The Non Dominated Set in Bayesian Decision Problems with Convex Loss Functions

Bayesian decision problems require subjective elicitation of the inputs: beliefs and preferences. Sometimes, elicitation methods may not represent perfectly the judgements of the decision maker. Several foundations propose to overlay this problem using robust approaches. In these models, beliefs are modelled by a class of probability distributions and preferences by a class of loss functions. Then, we are in the conditions of a Pareto order. Hence the solution concept is the set of non dominated alternatives. In this article we focus on the computation of the efficient set when the preferences are modeled by a class of convex loss functions.     

Alternative Procedures to Discriminate Non Nested Multivariate Linear Regression Models

ABSTRACT

This article builds classical and Bayesian testing procedures for choosing between non nested multivariate regression models. Although there are several classical tests for discriminating univariate regressions, only the Cox test is able to consistently handle the multivariate case. We then derive the limiting distribution of the Cox statistic in such a context, correcting an earlier derivation in the literature. Further, we show how to build alternative Bayes factors for the testing of nonnested multivariate linear regression models. In particular, we compute expressions for the posterior Bayes factor, the fractional Bayes factor, and the intrinsic Bayes factor.     

Comparison of two sampling schemes for generating record-breaking data from the proportional hazard rate models

ABSTRACT

In this article, we consider a sampling scheme in record-breaking data set-up, as record ranked set sampling. We compare the proposed sampling with the well-known sampling scheme in record values known as inverse sampling scheme when the underlying distribution follows the proportional hazard rate model. Various point estimators are obtained in each sampling schemes and compared with respect to mean squared error and Pitman measure of closeness criteria. It is observed in most of the situations that the new sampling scheme provides more efficient estimators than their counterparts. Finally, one data set has been analyzed for illustrative purposes.     

Relative Potency Estimation in Parallel-Line Assays – Method Comparison and Some Extensions

Relative potency estimations in both multiple parallel-line and slope-ratio assays involve construction of simultaneous confidence intervals for ratios of linear combinations of general linear model parameters. The key problem here is that of determining multiplicity adjusted percentage points of a multivariate t-distribution, the correlation matrix R of which depends on the unknown relative potency parameters. Several methods have been proposed in the literature on how to deal with R . In this article, we introduce a method based on an estimate of R (also called the plug-in approach) and compare it with various methods including conservative procedures based on probability inequalities. Attention is restricted to parallel-line assays though the theory is applicable for any ratios of coefficients in the general linear model. Extension of the plug-in method to linear mixed effect models is also discussed. The methods will be compared with respect to their simultaneous coverage probabilities via Monte Carlo simulations. We also evaluate the methods in terms of confidence interval width through application to data on multiple parallel-line assay.     

COMPARING TESTS OF AUTOREGRESSIVE VERSUS MOVING AVERAGE ERRORS IN REGRESSION MODELS USING BAHADUR’S ASYMPTOTIC RELATIVE EFFICIENCY

ABSTRACT

The purpose of this paper is to use Bahadur's asymptotic relative efficiency measure to compare the performance of various tests of autoregressive (AR) versus moving average (MA) error processes in regression models. Tests to be examined include non-nested procedures of the models against each other, and classical procedures based upon testing both the AR and MA error processes against the more general autoregressive-moving average model.     

Comparison of different computational implementations on fitting generalized linear mixed-effects models for repeated count measures

ABSTRACT

In modelling repeated count outcomes, generalized linear mixed-effects models are commonly used to account for within-cluster correlations. However, inconsistent results are frequently generated by various statistical R packages and SAS procedures, especially in case of a moderate or strong within-cluster correlation or overdispersion. We investigated the underlying numerical approaches and statistical theories on which these packages and procedures are built. We then compared the performance of these statistical packages and procedures by simulating both Poisson-distributed and overdispersed count data. The SAS NLMIXED procedure outperformed the others procedures in all settings.     

