On characterizations of distributions by regression of adjacent generalized order statistics

Let , r ≥ 1, denote generalized order statistics, with arbitrary parameters , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. where are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some .     

Probabilistic frontier regression model for multinomial ordinal type output data

Journal of Productivity Analysis - This paper proposes a probabilistic frontier regression model for multinomial ordinal type output data. We consider some of the output categories as...     

Multivariate regression models for panel data

The paper examines the relationship between heterogeneity bias and strict exogeneity in a distributed lag regression of y on x. The relationship is very strong when x is continuous, weaker when x is discrete, and non-existent as the order of the distributed lag becomes infinite. The individual specific random variables introduce nonlinearity and heteroskedasticity; so the paper provides an appropriate framework for the estimation of multivariate linear predictors. Restrictions are imposed using a minimum distance estimator. It is generally more efficient than the conventional estimators such as quasi-maximum likelihood. There are computationally simple generalizations of two- and three-stage least squares that achieve this efficiency gain. Some of these ideas are illustrated using the sample of Young Men in the National Longitudinal Survey. The paper reports regressions on the leads and lags of variables measuring union coverage, SMSA, and region. The results indicate that the leads and lags could have been generated just by a random intercept. This gives some support for analysis of covariance type estimates; these estimates indicate a substantial heterogeneity bias in the union, SMSA, and region coefficients.     

Determining of threat perceptions affecting direction of the US defense expenditures by using ordinal regression models

The article examines whether the US threat perceptions defined in terms of federal government national defense outlays in billions of constant (FY 2000) dollars change along with periodical changes in international politics between 1945 and 2007. Three different models affecting direction of the US defense expenditures are developed. The first model are estimated by using five link functions even though results of only two of them, complementary log–log and cauchit, are presented. As complementary log–log produced the best results, others models are predicted by using only this function. The parameter estimates of complementary log–log function for the first model indicate that four of these variables (Ford, Carter, Reagan and Bush Sr.) out of eleven are significant in the category of presidents. “Truman Docrtrine/Cominform”, “Korean War”, “Vietnam War”, and “Invasion of Iraq” also seem to be the important independent variables on empirical grounds for the first model. While “Party”, “Invasion of Iraq”, “Vietnam War”, “Korean War”, and “Cuban Missile Crisis” constitute the important independent variables on empirical grounds for the second model, “Korean War”, “Vietnam War”, “Invasion of Iraq”, “Truman Docrtrine/Cominform”, “The Cold War and New World Order”, and “Cuban Missile Crisis” are important independent variables on empirical grounds for the third model. Estimations based on these three models therefore suggest that aforementioned independent variables do indeed have effect on the US defense expenditures.     

Specification testing for regression models with dependent data

We examine a consistent test for the correct specification of a regression function with dependent data. The test is based on the supremum of the difference between the parametric and nonparametric estimates of the regression model. Rather surprisingly, the behaviour of the test depends on whether the regressors are deterministic or stochastic. In the former situation, the normalization constants necessary to obtain the limiting Gumbel distribution are data dependent and difficult to estimate, so it may be difficult to obtain valid critical values, whereas, in the latter, the asymptotic distribution may not be even known. Because of that, under very mild regularity conditions, we describe a bootstrap analogue for the test, showing its asymptotic validity and finite sample behaviour in a small Monte-Carlo experiment.     

Competing risks models in discrete time with nominal or ordinal categories of response

Two principle approaches to the modelling of competing risks in discrete time are considered. In the first approach which is based on the separation between failure and cause specific response only the causes of failure are considered as ordered. The second approach which is based on the conditional response given interval [a _t-1,a_r) is reached allows for an ordering of causes of failureand the category ‘no failure’. The latter approach is shown to be more general. It is shown that the considered competing risks models may be estimated within the framework of generalized linear models. A data set concerning duration of unemployment illustrates the approaches.     

