLUP in Mixed-Effects Models: A Comparison with ML and REML

The Best Linear Unbiased Predictor (BLUP) in mixed models is a function of the variance components and they are estimated using maximum likelihood (ML) or restricted ML methods. Nonconvergence of BLUP would occur due to a drawback of the standard likelihood-based approaches. In such situations, ML and REML either do not provide any BLUPs or all become equal. To overcome this drawback, we provide a generalized estimate (GE) of BLUP that does not suffer from the problem of negative or zero variance components, and compare its performance against the ML and REML estimates of BLUP. Simulated and published data are used to compare BLUP.     

Accurate Confidence Interval Estimation of Small Area Parameters Under the Fay–Herriot Model

Small area estimation has long been a popular and important research topic due to its growing demand in public and private sectors. We consider here the basic area level model, popularly known as the Fay–Herriot model. Although much of current research is predominantly focused on second order unbiased estimation of mean squared prediction errors, we concentrate on developing confidence intervals (CIs) for the small area means that are second order correct. The corrected CI can be readily implemented, because it only requires quantities that are already estimated as part of the mean squared error estimation. We extend the approach to a CI for the difference of two small area means. The findings are illustrated with a simulation study.