Variable selection in finite mixture of semi-parametric regression models

Abstract

In this paper we are concerned with variable selection in finite mixture of semiparametric regression models. This task consists of model selection for non parametric component and variable selection for parametric part. Thus, we encountered separate model selections for every non parametric component of each sub model. To overcome this computational burden, we introduced a class of variable selection procedures for finite mixture of semiparametric regression models using penalized approach for variable selection. It is shown that the new method is consistent for variable selection. Simulations show that the performance of proposed method is good, and it consequently improves pervious works in this area and also requires much less computing power than existing methods.     

Goodness-of-link tests for multivariate regression models

ABSTRACT

This note presents an approximation to multivariate regression models which is obtained from a first-order series expansion of the multivariate link function. The proposed approach yields a variable-addition approximation of regression models that enables a multivariate generalization of the well-known goodness-of-link specification test, available for univariate generalized linear models. Application of this general methodology is illustrated with models of multinomial discrete choice and multivariate fractional data, in which context it is shown to lead to well-established approximation and testing procedures.     

Change Points with Linear Trend Followed by Abrupt Change for the Exponential Distribution

We discuss testing procedures to detect if a random sequence of exponentially distributed random variables has been subjected to a linear trend change followed by an abrupt change. We propose three statistics and explore their distribution theories. As an illustration, we applied these tests to Stanford heart transplant data and airport inter arrival data.     

Biases in bias elicitation

Abstract

We consider the biases that can arise in bias elicitation when expert assessors make random errors. After presenting a general framework of the phenomenon, we illustrate it for two examples: the case of omitting variables bias and that of the bias arising in adjusting relative risks. Results show that, even when assessors’ elicitations of bias have desirable properties, the nonlinear nature of biases can lead to elicitations of bias that are, themselves, biased. We show the corrections which can be made to remove this bias and discuss the implications for the applied literature which employs these methods.     

Homogeneity test based on ranked set samples

Abstract

The homogeneity hypothesis is investigated in a location family of distributions. A moment-based test is introduced based on data collected from a ranked set sampling scheme. The asymptotic distribution of the proposed test statistic is determined and the performance of the test is studied via simulation. Furthermore, for small sample sizes, the bootstrap procedure is used to distinguish the homogeneity of data. An illustrative example is also presented to explain the proposed procedures in this paper.     

On double hysteretic heteroskedastic model

ABSTRACT

This paper proposes a hysteretic autoregressive model with GARCH specification and a skew Student's t-error distribution for financial time series. With an integrated hysteresis zone, this model allows both the conditional mean and conditional volatility switching in a regime to be delayed when the hysteresis variable lies in a hysteresis zone. We perform Bayesian estimation via an adaptive Markov Chain Monte Carlo sampling scheme. The proposed Bayesian method allows simultaneous inferences for all unknown parameters, including threshold values and a delay parameter. To implement model selection, we propose a numerical approximation of the marginal likelihoods to posterior odds. The proposed methodology is illustrated using simulation studies and two major Asia stock basis series. We conduct a model comparison for variant hysteresis and threshold GARCH models based on the posterior odds ratios, finding strong evidence of the hysteretic effect and some asymmetric heavy-tailness. Versus multi-regime threshold GARCH models, this new collection of models is more suitable to describe real data sets. Finally, we employ Bayesian forecasting methods in a Value-at-Risk study of the return series.     

Frequentist,Bayes, or Other?

ABSTRACT

Both philosophically and in practice, statistics is dominated by frequentist and Bayesian thinking. Under those paradigms, our courses and textbooks talk about the accuracy with which true model parameters are estimated or the posterior probability that they lie in a given set. In nonparametric problems, they talk about convergence to the true function (density, regression, etc.) or the probability that the true function lies in a given set. But the usual paradigms' focus on learning the true model and parameters can distract the analyst from another important task: discovering whether there are many sets of models and parameters that describe the data reasonably well. When we discover many good models we can see in what ways they agree. Points of agreement give us more confidence in our inferences, but points of disagreement give us less. Further, the usual paradigms’ focus seduces us into judging and adopting procedures according to how well they learn the true values. An alternative is to judge models and parameter values, not procedures, and judge them by how well they describe data, not how close they come to the truth. The latter is especially appealing in problems without a true model.     

