On the Restricted Liu Estimator in the Gauss–Markov Model

In this paper, we compare two estimators, the RLE (restricted Liu estimator) and the RLSE (restricted least squares estimator) of parameters in linear models under Gauss–Markov models. Using generalized inverse of matrices, we found some equivalency conditions for the superiority of the RLE with respect to the MSE criterion.     

Bayesian Model Selection: Measuring the χ2 Discrepancy with the Uniform Distribution

In the last years, many articles have been written about Bayesian model selection. In this article, a different and easier method is proposed and analyzed. The key idea of this method is based on the well-known property that, under the true model, the cumulative distribution function is distributed as a uniform distribution over the interval (0, 1). The method is first introduced for the continuous case and then for the discrete case by smoothing the cumulative distribution function. Some asymptotical properties of the method are obtained by developing an alternative to Helly's theorems. Finally, the performance of the method is evaluated by simulation, showing a good behavior.     

Harrison–Stevens Forecasting and the Multiprocess Dynamic Linear Model

This is an expository article. The Harrison–Stevens forecasting algorithm using the multiprocess dynamic linear model is a robust method for forecasting in a nonstationary time series. The purpose of this article is to help statisticians become familiar with the method.     

Inference for the Birnbaum–Saunders Lifetime Regression Model with Applications

The Birnbaum–Saunders distribution is a positively skewed distribution that is frequently used for analyzing lifetime data. Regression analysis is widely used in this context when some covariates are involved in the life-test. In this article, we discuss the maximum likelihood estimation of the model parameters and associated inference. We discuss the likelihood-ratio tests for some hypotheses of interest as well as some interval estimation methods. A Monte Carlo simulation study is then carried out to examine the performance of the proposed estimators and the interval estimation methods. Finally, some numerical data analyses are done for illustrating all the inferential methods developed here.     

A Cautionary Note on the Use of the Grønnesby and Borgan Goodness-of-Fit Test for the Cox Proportional Hazards Model

Grønnesby and Borgan (1996, Lifetime Data Analysis 2, 315–328) propose an omnibus goodness-of-fit test for the Cox proportional hazards model. The test is based on grouping the subjects by their estimated risk score and comparing the number of observed and a model based estimated number of expected events within each group. We show, using extensive simulations, that even for moderate sample sizes the choice of number of groups is critical for the test to attain the specified size. In light of these results we suggest a grouping strategy under which the test attains the correct size even for small samples. The power of the test statistic seems to be acceptable when compared to other goodness-of-fit tests.     

Linear Prediction Sufficiency for New Observations in the General Gauss–Markov Model

We consider the prediction of new observations in a general Gauss–Markov model. We state the fundamental equations of the best linear unbiased prediction, BLUP, and consider some properties of the BLUP. Particularly, we focus on such linear statistics, which preserve enough information for obtaining the BLUP of new observations as a linear function of them. We call such statistics linearly prediction sufficient for new observations, and introduce some equivalent characterizations for this new concept.     

Survival Estimation with Missing Censoring Times for the Generalized Koziol–Green Model

ABSTRACT

When analyzing time-to-event data, there are various situations in which right censoring times for unfailed units are missing. In that context, by taking a supplementary sample of a convenient percentage of unfailed units, we propose a semi-parametric method for estimating a survival function under the natural extension of the Koziol–Green model to double random censoring. Some large sample properties of this estimator are derived. We prove uniform strong consistency and asymptotic weak convergence to a Gaussian process. A simulation study is also presented in order to analyze the behavior of the proposed estimator.     

Likelihood Function and Canonical Correlation Analysis of the Peña–Box Model

The Peña–Box model is a type of dynamic factor model whose factors try to capture the time-effect movements of a multiple time series. The Peña–Box model can be expressed as a vector autoregressive (VAR) model with constraints. This article derives the maximum likelihood estimates and the likelihood ratio test of the VAR model for Gaussian processes. Then a test statistic constructed by canonical correlation coefficients is presented and adjusted for conditional heteroscedasticity. Simulations confirm the validity of adjustments for conditional heteroscedasticity, and show that the proposed statistics perform better than the statistics used in the existing literature.     

