On the first time of ruin in two-dimensional discrete time risk model with dependent claim occurrences

This article is concerned with a two-dimensional discrete time risk model based on exchangeable dependent claim occurrences. In particular, we obtain a recursive expression for the finite time non ruin probability under such a dependence among claim occurrences. For an illustration, we define a bivariate compound beta-binomial risk model and present numerical results on this model by comparing the corresponding results of the bivariate compound binomial risk model.     

Minimax Empirical Bayes Estimators in Multivariate Mixed Linear Models with Unequal Replications

Predictive influence of explanatory variables has been studied in both univariate and multivariate distributions. In the Bayesian approach, the same problem is considered in absence of multicollinearity in the dataset. The aim of this article is to study the same in the presence of perfect multicollinearity. To do this, we first derived the predictive distributions for full model and reduced model using vague prior density. Then the discrepancies between these predictive distributions are measured by the Kullback–Leibler (K–L) directed measure of divergence to assess the influence of deleted explanatory variables. Finally, distribution of the discrepancies is derived and the test procedure is performed.     

The Bayes rule of the parameter in (0,1) under Zhang’s loss function with an application to the beta-binomial model

Abstract

For the restricted parameter space (0,1), we propose Zhang’s loss function which satisfies all the 7 properties for a good loss function on (0,1). We then calculate the Bayes rule (estimator), the posterior expectation, the integrated risk, and the Bayes risk of the parameter in (0,1) under Zhang’s loss function. We also calculate the usual Bayes estimator under the squared error loss function, and the Bayes estimator has been proved to underestimate the Bayes estimator under Zhang’s loss function. Finally, the numerical simulations and a real data example of some monthly magazine exposure data exemplify our theoretical studies of two size relationships about the Bayes estimators and the Posterior Expected Zhang’s Losses (PEZLs).     

Empirical Uncertain Bayes Methods in Area‐level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area‐level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus, it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one‐point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area‐level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson‐gamma and the binomial‐beta, and the normal‐normal (Fay–Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well‐known expectation‐maximization algorithm and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second‐order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.     

Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates

Abstract.  Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.     

Empirical Bayes Confidence Intervals for Means of Natural Exponential Family-Quadratic Variance Function Distributions with Application to Small Area Estimation

Abstract.  The paper develops empirical Bayes (EB) confidence intervals for population means with distributions belonging to the natural exponential family-quadratic variance function (NEF-QVF) family when the sample size for a particular population is moderate or large. The basis for such development is to find an interval centred around the posterior mean which meets the target coverage probability asymptotically, and then show that the difference between the coverage probabilities of the Bayes and EB intervals is negligible up to a certain order. The approach taken is Edgeworth expansion so that the sample sizes from the different populations need not be significantly large. The proposed intervals meet the target coverage probabilities asymptotically, and are easy to construct. We illustrate use of these intervals in the context of small area estimation both through real and simulated data. The proposed intervals are different from the bootstrap intervals. The latter can be applied quite generally, but the order of accuracy of these intervals in meeting the desired coverage probability is unknown.     

Tuning Parameter Selector for the Penalized Likelihood Method in Multivariate Generalized Linear Models

Variable selection is fundamental to high-dimensional multivariate generalized linear models. The smoothly clipped absolute deviation (SCAD) method can solve the problem of variable selection and estimation. The choice of the tuning parameter in the SCAD method is critical, which controls the complexity of the selected model. This article proposes a criterion to select the tuning parameter for the SCAD method in multivariate generalized linear models, which is shown to be able to identify the true model consistently. Simulation studies are conducted to support theoretical findings, and two real data analysis are given to illustrate the proposed method.     

Testing for Uncorrelated Residuals in Dynamic Count Models With an Application to Corporate Bankruptcy

This article proposes new model checks for dynamic count models. Both portmanteau and omnibus-type tests for lack of residual autocorrelation are considered. The resulting test statistics are asymptotically pivotal when innovations are uncorrelated but possibly exhibit higher order serial dependence. Moreover, the tests are able to detect local alternatives converging to the null at the parametric rate T^{? 1/2}, with T the sample size. The finite sample performance of the test statistics are examined by means of Monte Carlo experiments. Using a dataset on U.S. corporate bankruptcies, the proposed tests are applied to check if different risk models are correctly specified. Supplementary materials for this article are available online.     

Empirical likelihood for parameters in an additive partially linear errors-in-variables model with longitudinal data

Empirical likelihood inferences for the parameter component in an additive partially linear errors-in-variables model with longitudinal data are investigated in this article. A corrected-attenuation block empirical likelihood procedure is used to estimate the regression coefficients, a corrected-attenuation block empirical log-likelihood ratio statistic is suggested and its asymptotic distribution is obtained. Compared with the method based on normal approximations, our proposed method does not require any consistent estimator for the asymptotic variance and bias. Simulation studies indicate that our proposed method performs better than the method based on normal approximations in terms of relatively higher coverage probabilities and smaller confidence regions. Furthermore, an example of an air pollution and health data set is used to illustrate the performance of the proposed method.     

Testing for spurious regression in a panel data model with the individual number and time length growing

This article shows a test for the spurious regression problem in a panel data model with a growing individual number and time series length. In the estimation, tapers are used and the integrated order for the remainder disturbance is extended to a real number; at the same time, the spurious regression problem can be detected without prior knowledge. Through Monte Carlo experiments, we examine the consistent estimators by various sizes of time length and individual number, in which the remainder disturbance is assumed to be either stationary or non-stationary. In addition, the asymptotic normality properties are discussed with a quasi log-likelihood function. From the power tests we can see that the estimators are quite successful and powerful.     

EM algorithm-based likelihood estimation for a generalized Gompertz regression model in presence of survival data with long-term survivors: an application to uterine cervical cancer data

In this paper we develop a regression model for survival data in the presence of long-term survivors based on the generalized Gompertz distribution introduced by El-Gohary et al. [The generalized Gompertz distribution. Appl Math Model. 2013;37:13–24] in a defective version. This model includes as special case the Gompertz cure rate model proposed by Gieser et al. [Modelling cure rates using the Gompertz model with covariate information. Stat Med. 1998;17:831–839]. Next, an expectation maximization algorithm is then developed for determining the maximum likelihood estimates (MLEs) of the parameters of the model. In addition, we discuss the construction of confidence intervals for the parameters using the asymptotic distributions of the MLEs and the parametric bootstrap method, and assess their performance through a Monte Carlo simulation study. Finally, the proposed methodology was applied to a database on uterine cervical cancer.