A Note on Nonlinear Cointegration,Misspecification, and Bimodality

We derive the asymptotic distribution of the ordinary least squares estimator in a regression with cointegrated variables under misspecification and/or nonlinearity in the regressors. We show that, under some circumstances, the order of convergence of the estimator changes and the asymptotic distribution is non-standard. The t-statistic might also diverge. A simple case arises when the intercept is erroneously omitted from the estimated model or in nonlinear-in-variables models with endogenous regressors. In the latter case, a solution is to use an instrumental variable estimator. The core results in this paper also generalise to more complicated nonlinear models involving integrated time series.     

Linear statistics in change-point estimation and their asymptotic behaviour

ABSTRACT The limiting behaviour of Bayes procedures in the asymptotic setting of the change-point estimation problem is studied. It is shown that the distribution of the difference between the Bayes estimator and the parameter converges to the distribution of a fairly complicated random variable. A class of linear statistics is introduced, and the form of the Bayes estimator within this class is deduced. The asymptotic properties of this linear estimator are investigated in two different settings for the prior distribution.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Estimation for the autoregressive moving average process observed with noise

SUMMARY The autoregressive moving average process ARMA (p,q) observed with noise has another ARMA (p,k) representation, where k = max (p,q). Parameters for the ARMA (p,k) representation satisfy some non-linear restrictions. We develop restricted Newton-Raphson estimators of the ARMA (p,k) process which takes advantage of the information given in the non-linear restrictions. The asymptotic relative efficiency of the estimators indicates that the proposed restricted Newton-Raphson estimator is more efficient than the unrestricted Newton-Raphson estimator. In a Monte Carlo experiment, the proposed estimator is shown to perform better than the unrestricted estimator of the ARMA (p,k) process.     

The Asymptotic Covariance Matrix of the Least Squares Estimator in the Stochastic Linear Regression Model: The Case of Elliptically Symmetric Distribution

This article considers the unconditional asymptotic covariance matrix of the least squares estimator in the linear regression model with stochastic explanatory variables. The asymptotic covariance matrix of the least squares estimator of regression parameters is evaluated relative to the standard asymptotic covariance matrix when the joint distribution of the dependent and explanatory variables is in the class of elliptically symmetric distributions. An empirical example using financial data is presented. Numerical examples and simulation experiments are given to illustrate the difference of the two asymptotic covariance matrices.     

On estimating the regression parameter in the simple errors-in-variables model without constant term

Starting from the criterion of minimum asymptotic mean-squared error, this note compares the least squares estimator with the consistent estimator [ybar]/[xbar] where [ybar]and [xbar] are the sample means of the dependent and the independent variable respectively.     

Regression modelling of the flows in an input–output table with accounting constraints

This paper details a method for estimating the unknown parameters of a regression model when the estimates of the dependent variable should be embedded in an input–output table with accounting constraints. Since in regression modelling the dependent variable is usually transformed either to achieve homoscedasticity of the residuals or for a better interpretation of the model, the estimating procedure becomes an optimization problem of an opportunely defined Lagrangian function with non-linear constraints. After detailing the algorithm and deriving the asymptotic distribution of the restricted estimator, the methodology is applied to estimate the flows of tourism within and between Italian regions with a gravity model. The procedure can be seen as an extension of Byron’s (J R Stat Soc Ser A 141:359–367, 1978) balancing method.     

Local linear variable bandwidth hazard rate estimation

A new hazard rate estimator under the random right censorship model is proposed in this article. The estimator arises naturally as a combination of the local linear fitting and variable bandwidth methods. As a consequence, it also inherits the benefits of both approaches. The asymptotic properties of the estimate in the boundary and in the interior of the region of estimation are provided and its asymptotic distribution is established. In addition, an automatic data-driven bandwidth selection procedure is proposed and evaluated via Monte Carlo simulations. Further numerical studies compare the performance of the proposed estimate with that of estimates with similar asymptotic properties.     

Weak convergence for the conditional distribution function in a Koziol–Green model under dependent censoring

In this paper we consider the conditional Koziol–Green model of Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] in which they generalized the Koziol–Green model of Veraverbeke and Cadarso Suárez [2000. Estimation of the conditional distribution in a conditional Koziol–Green model. Test 9, 97–122] by assuming that the association between a censoring time and a time until an event is described by a known Archimedean copula function. They got in this way, an informative censoring model with two different types of informative censoring. Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] derived in this model a non-parametric Koziol–Green estimator for the conditional distribution function of the time until an event, for which they showed the uniform consistency and the asymptotic normality. In this paper we extend their results and prove the weak convergence of the process associated to this estimator. Furthermore we show that the conditional Koziol–Green estimator is asymptotically more efficient in this model than the general copula-graphic estimator of Braekers and Veraverbeke [2005. A copula-graphic estimator for the conditional survival function under dependent censoring. Canad. J. Statist. 33, 429–447]. As last result, we construct an asymptotic confidence band for the conditional Koziol–Green estimator. Through a simulation study, we investigate the small sample properties of this asymptotic confidence band. Afterwards we apply this estimator and its confidence band on a practical data set.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Some Results About Standardization for a Non Confounder in Estimators of (log) Relative Risk

Confounding is very fundamental to the design and analysis of studies of causal effects. A variable is not a confounder if it is not a risk factor to disease or if it has the same distribution in the exposed and unexposed population. Whether or not to adjust for a non confounder to improve the precision of estimation has been argued by many authors. This article shows that if C is a non confounder, the pooled and standardized (log) relative risk estimators are asymptotic normal distributions with the mean being the true (log) relative risk, and that the asymptotic variance of the pooled (log) relative risk estimator is less than that of the stratified estimator.     

