Residual‐based IV estimation of dynamic panel data models with fixed effects

In dynamic panel regression, when the variance ratio of individual effects to disturbance is large, the system‐GMM estimator will have large asymptotic variance and poor finite sample performance. To deal with this variance ratio problem, we propose a residual‐based instrumental variables (RIV) estimator, which uses the residual from regressing Δy_i,t?1 on as the instrument for the level equation. The RIV estimator proposed is consistent and asymptotically normal under general assumptions. More importantly, its asymptotic variance is almost unaffected by the variance ratio of individual effects to disturbance. Monte Carlo simulations show that the RIV estimator has better finite sample performance compared to alternative estimators. The RIV estimator generates less finite sample bias than difference‐GMM, system‐GMM, collapsing‐GMM and Level‐IV estimators in most cases. Under RIV estimation, the variance ratio problem is well controlled, and the empirical distribution of its t‐statistic is similar to the standard normal distribution for moderate sample sizes.     

Estimation and testing of crossing‐points in fixed design regression

Many phenomena in the life sciences can be analyzed by using a fixed design regression model with a regression function m that exhibits a crossing‐point in the following sense: the regression function runs below or above its mean level, respectively, according as the input variable lies to the left or to the right of that crossing‐point, or vice versa. We propose a non‐parametric estimator and show weak and strong consistency as long as the crossing‐point is unique. It is defined as maximizing point arg max of a certain marked empirical process. For testing the hypothesis H₀ that the regression function m actually is constant (no crossing‐point), a decision rule is designed for the specific alternative H₁ that m possesses a crossing‐point. The pertaining test‐statistic is the ratio max/argmax of the maximum value and the maximizing point of the marked empirical process. Under the hypothesis the ratio converges in distribution to the corresponding ratio of a reflected Brownian bridge, for which we derive the distribution function. The test is consistent on the whole alternative and superior to the corresponding Kolmogorov–Smirnov test, which is based only on the maximal value max. Some practical examples of possible applications are given where a certain study about dental phobia is discussed in more detail.     

Robust explicit estimators of Weibull parameters

The Weibull distribution plays a central role in modeling duration data. Its maximum likelihood estimator is very sensitive to outliers. We propose three robust and explicit Weibull parameter estimators: the quantile least squares, the repeated median and the median/Q _n estimator. We derive their breakdown point, influence function, asymptotic variance and study their finite sample properties in a Monte Carlo study. The methods are illustrated on real lifetime data affected by a recording error.     

On the Generalized Bootstrap for Sample Surveys with Special Attention to Poisson Sampling

We study the generalized bootstrap technique under general sampling designs. We focus mainly on bootstrap variance estimation but we also investigate the empirical properties of bootstrap confidence intervals obtained using the percentile method. Generalized bootstrap consists of randomly generating bootstrap weights so that the first two (or more) design moments of the sampling error are tracked by the corresponding bootstrap moments. Most bootstrap methods in the literature can be viewed as special cases. We discuss issues such as the choice of the distribution used to generate bootstrap weights, the choice of the number of bootstrap replicates, and the potential occurrence of negative bootstrap weights. We first describe the generalized bootstrap for the linear Horvitz‐Thompson estimator and then consider non‐linear estimators such as those defined through estimating equations. We also develop two ways of bootstrapping the generalized regression estimator of a population total. We study in greater depth the case of Poisson sampling, which is often used to select samples in Price Index surveys conducted by national statistical agencies around the world. For Poisson sampling, we consider a pseudo‐population approach and show that the resulting bootstrap weights capture the first three design moments of the sampling error. A simulation study and an example with real survey data are used to illustrate the theory.     

Application of one‐step method to parameter estimation in ODE models

In this paper, we study application of Le Cam's one‐step method to parameter estimation in ordinary differential equation models. This computationally simple technique can serve as an alternative to numerical evaluation of the popular non‐linear least squares estimator, which typically requires the use of a multistep iterative algorithm and repetitive numerical integration of the ordinary differential equation system. The one‐step method starts from a preliminary ‐consistent estimator of the parameter of interest and next turns it into an asymptotic (as the sample size n →∞ ) equivalent of the least squares estimator through a numerically straightforward procedure. We demonstrate performance of the one‐step estimator via extensive simulations and real data examples. The method enables the researcher to obtain both point and interval estimates. The preliminary ‐consistent estimator that we use depends on non‐parametric smoothing, and we provide a data‐driven methodology for choosing its tuning parameter and support it by theory. An easy implementation scheme of the one‐step method for practical use is pointed out.     

