On a bivariate Markov process arising in the theory of single-server retrial queues

We introduce a bivariate Markov process which can be seen as the joint process of the channel state and the number of customers in orbit of a Markovian single-server retrial queue with state dependent intensities. We obtain a necessary and sufficient condition for the process to be regular, and necessary and sufficient conditions for ergodicity and recurrence. A product-form formula for the stationary distribution is obtained. Besides, we study the busy period, the number of served customers and other related quantities. We show that for all the above problems there exist "equivalent" birth-and-death processes. However, a "uniformly equivalent" birth-and-death process does not exist.     

Robust likelihood inferences about regression parameters for general bivariate continuous data

This paper introduces a way of modifying the bivariate normal likelihood function. One can use the adjusted likelihood to generate valid likelihood inferences for the regression parameter of interest, even if the bivariate normal assumption is fallacious. The retained asymptotic legitimacy requires no knowledge of the true underlying joint distributions so long as their second moments exist. The extension to the multivariate situations is straightforward in theory and yet appears to be arduous computationally. Nevertheless, it is illustrated that the implementation of this seemingly sophisticated procedure is almost effortless needing only outputs from existing statistical software. The efficacy of the proposed parametric approach is demonstrated via simulations.     

Estimating a generalized correlation coefficient for a generalized bivariate probit model

In this paper we consider semiparametric estimation of a generalized correlation coefficient in a generalized bivariate probit model. The generalized correlation coefficient provides a simple summary statistic measuring the relationship between the two binary decision processes in a general framework. Our semiparametric estimation procedure consists of two steps, combining semiparametric estimators for univariate binary choice models with the method of maximum likelihood for the bivariate probit model with nonparametrically generated regressors. The estimator is shown to be consistent and asymptotically normal. The estimator performs well in our simulation study.     

Failure rate of the minimum and maximum of a multivariate normal distribution

It is well known that if the parent distribution has a nonnegative support and has increasing failure rate (IFR), then all the order statistics have IFR. The result is not necessarily true in the case of bivariate distributions with dependent structures. In this paper we consider a multivariate normal distribution and prove that, the distributions of the minimum and maximum retain the IFR property. Received: September 1999     

Fisher information in order statistics and their concomitants in bivariate censored samples

We evaluate the Fisher information (FI) contained in a collection of order statistics and their concomitants from a bivariate random sample. Special attention is given to Type II censored samples. We present a general decomposition result and recurrence relations that are useful in finding the FI in all types of censored samples. We also obtain some asymptotic results for the FI. For the bivariate normal parent, we obtain explicit and asymptotic expressions for the elements of the FI matrix for Type II censored samples. We discuss implications of our findings on inference on the bivariate normal parameters, especially on the correlation. The first author’s research was supported in part by National Institutes of Health, USA, Grant # M01 RR00034 and the second author’s research was supported by a training grant from the Egyptian government     

Dose–response modelling for bivariate covariates with and without a spike at zero: theory and application to binary outcomes

In epidemiology and clinical research, there is often a proportion of unexposed individuals resulting in zero values of exposure, meaning that some individuals are not exposed and those exposed have some continuous distribution. Examples are smoking or alcohol consumption. We will call these variables with a spike at zero (SAZ). In this paper, we performed a systematic investigation on how to model covariates with a SAZ and derived theoretical odds ratio functions for selected bivariate distributions. We consider the bivariate normal and bivariate log normal distribution with a SAZ. Both confounding and effect modification can be elegantly described by formalizing the covariance matrix given the binary outcome variable Y. To model the effect of these variables, we use a procedure based on fractional polynomials first introduced by Royston and Altman (1994, Applied Statistics 43: 429–467) and modified for the SAZ situation (Royston and Sauerbrei, 2008, Multivariable model‐building: a pragmatic approach to regression analysis based on fractional polynomials for modelling continuous variables, Wiley; Becher et al., 2012, Biometrical Journal 54: 686–700). We aim to contribute to theory, practical procedures and application in epidemiology and clinical research to derive multivariable models for variables with a SAZ. As an example, we use data from a case–control study on lung cancer.     

