Limit theorems for multivariate discrete distributions

According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson's distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.     

A characterization of the gamma process by conditional moments

Summary The gamma process is determined by the form of conditional expectations and conditional variances. Also a new characterization of the gamma law is obtained and then applied to characterize the gamma process among the processes with independent increments.     

A unified approach to bivariate discrete distributions

Here we introduce a bivariate generalized hypergeometric factorial moment distribution (BGHFMD) through its probability generating function (p.g.f.) whose marginal distributions are the generalized hypergeometric factorial moment distributions introduced by Kemp and Kemp (Bull Int Stat Inst 43:336–338,1969). Well-known bivariate versions of distributions such as binomial, negative binomial and Poisson are special cases of this distribution. A genesis of the distribution and explicit closed form expressions for the probability mass function of the BGHFMD, its factorial moments and the p.g.f.’s of its conditional distributions are derived here. Certain recurrence relations for probabilities, moments and factorial moments of the bivariate distribution are also established.     

A bivariate Poisson count data model using conditional probabilities

The applied econometrics of bivariate count data predominantly focus on a bivariate Poisson density with a correlation structure that is very restrictive. The main limitation is that this bivariate distribution excludes zero and negative correlation. This paper introduces a new model which allows for a more flexible correlation structure. To this end the joint density is decomposed by means of the multiplication rule in marginal and conditional densities. Simulation experiments and an application of the model to recreational data are presented.     

Consistent estimation of a general nonparametric regression function in time series

We propose an estimator of the conditional distribution of X_t|X_t−1,X_t−2,…, and the corresponding regression function , where the conditioning set is of infinite order. We establish consistency of our estimator under stationarity and ergodicity conditions plus a mild smoothness condition.     

Bayesian modeling of joint and conditional distributions

In this paper, we study a Bayesian approach to flexible modeling of conditional distributions. The approach uses a flexible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution. We use a finite mixture of multivariate normals (FMMN) to estimate the joint distribution. The conditional distributions can then be assessed analytically or through simulations. The discrete variables are handled through the use of latent variables. The estimation procedure employs an MCMC algorithm. We provide a characterization of the Kullback–Leibler closure of FMMN and show that the joint and conditional predictive densities implied by the FMMN model are consistent estimators for a large class of data generating processes with continuous and discrete observables. The method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a Bayesian alternative to semi- and non-parametric models such as quantile and kernel regression. In experiments, the method compares favorably with classical nonparametric and alternative Bayesian methods.     

A tale of forecasting 1001 series : The Bayesian knight strikes again

In this paper it is pointed out that a Bayesian forecasting procedure performed better according to an average mean square error (MSE) criterion than the many other forecasting procedures utilized in the forecasting experiments reported in an extensive study by Makridakis et al. (1982). This fact was not mentioned or discussed by the authors. Also, it is emphasized that if criteria other than MSE are employed, Bayesian forecasts that are optimal relative to them should be employed. Specific examples are provided and analyzed to illustrate this point.     

Non‐parametric Analysis of Gap Times for Multiple Event Data: An Overview

Multiple event data are frequently encountered in medical follow‐up, engineering and other applications when the multiple events are considered as the major outcomes. They may be repetitions of the same event (recurrent events) or may be events of different nature. Times between successive events (gap times) are often of direct interest in these applications. The stochastic‐ordering structure and within‐subject dependence of multiple events generate statistical challenges for analysing such data, including induced dependent censoring and non‐identifiability of marginal distributions. This paper provides an overview of a class of existing non‐parametric estimation methods for gap time distributions for various types of multiple event data, where sampling bias from induced dependent censoring is effectively adjusted. We discuss the statistical issues in gap time analysis, describe the estimation procedures and illustrate the methods with a comparative simulation study and a real application to an AIDS clinical trial. A comprehensive understanding of challenges and available methods for non‐parametric analysis can be useful because there is no existing standard approach to identifying an appropriate gap time method that can be used to address research question of interest. The methods discussed in this review would allow practitioners to effectively handle a variety of real‐world multiple event data.     

Testing joint hypotheses when one of the alternatives is one-sided

We propose a class of statistics where the direction of one of the alternatives is incorporated. It is obtained by modifying a class of multivariate tests with elliptical confidence regions, not necessarily arising from normal-based distribution theory. The resulting statistics are easy to compute, they do not require the re-estimation of models subject to one-sided inequality restrictions, and their distributions do not require bounds-based inference. We derive explicit distribution and power functions, using them to prove some desirable properties of our class of modified tests. We then illustrate the relevance of the method by applying it to devising an improved test of random walks in autoregressive models with deterministic components. In this example, the usual alternative to a unit root is one-sided in the direction of stable roots, while deterministic components are allowed to go either way, and we show that it is beneficial to take the partially one-sided nature of the alternative into account.     

UMP unbiased tests for multiparameter testing problems with restricted alternatives

Applications of the standard theory of UMP unbiased tests depends on conditions which in general are difficult to verify. In the present paper, however, we suggest more simple rules for applying this theory for regular exponential families of distributions. This approach leads to UMP unbiased tests for various multiparameter testing problems with restricted alternatives, and is shown to give justification for conditional tests for testing symmetry, diagonals-parameter symmetry and independence in two-way contingency tables. The derived tests are shown to possess attractive small sample properties. Received: June 1998     

Some asymptotic results of a non‐parametric conditional mode estimator for functional time‐series data

We consider the estimation of the conditional mode function when the covariates take values in some abstract function space. The main goal of this paper was to establish the almost complete convergence and the asymptotic normality of the kernel estimator of the conditional mode when the process is assumed to be strongly mixing and under the concentration property over the functional regressors. Some applications are given. This approach can be applied in time‐series analysis to the prediction and confidence band building. We illustrate our methodology by using El Nio data.     