Nonlinear regression models with single‐index heteroscedasticity

We consider nonlinear heteroscedastic single‐index models where the mean function is a parametric nonlinear model and the variance function depends on a single‐index structure. We develop an efficient estimation method for the parameters in the mean function by using the weighted least squares estimation, and we propose a “delete‐one‐component” estimator for the single‐index in the variance function based on absolute residuals. Asymptotic results of estimators are also investigated. The estimation methods for the error distribution based on the classical empirical distribution function and an empirical likelihood method are discussed. The empirical likelihood method allows for incorporation of the assumptions on the error distribution into the estimation. Simulations illustrate the results, and a real chemical data set is analyzed to demonstrate the performance of the proposed estimators.     

On the Lp error of the Grenander-type estimator in the Cox model

We consider the Cox regression model and study the asymptotic global behavior of the Grenander-type estimator for a monotone baseline hazard function. This model is not included in the general setting of Durot (2007). However, we show that a similar central limit theorem holds for L_p-error of the Grenander-type estimator. As an illustration of application of our main result, we propose a test procedure for a Weibull baseline distribution, based on the L_p-distance between the Grenander estimator and a parametric estimator of the baseline hazard. Simulation studies are performed to investigate the performance of this test.     

Pseudo empirical likelihood method in the presence of missing data

In this paper, we propose an estimator for the population mean when some observations on the study and auxiliary variables are missing from the sample. The proposed estimator is valid for any unequal probability sampling design, and is based upon the pseudo empirical likelihood method. The proposed estimator is compared with other estimators in a simulation study.     

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.     

On the strong uniform consistency of a new kernel density estimator

Summary A new multivariate kernel probability density estimator is introduced and its strong uniform consistency is proved under certain regularity conditions. This result is then applied particularly to a kernel estimator whose mean vector and covariance matrix areμ _n andV _n, respectively, whereμ _n is an unspecified estimator of the mean vector andV _n, up to a multiplicative constant, the sample covariance matrix of the probability density to be estimated, respectively. Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec.     

Adaptive Estimation of a Conditional Density

The authors consider the problem of estimating a conditional density by a conditional kernel density estimate when the error associated with the estimate is measured by the L₁‐norm. On the basis of the combinatorial method of Devroye and Lugosi ( 1996 ), they propose a method for selecting the bandwidths adaptively and for providing a theoretical justification of the approach. They use simulated data to illustrate the finite‐sample performance of their estimator.     

Model‐Assisted Estimation of a Spatial Population Mean

This paper deals with the estimation of the mean of a spatial population. Under a design‐based approach to inference, an estimator assisted by a penalized spline regression model is proposed and studied. Proof that the estimator is design‐consistent and has a normal limiting distribution is provided. A simulation study is carried out to investigate the performance of the new estimator and its variance estimator, in terms of relative bias, efficiency, and confidence interval coverage rate. The results show that gains in efficiency over standard estimators in classical sampling theory may be impressive.     

Calibration methods for estimating quantiles

We propose a calibrated estimator of the quantiles of sample survey data and discuss the asymptotic theory behind it. This estimator is defined for any sampling design and uses the information available on J auxiliary variables. A simulation study based on a real population is used to compare the estimator with various methods proposed previously.     

Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading

We propose a multivariate realised kernel to estimate the ex-post covariation of log-prices. We show this new consistent estimator is guaranteed to be positive semi-definite and is robust to measurement error of certain types and can also handle non-synchronous trading. It is the first estimator which has these three properties which are all essential for empirical work in this area. We derive the large sample asymptotics of this estimator and assess its accuracy using a Monte Carlo study. We implement the estimator on some US equity data, comparing our results to previous work which has used returns measured over 5 or 10 min intervals. We show that the new estimator is substantially more precise.     