Analysis of a mixed model for ordinal data by iterative re-weighted REML

An estimation procedure will be presented for a class of threshold models for ordinal data. These models may include both fixed and random effects with associated components of variance on an underlying scale. The residual error distribution on the underlying scale may be rendered greater flexibility by introducing additional shape parameters, e.g. a kurtosis parameter or parameters to model heterogeneous residual variances as a function of factors and covariates. The estimation procedure is an extension of an iterative re-weighted restricted maximum likelihood procedure, originally developed for generalized linear mixed models. This procedure will be illustrated with a practical problem involving damage to potato tubers and with data from animal breeding and medical research from the literature.     

Some identification and estimation results for regression models with stochastically varying coefficients

Although various theoretical and applied papers have appeared in recent years concerned with the estimation and use of regression models with stochastically varying coefficients, little is available in the literature on the properties of the proposed estimators or the identifiability of the parameters of such models. The present paper derives sufficient conditions under which the maximum likelihood estimator is consistent and asymptotically normal and also provides sufficient conditions for the estimation of regression models with stationary stochastically varying coefficients. In many instances these requirements are found to have simple, intuitively appealing interpretations. Consistency and asymptotic normality is also proven for a two-step estimator and a method suggested by Rosenberg for generating initial estimates.     

Testing panel data regression models with spatial error correlation

This paper derives several lagrange multiplier (LM) tests for the panel data regression model with spatial error correlation. These tests draw upon two strands of earlier work. The first is the LM tests for the spatial error correlation model discussed in Anselin (Spatial Econometrics: Methods and Models, Kluwer Academic Publishers, Dordrecht; Rao's score test in spatial econometrics, J. Statist. Plann. Inference 97 (2001) 113) and Anselin et al. (Regional Sci. Urban Econom. 26 (1996) 77), and the second is the LM tests for the error component panel data model discussed in Breusch and Pagan (Rev. Econom. Stud. 47(1980) 239) and Baltagi et al. (J. Econometrics 54 (1992) 95). The idea is to allow for both spatial error correlation as well as random region effects in the panel data regression model and to test for their joint significance. Additionally, this paper derives conditional LM tests, which test for random regional effects given the presence of spatial error correlation. Also, spatial error correlation given the presence of random regional effects. These conditional LM tests are an alternative to the one-directional LM tests that test for random regional effects ignoring the presence of spatial error correlation or the one-directional LM tests for spatial error correlation ignoring the presence of random regional effects. We argue that these joint and conditional LM tests guard against possible misspecification. Extensive Monte Carlo experiments are conducted to study the performance of these LM tests as well as the corresponding likelihood ratio tests.     

A note on randomized response models for quantitative data

Standard randomized response (RR) models deal primarily with surveys which usually require a yes or a no response to a sensitive question, or a choice for responses from a set of nominal categories. As opposed to that, Eichhorn and Hayre (1983) have considered survey models involving a quantitative response variable and proposed an RR technique for it. Such models are very useful in studies involving a measured response variable which is highly sensitive in its nature. Eichhorn and Hayre obtained an unbiased estimate for the expectation of the quantitative response variable of interest. In this note we propose a procedure which uses a design parameter (controlled by the experimenter) that generalizes Eichhorn and Hayres results. Such a procedure yields an estimate for the desired expectation which has a uniformly smaller variance.Acknowledgements We are grateful to two referees for their valuable and constructive comments.     

On existence and uniqueness of maximum likelihood estimates in quantal and ordinal response models

Summary For quantal and ordinal response models, conditions on existence and uniqueness of maximum likelhood estimates are presented. Results are derived from general results on direction sets and spaces associated with a proper concave function. If each summand of the log likelihood is in any direction either strictly concave or affine, necessary and sufficient conditions are obtained. If all cell counts are strictly positive, then it is shown that estimates always exist, and that they are unique if all parameters are identifiable. If estimates exist without being unique, results on uniquely estimable linear functions are given, paralleling corresponding results in linear regression. An extension of the maximum likelihood principle is outlined yielding similar results even if the likelihood does not attain its supremum. The logit model, the linear probability model, cumulative and sequential models and binomial response models are considered in detail.     

Moment inequalities for order statistics with applications to characterizations of distributions

General inequalities of Hölder type between moments of order statistics and moments of record values respectively are derived. Special choices of the involved sample sizes and ranks and discussions of when equality is attained in these inequalities yield several characterizations of well known distributions, such as the uniform, polynomial, Pareto, reflected Pareto, exponential, Weibull distribution and some others.     

