A choice probability characterization of generalized extreme value models

A set of necessary and sufficient conditions is established for the representability of choice probabilities by additive random utility models with generalized extreme value (GEV) distributions of utilities. These conditions yield an operational testing procedure for GEV-representability which does not require explicit construction of the underlying distribution of utilities. In addition, this characterization of GEV models reveals a number of their underlying behavioral features.     

Bankruptcy prediction by generalized additive models

We compare several accounting‐based models for bankruptcy prediction. The models are developed and tested on large data sets containing annual financial statements for Norwegian limited liability firms. Out‐of‐sample and out‐of‐time validation shows that generalized additive models significantly outperform popular models like linear discriminant analysis, generalized linear models and neural networks at all levels of risk. Further, important issues like default horizon and performance depreciation are examined. We clearly see a performance depreciation as the default horizon is increased and as time goes by. Finally a multi‐year model, developed on all available data from three consecutive years, is compared with a one‐year model, developed on data from the most recent year only. The multi‐year model exhibits a desirable robustness to yearly fluctuations that is not present in the one‐year model. Copyright © 2006 John Wiley & Sons, Ltd.     

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

This paper develops a robust and efficient estimation procedure for quantile partially linear additive models with longitudinal data, where the nonparametric components are approximated by B spline basis functions. The proposed approach can incorporate the correlation structure between repeated measures to improve estimation efficiency. Moreover, the new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations method for non-normal correlated random errors. However, the proposed estimating functions are non-smooth and non-convex. In order to reduce computational burdens, we apply the induced smoothing method for fast and accurate computation of the parameter estimates and its asymptotic covariance. Under some regularity conditions, we establish the asymptotically normal distribution of the estimators for the parametric components and the convergence rate of the estimators for the nonparametric functions. Furthermore, a variable selection procedure based on smooth-threshold estimating equations is developed to simultaneously identify non-zero parametric and nonparametric components. Finally, simulation studies have been conducted to evaluate the finite sample performance of the proposed method, and a real data example is analyzed to illustrate the application of the proposed method.     

Jackknifing in generalized linear models

In a generalized linear model, the jackknife estimator of the asymptotic covariance matrix of the maximum likelihood estimator is shown to be consistent. The corresponding jackknife studentized statistic is asymptotically normal. In addition, these results remain true even if there exist unequal dispersion parameters in the model. On the other hand, the variance estimator and the studentized statistic based on the standard method (substitution and linearization) do not enjoy this robustness property against the presence of unequal dispersion parameters.This research was supported by an Operating Grant from the Natural Science and Engineering Research Council of Canada.     

Prediction and calibration in generalized linear models

This paper concerns prediction and calibration in generalized linear models. A new predictive procedure, giving improved prediction intervals, is briefly reviewed and further theoretical results, useful for calculations, are presented. Indeed, the calibration problem is faced within the classical approach and a suitable solution is obtained by inverting the associated improved prediction procedure. This calibration technique gives accurate confidence regions and it constitutes a substantial improvement over both the estimative solution and the naive solution, which involves, even for non-linear and non-normal models, the results available for the linear Gaussian case. Finally, some useful explicit formulae for the construction of prediction and calibration intervals are presented, with regard to generalized linear models with alternative error terms and link functions. This research was partially supported by a grant from Ministero dell’Instruzione, dell’Università e della Ricerca, Italy.     

Fuzzy clusterwise quasi-likelihood generalized linear models

The quasi-likelihood method has emerged as a useful approach to the parameter estimation of generalized linear models (GLM) in circumstances where there is insufficient distributional information to construct a likelihood function. Despite its flexibility, the quasi-likelihood approach to GLM is currently designed for an aggregate-sample analysis based on the assumption that the entire sample of observations is taken from a single homogenous population. Thus, this approach may not be suitable when heterogeneous subgroups exist in the population, which involve qualitatively distinct effects of covariates on the response variable. In this paper, the quasi-likelihood GLM approach is generalized to a fuzzy clustering framework which explicitly accounts for such cluster-level heterogeneity. A simple iterative estimation algorithm is presented to optimize the regularized fuzzy clustering criterion of the proposed method. The performance of the proposed method in recovering parameters is investigated based on a Monte Carlo analysis involving synthetic data. Finally, the empirical usefulness of the proposed method is illustrated through an application to actual data on the coupon usage behaviour of a sample of consumers.     

Conditional extreme value models: fallacies and pitfalls

Conditional extreme value models have been introduced by Heffernan and Resnick (Ann. Appl. Probab., 17, 537–571, 2007) to describe the asymptotic behavior of a random vector as one specific component becomes extreme. Obviously, this class of models is related to classical multivariate extreme value theory which describes the behavior of a random vector as its norm (and therefore at least one of its components) becomes extreme. However, it turns out that this relationship is rather subtle and sometimes contrary to intuition. We clarify the differences between the two approaches with the help of several illuminative (counter)examples. Furthermore, we discuss marginal standardization, which is a useful tool in classical multivariate extreme value theory but, as we point out, much less straightforward and sometimes even obscuring in conditional extreme value models. Finally, we indicate how, in some situations, a more comprehensive characterization of the asymptotic behavior can be obtained if the conditions of conditional extreme value models are relaxed so that the limit is no longer unique.     

Robust and accurate inference for generalized linear models

In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022–1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154–1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically χ²-distributed as the three classical tests but with a relative error of order O(n⁻¹). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.     