On the shape of the cross-ratio function in bivariate survival models induced by truncated and folded normal frailty distributions

In shared frailty models for bivariate survival data the frailty is identifiable through the cross-ratio function (CRF), which provides a convenient measure of association for correlated survival variables. The CRF may be used to compare patterns of dependence across models and data sets. We explore the shape of the CRF for the families of one-sided truncated normal and folded normal frailty distributions.     

Semiparametric estimation of a bivariate Tobit model

The existing semiparametric estimation literature has mainly focused on univariate Tobit models and no semiparametric estimation has been considered for bivariate Tobit models. In this paper, we consider semiparametric estimation of the bivariate Tobit model proposed by Amemiya (1974), under the independence condition without imposing any parametric restriction on the error distribution. Our estimator is shown to be consistent and asymptotically normal, and simulation results show that our estimator performs well in finite samples. It is also worth noting that while Amemiya’s (1974) instrumental variables estimator (IV) requires the normality assumption, our semiparametric estimator actually outperforms his IV estimator even when normality holds. Our approach can be extended to higher dimensional multivariate Tobit models.     

Sulla dominaza stocastical multivariata

Summary In this paper a sufficient condition for multivariate stochastic dominance is proposed. Such a condition is particularly intersesting in the bivariate case because of its simplicity. A numerical example is presented and worked out in detail.

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Lavoro eseguito nell'ambito del Contributo C.N.R. n. 84.00593.10     

On a recent characterization of the bivariate normal distribution

Summary A recent characterization of the bivariate normal distribution is recalled and a few remarks regarding this characterization and its extension to the multivariate case are given.     

Estimation in a generalization of bivariate probit models with dummy endogenous regressors

The purpose of this paper is to provide guidelines for empirical researchers who use a class of bivariate threshold crossing models with dummy endogenous variables. A common practice employed by the researchers is the specification of the joint distribution of unobservables as a bivariate normal distribution, which results in a bivariate probit model. To address the problem of misspecification in this practice, we propose an easy‐to‐implement semiparametric estimation framework with parametric copula and nonparametric marginal distributions. We establish asymptotic theory, including root‐n normality, for the sieve maximum likelihood estimators that can be used to conduct inference on the individual structural parameters and the average treatment effect (ATE). In order to show the practical relevance of the proposed framework, we conduct a sensitivity analysis via extensive Monte Carlo simulation exercises. The results suggest that estimates of the parameters, especially the ATE, are sensitive to parametric specification, while semiparametric estimation exhibits robustness to underlying data‐generating processes. We then provide an empirical illustration where we estimate the effect of health insurance on doctor visits. In this paper, we also show that the absence of excluded instruments may result in identification failure, in contrast to what some practitioners believe.     

Bivariate and multivariate scaled association models. An application to homogamy of social origin and education in Hungary between 1930 and 1979

In this article we compare bivariate and multivariate models for homogamy of social origin and education to test whether bivariate models of homogamy lead to biased results. We use data on Hungarian couples married between 1930 and 1979 and loglinear models of scaled association. The results indicate some differences between bivariate and multivariate analyses. At each point of time bivariate models overestimate homogamy, both with respect to education and social origin. However, results on trends in time do not differ much between the two analyses. The exception is the period 1940–1959, in which bivariate analysis showed decreasing educational homogamy, and multivariate analysis showed an increasing trend. The latter finding can be explained by declining homogamy of social origin, as well as the weaker reproduction and cross-effects in this period.     

泊松积分在概率论中的应用

文章介绍了泊松积分在正态分布、二维正态分布、对数正态分布等问题中常见的应用,并给出了具体的例子。     

On extremes of bivariate residual lifetimes from generalized Marshall–Olkin and time transformed exponential models

We study here extremes of residuals of the bivariate lifetime and the residual of extremes of the two lifetimes. In the case of generalized Marshall–Olkin model and the total time transformed exponential model, we first present some sufficient conditions for the extremes of residuals to be stochastically larger than the residual of the corresponding extremes, and then investigate the stochastic order of the residual of extremes of the two lifetimes based on the majorization of the age vector of the residuals.     