A class of binary response models for grouped duration data

This paper explores the relationship between conventional models for binary response such as the probit and logit, and the proportional hazard (PH) and related specifications for grouped duration data. I outline a general class of hazard models for grouped duration data based upon the choice of period-specific distribution functions, facilitating a thorough analysis of the implications of various specifications and consideration of various issues of model identification. This class of models nests, among others, the proportional hazard, probit, and logit specifications for interval survival. I consider the implications of various specifications for hazard behaviour, focusing on familiar specifications. While the specifications will generally yield results that are quite similar along a number of dimensions, there are significant differences. The probit model generates non-proportional effects of variables on the discrete hazard, while the logit and PH tend to show only slight non-proportionality. Furthermore, while the effects of variables on the derivatives are considerably larger for the probit specification, the time-pattern of the probit effects is relatively insensitive to changes in explanatory variables. I illustrate these issues by providing an example taken from Katz's (1986) unemployment data from the Panel Study of Income Dynamics.     

Nonparametric density estimation for symmetric distributions by contaminated data

A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed symmetric. The distribution of the other component (admixture) is known. We consider three estimates for the pdf of primary component: a naive one, a symmetrized naive estimate and a symmetrized estimate with adaptive weights. Asymptotic behavior and small sample performance of the estimates are investigated. Some rules of thumb for bandwidth selection are discussed.     

Conditional distributions of generalized order statistics and some characterizations

Generalized order statistics have been introduced in Kamps (1995a). They enable a unified approach to several models of ordered random variables, e.g. (ordinary) order statistics, record values, sequential order statistics, record values from non-identical distributions. The purpose of this paper is to develop conditional distributions of one generalized order statistic given another and to characterize the underlying continuous distribution by different conditional expectations. Well-known results for ordinary order statistics and record values are extended to generalized order statistics. Received: July 1997     

Functional regression of continuous state distributions

In this paper, we consider a regression model to study the distributional relationship between economic variables. Unlike the classical regression dealing exclusively with mean relationship, our model can be used to analyze the entire dependent structure in distribution. Technically, we treat density functions as random elements and represent the regression relationship as a compact linear operator in the Hilbert spaces of square integrable functions. We propose a consistent estimation procedure for our model, and develop a test to investigate the dependent structure of moments. An empirical example is provided to illustrate how our methodology can be implemented in practical applications.     

Evaluating estimation methods for ordinal data in structural equation modeling

This study examined the performance of two alternative estimation approaches in structural equation modeling for ordinal data under different levels of model misspecification, score skewness, sample size, and model size. Both approaches involve analyzing a polychoric correlation matrix as well as adjusting standard error estimates and model chi-squared, but one estimates model parameters with maximum likelihood and the other with robust weighted least-squared. Relative bias in parameter estimates and standard error estimates, Type I error rate, and empirical power of the model test, where appropriate, were evaluated through Monte Carlo simulations. These alternative approaches generally provided unbiased parameter estimates when the model was correctly specified. They also provided unbiased standard error estimates and adequate Type I error control in general unless sample size was small and the measured variables were moderately skewed. Differences between the methods in convergence problems and the evaluation criteria, especially under small sample and skewed variable conditions, were discussed.     

Semiparametric regression models and sensitivity analysis of longitudinal data with nonrandom dropouts

We propose a family of regression models to adjust for nonrandom dropouts in the analysis of longitudinal outcomes with fully observed covariates. The approach conceptually focuses on generalized linear models with random effects. A novel formulation of a shared random effects model is presented and shown to provide a dropout selection parameter with a meaningful interpretation. The proposed semiparametric and parametric models are made part of a sensitivity analysis to delineate the range of inferences consistent with observed data. Concerns about model identifiability are addressed by fixing some model parameters to construct functional estimators that are used as the basis of a global sensitivity test for parameter contrasts. Our simulation studies demonstrate a large reduction of bias for the semiparametric model relatively to the parametric model at times where the dropout rate is high or the dropout model is misspecified. The methodology's practical utility is illustrated in a data analysis.