ß-expectation tolerance region for the heteroscedastic multiple regression model with multivariate Student-t error

A ß-expectation tolerance region has been constructed for the multivariate regression model with heteroscedastic errors which follow a multivariate Student-t distribution with an unknown number of degrees of freedom. The ß-expectaion tolerance region obtained in this paper is optimal in the sense of having minimum enclosure among all such tolerance regions that guarantees that it would cover any preassigned proportions, namely, ß×100 percent of the future responses from the model.     

On characterizations and variance bounds of discrete α-unimodality

Characterizations of α-unimodality for integer-valued random variables about a specific mode are established in terms of their probability mass functions, distribution functions and characteristic functions. Using these characterizations variance lower bounds in terms of α and the mode are derived. For α=1 all these results are reduced to ordinary unimodality. The new variance lower bounds for discrete unimodality is sharper than its continuous counterpart. An upper bound for the variance of discrete unimodal distribution defined on a finite support is discussed.     

Application of Stein-type estimation in combining regression estimates from replicated experiments

This paper considers the application of Stein-type estimation procedure for the coefficients in a linear regression model when data are available from replicated experiment. Two families of estimators characterized by a single scalar are proposed and their large sample asymptotic properties are derived. These are utilized for comparing the performances of the two estimators along with the conventional estimator and conditions for the superiority of one estimator over the other are deduced.     

Distributional aspects in latent variable models

For observable indicators with ordered categories one can assume underlying latent variables following certain marginal distributions. Transforming the latent variables changes its marginal distributions but not the observable qualitative indicators. The joint distribution of the latent variables can be constructed from the marginal distributions. There is a broad class of multivariate distributions for which the observable indicators are equivalent. By choosing the multivariate normal distribution from this class we can analyse a linear relationship between the transformed latent variables. This leads to latent structural equation models. Estimation of these latter models is therefore more general than the distributional assumption might initially suggest. Robustness of the estimation procedure is also discussed for deviations from this distribution family. Using ordinal business survey data of the German Ifo-institute we test the efficiency of firms' price expectations implied by the rational expectation hypothesis.     

Consistency,asymptotic unbiasedness and bounds on the bias of s2 in the linear regression model with error component disturbances

The OLS estimator of the disturbance variance in the linear regression model with error component disturbances is shown to be weakly consistent and asymptotically unbiased without any restrictions on the regressor matrix. Also, simple exact bounds on the expected value of s² are given for both the one-way and two-way error component models.     

Estimation for 3-parameter lognormal distribution with unknown shifted origin

A new reparameterization of a 3-parameter lognormal distribution with unknown shifted origin is presented by using a dimensionless parameter. We avoid, in this article, the application of logarithmic and exponential transformations to a value which has a physical dimension. The distribution function contains two dimensional parameters and one dimensionless parameter. Modified moment estimators and maximum likelihood estimators are presented. The presented modified moment estimators and maximum likelihood estimators are confronted with some actual data.     

Partial residuals in cumulative regression models for ordinal data

We are concerned with cumulative regression models for an ordered categorical response variable Y. We propose two methods to build partial residuals from regression on a subset Z₁ of covariates Z., which take into regard the ordinal character of the response. The first method makes use of a multivariate GLM-representation of the model and produces residual measures for diagnostic purposes. The second uses a latent continuous variable model and yields new (adjusted) ordinal data Y^*. Both methods are illustrated by a data set from forestry.     

Jackknife variances and confidence intervals of the Mantel-Haenszel estimator

In the situation of stratified 2×2 tables, consitency of two different jackknife variances of the Mantel-Haenszel estimator is discussed in the case of increasing sample sizes, but a fixed number of strata. Different principles for constructing confidence limits for the common odds ratio are investigated from a theoretical point of view with regard to the position and the length of the resulting intervals. Monte Carlo experiments compare the finite sample performance of the consistent jackknife variance with that of other noniterative variance estimators. In addition, the properties of these variance estimators are investigated when used for confidence interval estimation.     

Monte Carlo integration: An application to labour market states and transitions

This paper is concerned with the application of simulation estimation methods to micro-econometric labour market models. Based on a multi-period probit model for direct job changes and unemployment, estimators for the likelihood of individual employment histories are obtained by Monte Carlo integration and employed in a standard ML-procedure. The results for West German panel data suggest that dynamic effects are largely prevalent on labour markets and that in particular, past unemployment has drastic negative effects on future employment chances. Further, there are no indications that foreigners have a different labour market performance, nor that they are crowding natives out into unemployment.     