Efficient Estimation for Semi-varying Coefficient Model with An Invertible Linear Process Error

For semi-varying-coefficient model with an invertible linear process error, we propose an efficient estimator procedure. This procedure is based on a pre-whitening transformation of the dependent variable that must be estimated from the data. We establish the proposed estimations’ asymptotic normalities, and assess their finite sample performance. Monte Carlo simulation suggest that the efficiency gain can be achieved in moderate-sized samples.     

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.     

Copula-graphic estimation with left-truncated and right-censored data

In Survival Analysis and related fields of research right-censored and left-truncated data often appear. Usually, it is assumed that the right-censoring variable is independent of the lifetime of ultimate interest. However, in particular applications dependent censoring may be present; this is the case, for example, when there exist several competing risks acting on the same individual. In this paper we propose a copula-graphic estimator for such a situation. The estimator is based on a known Archimedean copula function which properly represents the dependence structure between the lifetime and the censoring time. Therefore, the current work extends the copula-graphic estimator in de Uña-Álvarez and Veraverbeke [Generalized copula-graphic estimator. Test. 2013;22:343–360] in the presence of left-truncation. An asymptotic representation of the estimator is derived. The performance of the estimator is investigated in an intensive Monte Carlo simulation study. An application to unemployment duration is included for illustration purposes.     

A small sample confidence interval for autoregressive parameters

This paper is concerned with interval estimation of an autoregressive parameter when the parameter space allows for magnitudes outside the unit interval. In this case, intervals based on the least-squares estimator tend to require a high level of numerical computation and can be unreliable for small sample sizes. Intervals based on the asymptotic distribution of instrumental variable estimators provide an alternative. If the instrument is taken to be the sign function, the interval is centered at the Cauchy estimator and a large sample interval can be created by estimating the standard error of this estimator. The interval proposed in this paper avoids estimating this standard error and results in a small sample improvement in coverage probability. In fact, small sample coverage is exact when the innovations come from a normal distribution.     

Smooth Plug‐in Inverse Estimators in the Current Status Continuous Mark Model

Abstract. We consider the problem of estimating the joint distribution function of the event time and a continuous mark variable when the event time is subject to interval censoring case 1 and the continuous mark variable is only observed in case the event occurred before the time of inspection. The non‐parametric maximum likelihood estimator in this model is known to be inconsistent. We study two alternative smooth estimators, based on the explicit (inverse) expression of the distribution function of interest in terms of the density of the observable vector. We derive the pointwise asymptotic distribution of both estimators.     

Consistency and asymptotic distribution of the Theil–Sen estimator

In this paper, we obtain the strong consistency and asymptotic distribution of the Theil–Sen estimator in simple linear regression models with arbitrary error distributions. We show that the Theil–Sen estimator is super-efficient when the error distribution is discontinuous and that its asymptotic distribution may or may not be normal when the error distribution is continuous. We give an example in which the Theil–Sen estimator is not asymptotically normal. A small simulation study is conducted to confirm the super-efficiency and the non-normality of the asymptotic distribution.     

Parametric Inference in Stationary Time Series Models with Dependent Errors

This article is concerned with inference for the parameter vector in stationary time series models based on the frequency domain maximum likelihood estimator. The traditional method consistently estimates the asymptotic covariance matrix of the parameter estimator and usually assumes the independence of the innovation process. For dependent innovations, the asymptotic covariance matrix of the estimator depends on the fourth‐order cumulants of the unobserved innovation process, a consistent estimation of which is a difficult task. In this article, we propose a novel self‐normalization‐based approach to constructing a confidence region for the parameter vector in such models. The proposed procedure involves no smoothing parameter, and is widely applicable to a large class of long/short memory time series models with weakly dependent innovations. In simulation studies, we demonstrate favourable finite sample performance of our method in comparison with the traditional method and a residual block bootstrap approach.     

A non-parametric estimator for the doubly periodic Poisson intensity function

In a series of papers, J. Garrido and Y. Lu have proposed and investigated a doubly periodic Poisson model, and then applied it to analyze hurricane data. The authors have suggested several parametric models for the underlying intensity function. In the present paper we construct and analyze a non-parametric estimator for the doubly periodic intensity function. Assuming that only a single realization of the process is available in a bounded window, we show that the estimator is consistent and asymptotically normal when the window expands indefinitely. In addition, we calculate the asymptotic bias and variance of the estimator, and in this way gain helpful information for optimizing the performance of the estimator.