Nonparametric identification and estimation of current status data in the presence of death

We present a nonparametric study of current status data in the presence of death. Such data arise from biomedical investigations in which patients are examined for the onset of a certain disease, for example, tumor progression, but may die before the examination. A key difference between such studies on human subjects and the survival–sacrifice model in animal carcinogenicity experiments is that, due to ethical and perhaps technical reasons, deceased human subjects are not examined, so that the information on their disease status is lost. We show that, for current status data with death, only the overall and disease‐free survival functions can be identified, whereas the cumulative incidence of the disease is not identifiable. We describe a fast and stable algorithm to estimate the disease‐free survival function by maximizing a pseudo‐likelihood with plug‐in estimates for the overall survival rates. It is then proved that the global rate of convergence for the nonparametric maximum pseudo‐likelihood estimator is equal to O_p(n^?1/3) or the convergence rate of the estimated overall survival function, whichever is slower. Simulation studies show that the nonparametric maximum pseudo‐likelihood estimators are fairly accurate in small‐ to medium‐sized samples. Real data from breast cancer studies are analyzed as an illustration.     

Sampling with probabilities proportional to the variable of interest

To estimate the mean sojourn time, a sample of Tilburg fair visitors was asked for the duration of their stay on the fair grounds. The longer a visitor's sojourn, the larger his/her probability of being interviewed will be; therefore, longer sojourn times will be overrepresented in the sample. As a consequence, the arithmetic sample mean is not a good estimator.
The paper places this problem against a theoretical background. Sampling with unequal probabilities is considered in a general context. The special case that the sampling probabilities are a function of the variable under investigation, is discussed in detail. As a better estimator the harmonic mean of the observations is presented. Most properties of this estimator are difficult to derive analytically, but a suitable variance estimator is derived. The behavior of estimator and variance estimator is studied in a number of quite different examples.     

Design‐based Estimators Calibrated on Estimated Totals from Multiple Surveys

The use of auxiliary variables to improve the efficiency of estimators is a well‐known strategy in survey sampling. Typically, the auxiliary variables used are the totals of appropriate measurement that are exactly known from registers or administrative sources. Increasingly, however, these totals are estimated from surveys and are then used to calibrate estimators and improve their efficiency. We consider different types of survey structures and develop design‐based estimators that are calibrated on known as well as estimated totals of auxiliary variables. The optimality properties of these estimators are studied. These estimators can be viewed as extensions of the Montanari generalised regression estimator adapted to the more complex situations. The paper studies interesting special cases to develop insights and guidelines to properly manage the survey‐estimated auxiliary totals.     

Simultaneous estimation of hazard rates of several exponential populations

Suppose independent random samples are drawn from k (2) populations with a common location parameter and unequal scale parameters. We consider the problem of estimating simultaneously the hazard rates of these populations. The analogues of the maximum likelihood (ML), uniformly minimum variance unbiased (UMVU) and the best scale equivariant (BSE) estimators for the one population case are improved using Rao‐Blackwellization. The improved version of the BSE estimator is shown to be the best among these estimators. Finally, a class of estimators that dominates this improved estimator is obtained using the differential inequality approach.     

A general class of estimators for estimating population mean using auxiliary information

Summary A general class of estimators for estimating the population mean of the character under study which make use of auxiliary information is proposed. Under simple random sampling without replacement (SRSWOR), the expressions of Bias and Mean Square Error (MSE), up to the first and the second degrees of approximation are derived. General conditions, up to the first order approximation, are also obtained under which any member of this class performs more efficiently than the mean per unit estimator, the ratio estimator and the product estimator. The class of estimators in its optimum case, under the first degree approximation, is discussed. It is shown that it is not possible to obtain optimum values of parameters “a”, “b” and “p”, that are independent of each other. However, the optimum relation among them is given by (b−a)p=ρ C _y/C _x. Under this condition, the expression of MSE of the class is that of the linear regression estimator.     

Shrinkage estimation of the proportion in randomized response

This article considers the asymptotic estimation theory for the proportion in randomized response survey usinguncertain prior information (UPI) about the true proportion parameter which is assumed to be available on the basis of some sort of realistic conjecture. Three estimators, namely, the unrestricted estimator, the shrinkage restricted estimator and an estimator based on a preliminary test, are proposed. Their asymptotic mean squared errors are derived and compared. The relative dominance picture of the estimators is presented.