Generalized Concept of Relative Risk and Wider Applications of the Proportional Hazards Model and the Kaplan–Meier Estimator

The concepts of relative risk and hazard ratio are generalized for ordinary ordinal and continuous response variables, respectively. Under the generalized concepts, the Cox proportional hazards model with the Breslow's and Efron's methods can be regarded as generalizations of the Mantel–Haenszel estimator for dealing with broader types of covariates and responses. When ordinal responses can be regarded as discretized observations of a hypothetical continuous variable, the estimated relative risks from the Cox model reflect the associations between the responses and covariates. Examples are given to illustrate the generalized concepts and wider applications of the Cox model and the Kaplan–Meier estimator.     

Conditional Density Estimation in the Single Functional Index Model for α-Mixing Functional Data

In this article, we investigate the nonparametric estimation of the conditional density of a scalar response variable Y, given the explanatory variable X taking value in a Hilbert space when the observations are linked with a single index structure. The goal of this article is to present the asymptotic results such as pointwise almost complete consistency and the uniform almost complete convergence of the kernel estimation with rate for the conditional density in the setting of the α-mixing functional data, which extend the i.i.d case in Attaoui et al. (2011 Attaoui , S. , Laksaci , A. , Ould-Said , E. ( 2011 ). A note on the conditional density estimate in the single functional index model . Statist. Probab. Lett. 81 ( 1 ): 45 – 53 .[Crossref], [Web of Science ®] , [Google Scholar]) to the dependence setting. As an application, the convergence rate of the kernel estimation for the conditional mode is also obtained.     

Superiority of the r–k Class Estimator Over Some Estimators In A Linear Model

In regression analysis, to overcome the problem of multicollinearity, the r ? k class estimator is proposed as an alternative to the ordinary least squares estimator which is a general estimator including the ordinary ridge regression estimator, the principal components regression estimator and the ordinary least squares estimator. In this article, we derive the necessary and sufficient conditions for the superiority of the r ? k class estimator over each of these estimators under the Mahalanobis loss function by the average loss criterion. Then, we compare these estimators with each other using the same criterion. Also, we suggest to test to verify if these conditions are indeed satisfied. Finally, a numerical example and a Monte Carlo simulation are done to illustrate the theoretical results.     

Authors’ responses to the discussion on “Predicting losses of residential structures in the state of Florida by the Public Hurricane Loss Evaluation Model”

Hurricanes threaten the Florida coast line every year and are capable of causing catastrophic losses. The Public Hurricane Loss Model was developed in response to the need for having an open transparent model to predict the above losses. The results were summarized in the paper “Predicting Losses of Residential Structures in the State of Florida by the Public Hurricane Loss Model” which was subsequently a subject of discussion by experts in Meteorology, Engineering, Actuarial Sciences and Statistics. The present paper presents a response to the discussions on the above mentioned article.     

Distribution of the Λ-Criterion for Sphericity Test and Its Connection to a Generalized Dirichlet Model

It is shown that the exact null distribution of the likelihood ratio criterion for sphericity test in the p-variate normal case and the marginal distribution of the first component of a (p ? 1)-variate generalized Dirichlet model with a given set of parameters are identical. The exact distribution of the likelihood ratio criterion so obtained has a general format for every p. A novel idea is introduced here through which the complicated exact null distribution of the sphericity test criterion in multivariate statistical analysis is converted into an easily tractable marginal density in a generalized Dirichlet model. It provides a direct and easiest method of computation of p-values. The computation of p-values and a table of critical points corresponding to p = 3 and 4 are also presented.     

An Empirical Investigation of the “Dynamic McFadden” Model of Purchase Timing and Brand Choice: Implications for Market Structure

Purchase timing and brand-choice decisions of households are jointly investigated using the “dynamic McFadden” model of Heckman and Singer. The hazard of brand purchase is decomposed into the category purchase hazard and the probability of brand choice conditional on category purchase. The former is modeled using the hazard-function approach and the latter using a logit model. Unobserved heterogeneity in brand preferences, marketing effects, and baseline hazard parameters is accounted for in the empirical analysis. The distribution of preference heterogeneity identifies the locations of brands in multiattribute perceptual space and the distribution of attribute importance weights across households.     

