Kernel spatial density estimation in infinite dimension space

In this paper, we propose a nonparametric method to estimate the spatial density of a functional stationary random field. This latter is with values in some infinite dimensional normed space and admitted a density with respect to some reference measure. We study both the weak and strong consistencies of the considered estimator and also give some rates of convergence. Special attention is paid to the links between the probabilities of small balls and the rates of convergence of the estimator. The practical use and the behavior of the estimator are illustrated through some simulations and a real data application.     

Robust and consistent estimation of nonlinear errors-in-variables models

This paper considers the consistent estimation of nonlinear errors-in-variables models. It adopts the functional modeling approach by assuming that the true but unobserved regressors are random variables but making no parametric assumption on the distribution from which the latent variables are drawn. This paper shows how the information extracted from the replicate measurements can be used to identify and consistently estimate a general nonlinear errors-in-variables model. The identification is established through characteristic functions. The estimation procedure involves nonparametric estimation of the conditional density of the latent variables given the measurements using the identification results at the first stage, and at the second stage, a semiparametric nonlinear least-squares estimator is proposed. The consistency of the proposed estimator is also established. Finite sample performance of the estimator is investigated through a Monte Carlo study.     

A general asymptotic theory for time-series models

This paper develops a general asymptotic theory for the estimation of strictly stationary and ergodic time–series models. Under simple conditions that are straightforward to check, we establish the strong consistency, the rate of strong convergence and the asymptotic normality of a general class of estimators that includes LSE, MLE and some M-type estimators. As an application, we verify the assumptions for the long-memory fractional ARIMA model. Other examples include the GARCH(1,1) model, random coefficient AR(1) model and the threshold MA(1) model.     

Subsampling the distribution of diverging statistics with applications to finance

In this paper we propose a subsampling estimator for the distribution of statistics diverging at either known or unknown rates when the underlying time series is strictly stationary and strong mixing. Based on our results we provide a detailed discussion of how to estimate extreme order statistics with dependent data and present two applications to assessing financial market risk. Our method performs well in estimating Value at Risk and provides a superior alternative to Hill's estimator in operationalizing Safety First portfolio selection.     

Reduced rank regression in cointegrated models

The coefficient matrix of a cointegrated first-order autoregression is estimated by reduced rank regression (RRR), depending on the larger canonical correlations and vectors of the first difference of the observed series and the lagged variables. In a suitable coordinate system the components of the least-squares (LS) estimator associated with the lagged nonstationary variables are of order 1/T, where T is the sample size, and are asymptotically functionals of a Brownian motion process; the components associated with the lagged stationary variables are of the order T^−1/2 and are asymptotically normal. The components of the RRR estimator associated with the stationary part are asymptotically the same as for the LS estimator. Some components of the RRR estimator associated with nonstationary regressors have zero error to order 1/T and the other components have a more concentrated distribution than the corresponding components of the LS estimator.     

Variable selection via composite quantile regression with dependent errors

We propose composite quantile regression for dependent data, in which the errors are from short‐range dependent and strictly stationary linear processes. Under some regularity conditions, we show that composite quantile estimator enjoys root‐n consistency and asymptotic normality. We investigate the asymptotic relative efficiency of composite quantile estimator to both single‐level quantile regression and least‐squares regression. When the errors have finite variance, the relative efficiency of composite quantile estimator with respect to the least‐squares estimator has a universal lower bound. Under some regularity conditions, the adaptive least absolute shrinkage and selection operator penalty leads to consistent variable selection, and the asymptotic distribution of the non‐zero coefficient is the same as that of the counterparts obtained when the true model is known. We conduct a simulation study and a real data analysis to evaluate the performance of the proposed approach.     

Asymptotic properties of estimators for the volume fractions of jointly stationary random sets

In the present paper, we show how a consistent estimator can be derived for the asymptotic covariance matrix of stationary 0–1-valued vector fields in R ^d , whose supports are jointly stationary random closed sets. As an example, which is of particular interest for statistical applications, we consider jointly stationary random closed sets associated with the Boolean model in R ^d such that the components indicate the frequency of coverage by the single grains of the Boolean model. For this model, a representation formula for the entries of the covariance matrix is obtained.     