Model averaging for semiparametric additive partial linear models

To improve the prediction accuracy of semiparametric additive partial linear models(APLM) and the coverage probability of confidence intervals of the parameters of interest,we explore a focused information criterion for model selection among ALPM after we estimate the nonparametric functions by the polynomial spline smoothing,and introduce a general model average estimator.The major advantage of the proposed procedures is that iterative backfitting implementation is avoided,which thus results in gains in co...     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

Adaptive quasi-likelihood estimate in generalized linear models

This paper gives a thorough theoretical treatment on the adaptive quasi-likelihood estimate of the parameters in the generalized linear models. The unknown covariance matrix of the response variable is estimated by the sample. It is shown that the adaptive estimator defined in this paper is asymptotically most efficient in the sense that it is asymptotic normal, and the covariance matrix of the limit distribution coincides with the one for the quasi-likelihood estimator for the case that the covariance matrix of the response variable is completely known.     

双重广义线性模型的经验似然推断

基于截面经验似然方法,将双重广义线性模型的拟似然估计方程作为截面经验似然比函数的约束条件,构造了均值模型和散度模型未知参数的置信区间.最后通过数据模拟,将该方法与正态逼近方法比较,说明了该方法是有效和可行的.     

Heavy-traffic extreme value limits for Erlang delay models

We consider the maximum queue length and the maximum number of idle servers in the classical Erlang delay model and the generalization allowing customer abandonment—the M/M/n+M queue. We use strong approximations to show, under regularity conditions, that properly scaled versions of the maximum queue length and maximum number of idle servers over subintervals [0,t] in the delay models converge jointly to independent random variables with the Gumbel extreme value distribution in the quality-and-efficiency-driven (QED) and ED many-server heavy-traffic limiting regimes as n and t increase to infinity together appropriately; we require that t _n →∞ and t _n =o(n ^1/2?ε ) as n→∞ for some ε>0.     

Bayesian hierarchical linear mixed models for additive smoothing splines

Bayesian hierarchical models have been used for smoothing splines, thin-plate splines, and L-splines. In analyzing high dimensional data sets, additive models and backfitting methods are often used. A full Bayesian analysis for such models may include a large number of random effects, many of which are not intuitive, so researchers typically use noninformative improper or nearly improper priors. We investigate propriety of the posterior for these cases. Our findings extend known results for normal linear mixed models to certain cases with Bayesian additive smoothing spline models. Supported by National Science Foundation grant SES-0351523 and by National Institutes of Health grants R01-CA100760 and R01-MH071418.     

Variable selection via generalized SELO-penalized linear regression models

The seamless-L_0(SELO) penalty is a smooth function on [0, ∞) that very closely resembles the L_0 penalty, which has been demonstrated theoretically and practically to be effective in nonconvex penalization for variable selection. In this paper, we first generalize SELO to a class of penalties retaining good features of SELO, and then propose variable selection and estimation in linear models using the proposed generalized SELO(GSELO) penalized least squares(PLS) approach. We show that the GSELO-PLS procedure possesses the oracle property and consistently selects the true model under some regularity conditions in the presence of a diverging number of variables. The entire path of GSELO-PLS estimates can be efficiently computed through a smoothing quasi-Newton(SQN) method. A modified BIC coupled with a continuation strategy is developed to select the optimal tuning parameter. Simulation studies and analysis of a clinical data are carried out to evaluate the finite sample performance of the proposed method. In addition, numerical experiments involving simulation studies and analysis of a microarray data are also conducted for GSELO-PLS in the high-dimensional settings.     

Multivariate arma models with generalized autoregressive linear innovation

In this paper we develop, in a multivariate framework, an alternative approach to the classical non linear analysis of time series. The proposed class of stochastic processes, of which the bilinear model is a special case, is based on a generalized autoregressive modelling of linear innovations. The probability structure is analyzed under quite general conditions. Moreover an important subclass of bilinear processes is studied in greater details. Finally, the usefulness of the results is illustrated via a numerical study.     

Comparing extreme models when the sign of the extreme value index is known

In the literature on analyzing extremes, both generalized Pareto distributions and Pareto distributions are employed to infer the tail of a distribution with a known positive extreme value index. Similar studies exist for a known negative extreme value index. Intuitively, one should not employ the generalized Pareto distribution in the case of knowing the sign of the extreme value index. In this work, we show that fitting a generalized Pareto distribution is equivalent to the model in Hall (1982) in the case of a negative extreme value index, in both improving the rate of convergence and including the bias term of the asymptotic results of that reference. When the extreme value index is known to be positive, we show that fitting a generalized Pareto distribution may be preferred in some cases determined by a so-called second-order parameter and the extreme value index itself.     

Estimating bank default with generalised extreme value regression models

The paper proposes a novel model for the prediction of bank failures, on the basis of both macroeconomic and bank-specific microeconomic factors. As bank failures are rare, in the paper we apply a regression method for binary data based on extreme value theory, which turns out to be more effective than classical logistic regression models, as it better leverages the information in the tail of the default distribution. The application of this model to the occurrence of bank defaults in a highly bank dependent economy (Italy) shows that, while microeconomic factors as well as regulatory capital are significant to explain proper failures, macroeconomic factors are relevant only when failures are defined not only in terms of actual defaults but also in terms of mergers and acquisitions. In terms of predictive accuracy, the model based on extreme value theory outperforms classical logistic regression models.     

Conflict analysis using Bayesian neural networks and generalized linear models

The study of conflict analysis has recently become more important due to current world events. Despite numerous quantitative analyses on the study of international conflict, the statistical results are often inconsistent with each other. The causes of conflict, however, are often stable and replicable when the prior probability of conflict is large. As there has been much conjecture about neural networks being able to cope with the complexity of such interconnected and interdependent data, we formulate a statistical version of a neural network model and compare the results to those of conventional statistical models. We then show how to apply Bayesian methods to the preferred model, with the aim of finding the posterior probabilities of conflict outbreak and hence being able to plan for conflict prevention.