The Missing Model: A “Circle of Gifts”

Abstract

In spite of the search for electronic publishing models, there is apparently already early, energetic scholarly publishing on the networks. Without a great deal of adjustment and even without formal academic recognition, the Net's accessibility and excitement have provided academic "incentives." With thoughtful development, it could provide a highly effective and competitive path to delivering scholarly knowledge. Indeed, there is currently so much momentum that it is difficult to imagine a failure to proceed, short of a deliberate and concerted attempt by universities and scholars to suppress the "thousand flowers" blooming on the scholarly networks, a choice not to support and subsidize academic publishing on the Net.

An effective scholar-based publishing system must recognize and support the widely divers forms of communication necessary for research and scholarship. It should not discriminate against nonlucrative projects or disciplines. It complements the paper-based scholarly efforts of university presses and of academic libraries, which are already beginning to provide networked access and archiving for university-generated scholarship. It creates a linked structure of scholars, societies, and academic institutions. It aims to fulfill another vital objective: retaining for the academy some ownership of its authored intellectual property.

The Missing Model, fledgling and fuzzy, is alive and well and living on the Net. To thrive, the academy needs to thoughtfully nurture it. The Circle of Gifts concept formally entwines the scholarly community, research institutions, scholars, and learned societies in the age-old and time-honored enterprise in which ideas are shared, affordably, by the creators of the ideas.     

Model uncertainty and variable selection in Bayesian lasso regression

While Bayesian analogues of lasso regression have become popular, comparatively little has been said about formal treatments of model uncertainty in such settings. This paper describes methods that can be used to evaluate the posterior distribution over the space of all possible regression models for Bayesian lasso regression. Access to the model space posterior distribution is necessary if model-averaged inference—e.g., model-averaged prediction and calculation of posterior variable inclusion probabilities—is desired. The key element of all such inference is the ability to evaluate the marginal likelihood of the data under a given regression model, which has so far proved difficult for the Bayesian lasso. This paper describes how the marginal likelihood can be accurately computed when the number of predictors in the model is not too large, allowing for model space enumeration when the total number of possible predictors is modest. In cases where the total number of possible predictors is large, a simple Markov chain Monte Carlo approach for sampling the model space posterior is provided. This Gibbs sampling approach is similar in spirit to the stochastic search variable selection methods that have become one of the main tools for addressing Bayesian regression model uncertainty, and the adaption of these methods to the Bayesian lasso is shown to be straightforward.     

A Realized Stochastic Volatility Model With Box–Cox Transformation

This article presents a new class of realized stochastic volatility model based on realized volatilities and returns jointly. We generalize the traditionally used logarithm transformation of realized volatility to the Box–Cox transformation, a more flexible parametric family of transformations. A two-step maximum likelihood estimation procedure is introduced to estimate this model on the basis of Koopman and Scharth (2013 Koopman, S.J., Scharth, M. (2013), The Analysis of Stochastic Volatility in the Presence of Daily Realised Measures, Journal of Financial Econometrics, 11, 76–115.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results show that the two-step estimator performs well, and the misspecified log transformation may lead to inaccurate parameter estimation and certain excessive skewness and kurtosis. Finally, an empirical investigation on realized volatility measures and daily returns is carried out for several stock indices.     

The Self‐Similar and Multifractal Nature of a Network Traffic Model

Abstract

We look at a family of models for Internet traffic with increasing input rates and consider approximation models which exhibit self‐similarity at large time scales and multifractality at small time scales. Depending on whether the input rate is fast or slow, the total cumulative input traffic can be approximated by a self‐similar stable Lévy motion or a self‐similar Gaussian process. The stable Lévy limit does not depend on the behavior of the individual transmission schedules but the Gaussian limit does. Also, the models and their approximations show multifractal behavior at small time scales.     

An Approximate Closed Form Solution for a Quadratic Dose–Response Model*

The quadratic dose–response model is often used in radiobiology studies. Since the model is nonlinear, the least squares estimators of the parameters of the model are determined by an iterative procedure. We give a simple closed form approximation for estimating the parameters of the model.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.