The conditional maximum likelihood estimator of the shape parameter in the gamma distribution

Summary The conditional maximum likelihood estimator of the shape parameter in the gamma distribution is studied for a finite sample size in comparison with the (unconditional) maximum likelihood estimator. The former estimator is concluded to be strictly superior to the latter. The reasons for the conclusion include the undesirable behavior of the residual likelihood, the consistency and relatively less bias of the conditional maximum likelihood estimator. Simulation studies for risk comparisons also support the conclusion.     

Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting

We consider semiparametric frequency domain analysis of cointegration between long memory processes, i.e. fractional cointegration, allowing derivation of useful long-run relations even among stationary processes. The approach is due to Robinson (1994b. Annals of Statistics 22, 515–539) and uses a degenerating part of the periodogram near the origin to form a narrow-band frequency domain least squares (FDLS) estimator of the cointegrating relation, which is consistent for arbitrary short-run dynamics. We derive the asymptotic distribution theory for the FDLS estimator of the cointegration vector in the stationary long memory case, thus complementing Robinson's consistency result. An application to the relation between the volatility realized in the stock market and the associated implicit volatility derived from option prices is offered.     

Control variate method for stationary processes

The sample mean is one of the most natural estimators of the population mean based on independent identically distributed sample. However, if some control variate is available, it is known that the control variate method reduces the variance of the sample mean. The control variate method often assumes that the variable of interest and the control variable are i.i.d. Here we assume that these variables are stationary processes with spectral density matrices, i.e. dependent. Then we propose an estimator of the mean of the stationary process of interest by using control variate method based on nonparametric spectral estimator. It is shown that this estimator improves the sample mean in the sense of mean square error. Also this analysis is extended to the case when the mean dynamics is of the form of regression. Then we propose a control variate estimator for the regression coefficients which improves the least squares estimator (LSE). Numerical studies will be given to see how our estimator improves the LSE.     

Estimating an economic model of crime using panel data from North Carolina

This paper replicates the Cornwell and Trumbull ( 1994 ) estimation of a crime model using panel data on 90 counties in North Carolina over the period 1981–1987. While the Between and Within estimates are replicated, the fixed effects 2SLS as well as the 2SLS estimates are not. In fact, the fixed effects 2SLS estimates turn out to be insignificant for all important deterrent variables as well as legal opportunity variables. We argue that the usual Hausman test, based on the difference between fixed effects and random effects, may lead to misleading inference when endogenous variables of the conventional simultaneous equation type are among the regressors. We estimate the model using random effects 2SLS and perform a Hausman test based on the difference between fixed effects 2SLS and random effects 2SLS. We cannot reject the consistency of the random effects 2SLS estimator and this estimator yields plausible and significant estimates of the crime model. This result should be tempered by the legitimacy of the chosen instruments. Copyright © 2006 John Wiley & Sons, Ltd.     

Estimation of a survival curve with randomly censored data in the presence of a covariate

This paper deals with the estimation of a survival curve in models with random right censoring and dependent censoring mechanism. We consider a specific dependent censorship model in which conditional on a covariate, the survival and censoring times are assumed to be independent. We investigate asymptotic properties of a corrected version of a survival curve estimator introduced by Cheng (1989). In particular we show uniform strong consistency and weak convergence to a Gaussian process. Comparisons of this estimator with the well-known Kaplan-Meier-estimator are included. Finally, some examples illustrate how the estimator performs. Received: February 2000     

The ascending ladder height distribution for a certain class of dependent random walks

A random walk { S_n } with S_n = (X_l - Y_l) +…+ ( X_n - Y_n ) is considered where the X_n Y_n are non-negative random variables, the Y_n are exponentially distributed with rate δ and the X_n have common distribution function B . It is shown that the expression δ(1 - S (x)) for the density of the ascending ladder height distribution of (S_n), which is well-known for i.i.d. X_n , holds also when the X_n form a stationary sequence of not necessarily independent random variables.     