A moment-based test for extreme-value dependence

This paper proposes a new rank-based test of extreme-value dependence. The procedure is based on the first three moments of the bivariate probability integral transform of the underlying copula. It is seen that the test statistic is asymptotically normal and its finite- and large-sample variance are calculated explicitly. Consistent plug-in estimators for the variance are proposed, and a fast algorithm for their computation is given. Although it is shown via counterexamples that no test based on the probability integral transform can be consistent, the proposed procedure achieves good power against common alternatives, both in finite samples and asymptotically.     

On measures of technical inefficiency and production uncertainty in stochastic frontier production model with correlated error components

Analysis of the behavior of technical inefficiency with respect to parameters and variables of a stochastic frontier model is a neglected area of research in frontier literature. An attempt in this direction, however, has recently been made. It has been shown that in a “standard” stochastic frontier model that both the firm level technical inefficiency and the production uncertainty are monotonically decreasing with observational error. In this paper we show, considering a stochastic frontier model whose error components are jointly distributed as truncated bivariate normal, that this property holds if and only if the distribution of observational error is negatively skewed. We also derive a necessary and sufficient condition under which both firm level technical inefficiency and production uncertainty are monotonically increasing with noise-inefficiency correlation. We next propose a new measure of the industry level production uncertainty and establish the necessary and sufficient condition for firm level technical inefficiency and production uncertainty to be monotonically increasing with industry level production uncertainty. We also study the limiting probabilistic behavior of these conditions under different parametric configuration of our model. Finally we carry out Monte Carlo simulations to study the sample behavior of the population monotonic property of the firm level technical inefficiency and production uncertainty in our model.

Analyzing the Relationships between Individual Criminal Behavior and the Severity and Probability of Punishment

This paper deals with problems encountered in analyzing how an individual is deterred from committing a crime by the severity and probability of punishment. It is argued that it might be advantageous to base such an analysis on a model of maximization of expected utility. According to this model, the attractiveness of committing a crime is strongly affected by the product of the relative utility of committing the crime and being punished as compared with the utility of not committing the crime times the probability of the punishment. This implies that the bivariate linear relationships of choices to commit or not to commit crime and the severity and probability factors are dependent on the variations in both these factors and their mean values. In this paper, these bivariate relationships are analyzed in two ways – formally-algebraically and by numerical examples.     

Large sample properties for a class of copulas in bivariate survival analysis

This work is concerned with asymptotic properties of the bivariate survival function estimator using the functional relationship between marginal survival functions and a class of copulas for the dependence structure. Specifically, we study consistency and weak convergence of the bivariate survival function estimator obtained considering a two-step procedure of estimation. The obtained results are found from a key decomposition of the bivariate survival function in quantities that can be studied separately. In particular, we use relating results to almost sure and weak convergence of estimators, almost sure convergence of uniformly equicontinuous functions, and the delta method for functionals.     

Multivariate subset autoregressive modelling with zero constraints for detecting ‘overall causality’

The necessary and sufficient condition to test for ‘overall causality’, i.e., the presence of Granger- causality and instantaneous causal relations, in a bivariate and trivariate autoregressive model with recursive form is discussed. It is argued that the conventional AR model (the reduced form AR) is a more straightforward and effective means of testing for ‘overall causality’. To detect instanta- neous causality it is proposed to select the best subset system in a residual regression system in conjunction with model selection criteria. The Canadian money-income-bank rate system is re-examined in this way and by using a previously proposed algorithm we identify the optimum multivariate subset AR with constraints to detect whether there is ‘overall causality’ in that system.     

A new bivariate generalized Poisson distribution

In this paper, a new bivariate generalized Poisson distribution (GPD) that allows any type of correlation is defined and studied. The marginal distributions of the bivariate model are the univariate GPDs. The parameters of the bivariate distribution are estimated by using the moment and maximum likelihood methods. Some test statistics are discussed and one numerical data set is used to illustrate the applications of the bivariate model.