Semiparametric and nonparametric estimation of sample selection models under symmetry

This paper considers the semiparametric estimation of binary choice sample selection models under a joint symmetry assumption. Our approaches overcome various drawbacks associated with existing estimators. In particular, our method provides root-n$n$ consistent estimators for both the intercept and slope parameters of the outcome equation in a heteroscedastic framework, without the usual cross equation exclusion restriction or parametric specification for the error distribution and/or the form of heteroscedasticity. Our two-step estimators are shown to be consistent and asymptotically normal. A Monte Carlo simulation study indicates the usefulness of our approaches.     

On unordering Horvitz and Thompson'sT 1 class of linear estimators

Summary When elements of a finite population are sampled with varying probability selection at each draw,Horvitz andThompson [1952] have formulated certain classes of linear estimators to bear on the problem of providing a smaple appraisal of the population total.Horvitz andThompson's T ₁ class is an ordered one, which was examined by the present author [1967 b]. For some sampling procedures a best estimator exists for theT ₁ class. Subsequently the present author [1967 c] appliedMurthy's technique [Murthy 1967] of unordering an ordered estimator and derived a more efficient estimator. The present paper is concerned with applyingMurthy's technique to theT ₁ class itself, and examining the unorderedT ₁ class. Curiously enough, it is noted that the condition of unbiasedness is sufficient to completely specify the unorderedT ₁ class for the sampling procedure considered here.Research sponsored by Marathwada University, Aurangabad, India; under Grant No. Research-12-68-69/3314-16.     

The Asymptotics of MM-Estimators for Linear Regression with Fixed Designs

MM-estimators achieve simultaneous high efficiency and high breakdown point over contamination neighborhoods. Inference based on these estimators relies on their asymptotic properties, which have been studied for the case of random covariates. In this paper we show that, under relatively mild regularity conditions, MM-estimators for linear regression models are strongly consistent when the design is fixed. Moreover, their strong consistency allows us to show that these estimators are also asymptotically normal for non-random covariates. These results justify the use of a normal approximation to the finite-sample distribution of MM-estimators for linear regression with fixed explanatory variables. Additionally, these results have been used to extend the robust bootstrap (Salibian-Barrera and Zamar in Ann Stat 30:556–582, 2002) to the case of fixed designs [see Salibian-Barrera 2004, submitted].Research supported by an NSERC Research Grant (Individual)     

Some small sample properties of estimators and test statistics in the multivariate logit model

In this paper we compare three estimators for the multivariate logit model: two asymptotically efficient methods and a consistent method. The most interesting result is that at sample sizes of more than one hundred, the simple consistent estimator performs almost as well as the asymptotically efficient estimators.     

Best unbiased estimators for the parameters of a two-parameter Pareto distribution

Summary For a two-parameter Pareto distributionMalik [1970] has shown that the maximum likelihood estimators of the parameters are jointly sufficient. In this article the maximum likelihood estimators are shown to be jointly complete. Furthermore, unbiased estimators for the two parameters are obtained and are shown to be functions of the jointly complete sufficient statistics, thereby establishing them as the best unblased estimators of the two parameters.This research is a part of the first author's Ph.D. dissertation. The authors wish to thank Dr. Kenny S. Crump, for many helpful suggestions and a referee for improvements in the proofs.     

Bootstrapping cointegrating regressions

In this paper, we consider bootstrapping cointegrating regressions. It is shown that the method of bootstrap, if properly implemented, generally yields consistent estimators and test statistics for cointegrating regressions. For the cointegrating regression models driven by general linear processes, we employ the sieve bootstrap based on the approximated finite-order vector autoregressions for the regression errors and the first differences of the regressors. In particular, we establish the bootstrap consistency for OLS method. The bootstrap method can thus be used to correct for the finite sample bias of the OLS estimator and to approximate the asymptotic critical values of the OLS-based test statistics in general cointegrating regressions. The bootstrap OLS procedure, however, is not efficient. For the efficient estimation and hypothesis testing, we consider the procedure proposed by Saikkonen [1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] and Stock and Watson [1993. A simple estimator of cointegrating vectors in higher order integrating systems. Econometrica 61, 783–820] relying on the regression augmented with the leads and lags of differenced regressors. The bootstrap versions of their procedures are shown to be consistent, and can be used to do asymptotically valid inferences. A Monte Carlo study is conducted to investigate the finite sample performances of the proposed bootstrap methods.     

Approximate distribution of the maximum deviation of histograms

Summary We consider a histogram, based on order statistics, and density estimators which are closely related to the histogram.When investigating the distribution of the maximum absolute deviation of density estimators it turns out that an approximation by the distribution of the largest absolute value of a normal sample is asymptotically considerably better than an approximation by the limit distribution (which is the extreme value distribution). For the one-side deviation, a corresponding approximation is less accurate. The accuracy can be improved by using an asymptotic expansion.     

Statistical inference in simplicially contoured sample distributions

For generalizations of the n-dimensional two parameter exponential distribution with identical marginals with threshold and dispersion parameters the exact distributions of estimators and test statistics are given. Under certain conditions the consistency of the estimators and the rate of convergence is shown. Therefore generalized Gamma- and F-distributions are defined.     