Comparison of location parameters of two exponential distributions when scale parameters are different and unknown

Letx _i(1)≤x _i(2)≤…≤x _i(r_i) be the right-censored samples of sizesn _i from theith exponential distributions $\sigma _i^{ - 1} exp\{ - (x - \mu _i )\sigma _i^{ - 1} \} ,i = 1,2$ where μ_i and σ_i are the unknown location and scale parameters respectively. This paper deals with the posteriori distribution of the difference between the two location parameters, namely μ₂-μ₁, which may be represented in the form $\mu _2 - \mu _1 \mathop = \limits^\mathcal{D} x_{2(1)} - x_{1(1)} + F_1 \sin \theta - F_2 \cos \theta $ where $\mathop = \limits^\mathcal{D} $ stands for equal in distribution,F _i stands for the central F-variable with [2,2(r _i?1)] degrees of freedom and $\tan \theta = \frac{{n_2 s_{x1} }}{{n_1 s_{x2} }}, s_{x1} = (r_1 - 1)^{ - 1} \left\{ {\sum\limits_{j = 1}^{r_i - 1} {(n_i - j)(x_{i(j + 1)} - x_{i(j)} )} } \right\}$ The paper also derives the distribution of the statisticV=F ₁ sin σ?F ₂ cos σ and tables of critical values of theV-statistic are provided for the 5% level of significance and selected degrees of freedom.     

A Poisson geometric process approach for predicting drop-out and committed first-time blood donors

A Poisson geometric process (PGP) model is proposed to study individual blood donation patterns for a blood donor retention program. Extended from the geometric process (GP) model of Lam [16 Y. Lam, Geometric process and replacement problem, Acta Math. Appl. Sin. 4 (1988), pp. 366–377. doi: 10.1007/BF02007241[Crossref] , [Google Scholar]], the PGP model captures the rather pronounced trend patterns across clusters of donors via the ratio parameters in a mixture setting. Within the state-space modeling framework, it allows for overdispersion by equating the mean of the Poisson data distribution to a latent GP. Alternatively, by simply setting, the mean of the Poisson distribution to be the mean of a GP, it has equidispersion. With the group-specific mean and ratio functions, the mixture PGP model facilitates classification of donors into committed, drop-out and one-time groups. Based on only two years of observations, the PGP model nicely predicts donors’ future donations to foster timely recruitment decision. The model is implemented using a Bayesian approach via the user-friendly software WinBUGS.     

Orthogonalizing parametric link transformation families in binary regression analysis

This paper studies the application of the orthogonalization technique of Cox and Reid (1987) to parametric families of link functions used in binary regression analysis. The explicit form of Cox and Reid's condition (4), for orthogonality at a point, is derived for arbitrary link families. This condition is used to determine a transform of a family introduced by Burr (1942) and Prentice (1975, 1976) which is locally orthogonal when the regression parameter is zero. Thus the benefits of having orthogonal parameters are limited to “small” regression effects. The extent to which approximate orthogonality holds for nonzero regression coefficients is investigated for two data sets from the literature. Two specific issues considered are: (1) the ability of orthogonal reparametrization to reduce the variability of the regression parameters caused by estimation of the link parameter and (2) the improved numerical stability (and hence interpretability) of regression estimates corresponding to different link parameters.     

Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density

We investigate the issue of bandwidth estimation in a functional nonparametric regression model with function-valued, continuous real-valued and discrete-valued regressors under the framework of unknown error density. Extending from the recent work of Shang (2013 Shang, H.L. (2013), ‘Bayesian Bandwidth Estimation for a Nonparametric Functional Regression Model with Unknown Error Density’, Computational Statistics &; Data Analysis, 67, 185–198. doi: 10.1016/j.csda.2013.05.006[Crossref], [Web of Science ®] , [Google Scholar]) [‘Bayesian Bandwidth Estimation for a Nonparametric Functional Regression Model with Unknown Error Density’, Computational Statistics &; Data Analysis, 67, 185–198], we approximate the unknown error density by a kernel density estimator of residuals, where the regression function is estimated by the functional Nadaraya–Watson estimator that admits mixed types of regressors. We derive a likelihood and posterior density for the bandwidth parameters under the kernel-form error density, and put forward a Bayesian bandwidth estimation approach that can simultaneously estimate the bandwidths. Simulation studies demonstrated the estimation accuracy of the regression function and error density for the proposed Bayesian approach. Illustrated by a spectroscopy data set in the food quality control, we applied the proposed Bayesian approach to select the optimal bandwidths in a functional nonparametric regression model with mixed types of regressors.     

Model selection in quantile regression models

Lasso methods are regularisation and shrinkage methods widely used for subset selection and estimation in regression problems. From a Bayesian perspective, the Lasso-type estimate can be viewed as a Bayesian posterior mode when specifying independent Laplace prior distributions for the coefficients of independent variables [32 T. Park, G. Casella, The Bayesian Lasso, J. Amer. Statist. Assoc. 103 (2008), pp. 681–686. doi: 10.1198/016214508000000337[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. A scale mixture of normal priors can also provide an adaptive regularisation method and represents an alternative model to the Bayesian Lasso-type model. In this paper, we assign a normal prior with mean zero and unknown variance for each quantile coefficient of independent variable. Then, a simple Markov Chain Monte Carlo-based computation technique is developed for quantile regression (QReg) models, including continuous, binary and left-censored outcomes. Based on the proposed prior, we propose a criterion for model selection in QReg models. The proposed criterion can be applied to classical least-squares, classical QReg, classical Tobit QReg and many others. For example, the proposed criterion can be applied to rq(), lm() and crq() which is available in an R package called Brq. Through simulation studies and analysis of a prostate cancer data set, we assess the performance of the proposed methods. The simulation studies and the prostate cancer data set analysis confirm that our methods perform well, compared with other approaches.     