A COMPARISON STUDY OF SOME COMBINED CLASSIFIERS

ABSTRACT

In this article we consider two methods for combining a number of individual classifiers in order to construct more effective classification rules. The effectiveness of these methods, as measured by a comparison of their misclassification error rates with those of the individual classifiers, is assessed via a number of examples that involve simulated data. We also compare the results to those of two existing combining procedures.     

Overlapping group lasso for high-dimensional generalized linear models

Abstract

Structured sparsity has recently been a very popular technique to deal with the high-dimensional data. In this paper, we mainly focus on the theoretical problems for the overlapping group structure of generalized linear models (GLMs). Although the overlapping group lasso method for GLMs has been widely applied in some applications, the theoretical properties about it are still unknown. Under some general conditions, we presents the oracle inequalities for the estimation and prediction error of overlapping group Lasso method in the generalized linear model setting. Then, we apply these results to the so-called Logistic and Poisson regression models. It is shown that the results of the Lasso and group Lasso procedures for GLMs can be recovered by specifying the group structures in our proposed method. The effect of overlap and the performance of variable selection of our proposed method are both studied by numerical simulations. Finally, we apply our proposed method to two gene expression data sets: the p53 data and the lung cancer data.     

Characterizing Relationships Between Estimations Under a General Linear Model with Explicit and Implicit Restrictions by Rank of Matrix

In the investigation of the restricted linear model ? _r  = {y, X β | A β = b, σ² Σ}, the parameter constraints A β = b are often handled by transforming the model into certain implicitly restricted model. Any estimation derived from the explicitly and implicitly restricted models on the vector β and its functions should be equivalent, although the expressions of the estimation under the two models may be different. However, people more likely want to directly compare different expressions of estimations and yield a conclusion on their equivalence by using some algebraic operations on expressions of estimations. In this article, we give some results on equivalence of the well-known OLSEs and BLUEs under the explicitly and implicitly restricted linear models by using some expansion formulas for ranks of matrices.     

Modelling the evolution of a random set utilizing covariate information with applications to nowcasting

ABSTRACT

We propose models that allow us to capture the evolution of objects over time and more importantly, we provide forecasts to describe an object at future unobserved states utilizing information from the current state along with covariate information. We view objects as random sets and proceed to model them in a hierarchical Bayesian framework and estimate the model parameters using a Markov chain Monte Carlo scheme. We illustrate the methodology with an application to nowcasting of severe weather precipitation fields as obtained from weather radar images, where the severe storm cells are treated as random sets and the wind velocity is used to inform the distributions of the model parameters.     

Subjective elicitation of hyperparameters of a conjugate Dirichlet prior and the corresponding Bayes analysis

Abstract

The paper elicits subjectively the Dirichlet prior hyperparameters based on the realistic opinion collected from the experts. The procedure used for subjective elicitation considers several stages such as the choice of experts, formation of some relevant questions to be asked to the experts for getting their opinion, pooling of opinion, quantification of information and then the formation of exact prior distribution through quantile assessment based on an iterative procedure. The resulting prior distribution is used to provide the Bayes analysis assuming multinomial sampling plan. The results are illustrated by means of a data set involving two life style factors of gallbladder carcinoma patients. The results convey the message that matches closely with the opinion given by the medical experts.     

Bayesian analysis of correlated mixed categorical data by incorporating historical prior information

In this article, we develop statistical models for analysis of correlated mixed categorical (binary and ordinal) response data arising in medical and epidemi-ologic studies. There is evidence in the literature to suggest that models including correlation structure can lead to substantial improvement in precision of estimation or are more appropriate (accurate). We use a very rich class of scale mixture of multivariate normal (SMMVN) iink functions to accommodate heavy tailed distributions. In order to incorporate available historical information, we propose a unified prior elicitation scheme based on SMMVN-link models. Further, simulation-based techniques are developed to assess model adequacy. Finally, a real data example from prostate cancer studies is used to illustrate the proposed methodologies.