On truncated bivariate discrete distributions: A unified treatment

A unified treatment of three types of zero class truncation for bivariate discrete distributions is presented. Using the probability generating function approach, various properties of the truncated distributions are examined in association with the corresponding properties of the initial complete form of the distribution. Expressions for moments and conditional distributions are also obtained. Bivariate versions of the Thomas and the Intervened Poisson distributions are introduced and used as illustrative examples. Received November 2000/Revised March 2002     

Semiparametric time series models driven by latent factor

We introduce a class of semiparametric time series models (SemiParTS) driven by a latent factor process. The proposed SemiParTS class is flexible because, given the latent process, only the conditional mean and variance of the time series are specified. These are the primary features of SemiParTS: (i) no parametric form is assumed for the conditional distribution of the time series given the latent process; (ii) it is suitable for a wide range of data: non-negative, count, bounded, binary, and real-valued time series; (iii) it does not constrain the dispersion parameter to be known. The quasi-likelihood inference is employed in order to estimate the parameters in the mean function. Here, we derive explicit expressions for the marginal moments and for the autocorrelation function of the time series process so that a method of moments can be employed to estimate the dispersion parameter and also the parameters related to the latent process. Simulated results that aim to check the proposed estimation procedure are presented. Forecasting procedures are proposed and evaluated in simulated and real data. Analyses of the number of admissions in a hospital due to asthma and a total insolation time series illustrate the potential for practical situations that involve the proposed models.     

Bayesian median autoregression for robust time series forecasting

We develop a Bayesian median autoregressive (BayesMAR) model for time series forecasting. The proposed method utilizes time-varying quantile regression at the median, favorably inheriting the robustness of median regression in contrast to the widely used mean-based methods. Motivated by a working Laplace likelihood approach in Bayesian quantile regression, BayesMAR adopts a parametric model bearing the same structure as autoregressive models by altering the Gaussian error to Laplace, leading to a simple, robust, and interpretable modeling strategy for time series forecasting. We estimate model parameters by Markov chain Monte Carlo. Bayesian model averaging is used to account for model uncertainty, including the uncertainty in the autoregressive order, in addition to a Bayesian model selection approach. The proposed methods are illustrated using simulations and real data applications. An application to U.S. macroeconomic data forecasting shows that BayesMAR leads to favorable and often superior predictive performance compared to the selected mean-based alternatives under various loss functions that encompass both point and probabilistic forecasts. The proposed methods are generic and can be used to complement a rich class of methods that build on autoregressive models.     

Predicting loss given default (LGD) for residential mortgage loans: A two-stage model and empirical evidence for UK bank data

With the implementation of the Basel II regulatory framework, it became increasingly important for financial institutions to develop accurate loss models. This work investigates the loss given default (LGD) of mortgage loans using a large set of recovery data of residential mortgage defaults from a major UK bank. A Probability of Repossession Model and a Haircut Model are developed and then combined to give an expected loss percentage. We find that the Probability of Repossession Model should consist of more than just the commonly used loan-to-value ratio, and that the estimation of LGD benefits from the Haircut Model, which predicts the discount which the sale price of a repossessed property may undergo. This two-stage LGD model is shown to perform better than a single-stage LGD model (which models LGD directly from loan and collateral characteristics), as it achieves a better R² value and matches the distribution of the observed LGD more accurately.     

Complementary information for skewness measures

This paper introduces some new elements to measure the skewness of a probability distribution, suggesting that a given distribution can have both positive and negative skewness, depending on the centred sub‐interval of the support set being observed. A skewness function for positive reals is defined, from which a bivariate index of positive–negative skewness is obtained. Certain interesting properties of this new index are studied, and they are also obtained for some common discrete distributions. We show the advantages of their use as a complement to the information derived by traditional measures of skewness.     

A new condition for identifiability of finite mixture distributions

In this paper a sufficient condition for the identifiability of finite mixtures is given. This condition is less restrictive than Teicher’s condition Teicher H, Ann Math Stat 34:1265–1269 (1963) and therefore it can be applied to a wider range of families of mixtures. In particular, it applies to the classes of all finite mixtures of Log-gamma and of reversed Log-gamma distributions. These families have been already studied by Henna J Jpn Stat Soc 24:193–200 (1994) using another condition, different from Teicher’s, but more difficult to check in many cases. Furthermore, the result given in this paper is very appropiated for the case of mixtures of the union of different distribution families. To illustrate this an application to the class of all finite mixtures generated by the union of Lognormal, Gamma and Weibull distributions is given, where Teicher’s and Henna’s conditions are not applicable     

附条件不起诉的监督救济机制

附条件不起诉是人民检察院对刑事案件审查后作出的一种不起诉决定,是起诉裁量权的运用,也是起诉便宜原则的具体体现。本文在借鉴国内外经验基础上,重点分析了对附条件不起诉制度的监督救济机制。     

Locally D- and c-optimal designs for Poisson and negative binomial regression models

The research on optimal experimental designs for nonlinear regression models is of great interest because these models are used to characterize chemical, biological or agricultural phenomena. Much of them involve an exponential decay. In this paper, locally D- and c-optimal designs are derived analytically for Poisson and negative binomial regression models.     

An identity for the product moments of order statistics

A general identity for the product moments of successive order statistics is given, which is valid in a class of probability distributions including Weibull, Pareto, exponential and Burr distributions.