Frequency domain estimation of temporally aggregated Gaussian cointegrated systems

This paper considers joint estimation of long run equilibrium coefficients and parameters governing the short run dynamics of a fully parametric Gaussian cointegrated system formulated in continuous time. The model allows the stationary disturbances to be generated by a stochastic differential equation system and for the variables to be a mixture of stocks and flows. We derive a precise form for the exact discrete analogue of the continuous time model in triangular error correction form, which acts as the basis for frequency domain estimation of the unknown parameters using discrete time data. We formally establish the order of consistency and the asymptotic sampling properties of such an estimator. The estimator of the cointegrating parameters is shown to converge at the rate of the sample size to a mixed normal distribution, while that of the short run parameters converges at the rate of the square root of the sample size to a limiting normal distribution.     

Local linear M-estimation for spatial processes in fixed-design models

We investigate the local linear M-estimation for regression in a fixed-design model when the errors are from a strongly mixing random field. We establish the weak and strong consistency as well as the asymptotic normality of the local linear M-estimator. The conditions on ρ(·) used in this paper are mild and allow many important special cases such as the least square estimator and the least absolute distance estimator.     

Strong representation of the presmoothed quantile function estimator for censored data

We consider lifetime data subject to right random censorship. In this context, this paper deals with the topic of estimating the distribution function of the lifetime and the corresponding quantile function. As it has been shown that the classical Kaplan–Meier estimator of the distribution function can be improved by means of presmoothing ideas, we introduce a quantile function estimator via the presmoothed distribution function estimator studied by Cao et al. [Journal of Nonparametric statistics, Vol. 17 (2005) pp. 31–56.] The main result of this paper is an almost sure representation of this presmoothed estimator. As a consequence, its strong consistency and asymptotic normality are established. The performance of this new quantile estimator is analyzed in a simulation study and applied to a real data example.     

Linear instrumental variable estimation of limited dependent variable models with endogenous explanatory variables

The purpose of this paper is to present and analyze an instrumental variables estimator for limited dependent variable models that does not require functional form assumptions for the distribution of disturbances. This estimator is a weighted instrumental variables estimator, where the weight is the ratio of a multivariate normal density to the actual density of the instrumental variables. A semi-non-parametric estimator of the weights is presented and some conjectures concerning the asymptotic distribution of the estimator are discussed.     

Model specification testing of time series regressions

Given the specification of the lag length and functional form of a (non)linear time series regression we shall propose a test of the null hypothesis that the expectation of the error conditional on the exogenous variables, all lagged exogenous variables and all lagged dependent variables equals zero with probability 1. In the case that the data-generating process is strictly stationary this test is consistent with respect to the alternative hypothesis that the null is false. The test is also applicable for a particular class of non-stationary time series regressions, although in that case consistency with respect to all possible alternatives is no longer guaranteed. The test involved is a generalization of a test proposed in Bierens (1982b). Moreover, we also present a similar but simpler test of the hypothesis that the errors are martingale differences.     

Nonparametric efficiency analysis: A multivariate conditional quantile approach

This paper focuses on nonparametric efficiency analysis based on robust estimation of partial frontiers in a complete multivariate setup (multiple inputs and multiple outputs). It introduces α-quantile efficiency scores. A nonparametric estimator is proposed achieving strong consistency and asymptotic normality. Then if α increases to one as a function of the sample size we recover the properties of the FDH estimator. But our estimator is more robust to the perturbations in data, since it attains a finite gross-error sensitivity. Environmental variables can be introduced to evaluate efficiencies and a consistent estimator is proposed. Numerical examples illustrate the usefulness of the approach.     

Goodness-of-fit tests for long memory moving average marginal density

This paper addresses the problem of fitting a known density to the marginal error density of a stationary long memory moving average process when its mean is known and unknown. In the case of unknown mean, when mean is estimated by the sample mean, the first order difference between the residual empirical and null distribution functions is known to be asymptotically degenerate at zero, and hence can not be used to fit a distribution up to an unknown mean. In this paper we show that by using a suitable class of estimators of the mean, this first order degeneracy does not occur. We also investigate the large sample behavior of tests based on an integrated square difference between kernel type error density estimators and the expected value of the error density estimator based on errors. The asymptotic null distributions of suitably standardized test statistics are shown to be chi-square with one degree of freedom in both cases of the known and unknown mean. In addition, we discuss the consistency and asymptotic power against local alternatives of the density estimator based test in the case of known mean. A finite sample simulation study of the test based on residual empirical process is also included.