A Computational Bayesian Method for Generalized Semiparametric Regression Models

To better understand the power shift and the U.S. role compared to China and others regional actors, the Chicago Council on Global Affairs and the East Asia Institute (EAI) surveyed people in six countries - China, Japan, South Korea, Vietnam, Indonesian, and the United States - in the first half of 2008 about regional security and economic integration in Asia and about how these nations perceive each other (Bouton et al., 2010 Bouton, M., Steven, K., Benjamin, P., and Gregory, H. (2010). Soft power in Asia survey, 2008. ICPSR25342-v1. Ann Arbor, MI: Inter-university Consortium for Political and Social Research [distributor], 2010-04-05. doi:10.3886/ICPSR25342.v1[Crossref] , [Google Scholar]). There exists latent variance that cannot be adequately explained by parametric models. This is, in large part, due to the hidden structures and latent stories that from in unexpected ways. Therefore, a new Gibbs sampler is developed here in order to reveal preciously unseen structures and latent variances found in the survey dataset of Bouton et al. This new sampler is based upon the semiparametric regression, a well-known tool frequently utilized in order to capture the functional dependence between variables with fixed effect parametric and nonlinear regression. This is then extended to a generalized semiparametric regression for binary responses with logit and probit link function. The new sampler is then developed for the generalized linear mixed model with a nonparametric random effect. It is expressed as nonparametric regression with the multinomial-Dirichlet distribution for the number and positions of knots.     

Computational challenges and temporal dependence in Bayesian nonparametric models

Müller et al. (Stat Methods Appl, 2017) provide an excellent review of several classes of Bayesian nonparametric models which have found widespread application in a variety of contexts, successfully highlighting their flexibility in comparison with parametric families. Particular attention in the paper is dedicated to modelling spatial dependence. Here we contribute by concisely discussing general computational challenges which arise with posterior inference with Bayesian nonparametric models and certain aspects of modelling temporal dependence.     

Fiducial Inferences for the Truncated Exponential Distribution

Lindqvist and Taraldsen (2005 Lindqvist , B. H. , Taraldsen , G. ( 2005 ). Monte Carlo conditioning on a sufficient statistic . Biometrika 92 : 451 – 464 .[Crossref], [Web of Science ®] , [Google Scholar]) introduced an interesting parametric family of distributions in the unit interval. In this note, inference procedures are given, both from the classical and the Bayesian view point. It is shown numerically through various examples that the posterior distribution for the parameter and the induced fiducial distribution are almost equivalent. The parametric family under study is a regular member of the Natural Exponential Family and so use of this fact permits induction of a unique fiducial in terms of the minimal sufficient statistic.     

Tuning Parameter Selection and Various Good Fitting Characteristics for the Liu-Type Estimator in Linear Regression

Liu (2003 Liu , K. ( 2003 ). Using Liu-Type estimator to combat collinearity . Commun. Statist. Theor. Meth. 32 ( 5 ): 1009 – 1020 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the Liu-Type estimator (LTE) to combat the well-known multicollinearity problem in linear regression. In this article, various better fitting characteristics of the LTE than those of the ordinary ridge regression estimator (Hoerl and Kennard, 1970 Hoerl , A. E. , Kennard , R. W. ( 1970 ). Ridge regression: Biased estimation for non-orthogonal problems . Technometrics 12 : 55 – 67 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are considered. In particular, we derived two methods to determine the parameter d for the LTE and find that the ridge parameter k could serve for regularization of an ill-conditioned design matrix, while the other parameter d could be used for tuning the fit quality. In addition, the coefficients of regression, coefficient of multiple determination, residual error variance, and generalized cross validation (GCV) of the prediction quality are very stable, and as the ridge parameter increases they eventually reach asymptotic levels, which produces robust regression models. Furthermore, a Monte Carlo evaluation of these features is also given to illustrate some of the theoretical results.     

Second-Order Powers of a Class of Tests in the Presence of a Nuisance Parameter

The second-order local powers of a broad class of asymptotic chi-squared tests are considered in a composite case where both the parameter of interest and the nuisance parameter are possibly multidimensional for which no assumption has been made regarding global parametric orthogonality or curved exponentiality. The main result is that the second-order (point-by-point) local power identity holds if approximate third cumulants of a square-root version of the (modified) test statistic in the class vanish up to the second-order, which is an extension of Kakizawa (2010a Kakizawa , Y. ( 2010a ). Second-order power comparison of tests . Commun. Statist. Theor. Meth. 39 : 1424 – 1436 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in the absence of the nuisance parameter. It is also shown that in the presence of the nuisance parameter, such a third cumulant condition does not always imply the second-order local unbiasedness of the resulting test. Then, the adjusted likelihood ratio test by Mukerjee (1993b Mukerjee , R. ( 1993b ). An extension of the conditional likelihood ratio test to the general multiparameter case . Ann. Inst. Statist. Math. 45 : 759 – 771 .[Crossref], [Web of Science ®] , [Google Scholar]) can be interpreted as the second-order local unbiased modification after applying the third cumulant condition.