On selecting the best ofk lognormal distributions

Summary Fork lognormal populations, which differ only in one certain parameter Ϙ, the problem of finding the population with the largest value ofϑ is considered. For two-parameter lognormal families, several natural choices ofϑ are treated, where the problem can be solved, through logarithmic transformation of the observations, within the framework of estimating parameters ink, possibly restricted, normal populations. For three-parameter lognormal families, this standard approach of selecting in terms of natural estimators fails to work ifϑ is the “guaranteed lifetime”. For this case, a selection procedure is derived which is based on anL-statistic which has the smallest asymptotic variance. Of importance here is that it is location equivariant, whereas it does not matter what it actually estimates. Comparisons are made with other suitable selection rules, through the asymptotic relative efficiencies, as well as in an example of intermediate sample sizes. It is shown that only in the latter, the selection rule, which is based on the sample minima, compares favorably. The research of this author was supported by the Office of Naval Research Contract N00014-88-K-0170 and NSF Grant Number DMS-8606964 at Purdue University. Reproduction in whole or in part is permitted for any purpose of the United States Government. The research of this author was supported by the Air Force Office of Scientific Research Grant 85-0347 at the University of Illinois at Chicago.     

Usual stochastic ordering of the sample maxima from dependent distribution-free random variables

In this paper, we discuss stochastic comparison of the largest order statistics arising from two sets of dependent distribution-free random variables with respect to multivariate chain majorization, where the dependency structure can be defined by Archimedean copulas. When a distribution-free model with possibly two parameter vectors has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we obtain the first sample maxima is larger than the second sample maxima with respect to the usual stochastic order, based on certain conditions. Applications of our results for scale proportional reverse hazards model, exponentiated gamma distribution, Gompertz–Makeham distribution, and location-scale model, are also given. Meanwhile, we provide two numerical examples to illustrate the results established here.     

A further note on the asymptotic efficiency of friedman’s x r 2 -test

Summary For independently distributed error components, the asymptotic relative efficiency (A.R.E.) ofFriedman’sx _r ² -tests with respect to the classical analysis of variance test has been studied byElteren andNoether andSen [1967]. The present note extends these results to the case of correlated errors arising in some random-effects or mixed-effects models. Work supported by the U.S. Army Research Office, Durham, Grant DA-ARO-D-31-124-G 746.     

An application of approximate finite sample results to parameter estimation in a linear errors-in-variables model

In the simple errors-in-variables model the least squares estimator of the slope coefficient is known to be biased towards zero for finite sample size as well as asymptotically. In this paper we suggest a new corrected least squares estimator, where the bias correction is based on approximating the finite sample bias by a lower bound. This estimator is computationally very simple. It is compared with previously proposed corrected least squares estimators, where the correction aims at removing the asymptotic bias or the exact finite sample bias. For each type of corrected least squares estimators we consider the theoretical form, which depends on an unknown parameter, as well as various feasible forms. An analytical comparison of the theoretical estimators is complemented by a Monte Carlo study evaluating the performance of the feasible estimators. The new estimator proposed in this paper proves to be superior with respect to the mean squared error.     

A Note on Testing the Nested Structure in Multivariate Regression Models

In this article we propose a simple method of identifying, at an earlier stage of analysis, the nested structure among the coefficient matrices in multivariate regression models. When the limiting distribution of the estimators of the coefficient matrices are jointly normal, the Wald type statistics based on the proposed method is asymptotically a chi-squared random variable. A numerical example that arises in cointegration analysis is provided to illustrate the method and a small simulation study is provided to illustrate its effectiveness.     

A note on finite sample breakdown points of projection based multivariate location and scatter statistics

In this note, general results of finite sample breakdown point are obtained for two classes of projection based location and scatter statistics: the Stahel-Donoho statistics and the Maronna-Yohai statistics. It is shown that these projection based location and scatter statistics can achieve the maximum breakdown point of affine equivariant multivariate location and scatter statistics. General relationships between the finite sample breakdown point of these statistics and the uniform finite sample breakdown point of the sample median and a modified sample median absolute deviation are formally established. Received: May 1999     

On the variance of the ratio estimator for midzuno-sen sampling scheme

Summary For the sampling scheme ofMidzuno [3] andSen [4], which provides unbiased ratio estimators an expression for the variance of the estimator does not seem to be available in literature. An expression for the same is derived in this note.     

Approximate inference in multivariate nonlinear functional relationships

An estimation procedure based on estimating equations is presented for the parameters in a multivariate functional relationship model, where all observations are subject to error. The covariance matrix of the observational errors may be parametrized and is allowed to be different for different sets of observations. Estimators are defined for the unknown relation parameters and error parameters.
For linear models (i.e. where the model function is linear in the incidental parameters) the estimators are consistent and asymptotically normal. A consistent expression for the covariance matrix of the estimators is derived. The results are valid for general error distributions.
For nonlinear models the estimators are based on locally linear approximations to the model function. The afore mentioned properties of the estimators are now only approximately valid. The adequacy of the approximate inference, based on asymptotic theory for the linearized model, needs at least informal check. Some examples are given to illustrate the estimation procedure.     

Generalized empirical likelihood tests in time series models with potential identification failure

We introduce test statistics based on generalized empirical likelihood methods that can be used to test simple hypotheses involving the unknown parameter vector in moment condition time series models. The test statistics generalize those in Guggenberger and Smith [2005. Generalized empirical likelihood estimators and tests under partial, weak and strong identification. Econometric Theory 21 (4), 667–709] from the i.i.d. to the time series context and are alternatives to those in Kleibergen [2005a. Testing parameters in GMM without assuming that they are identified. Econometrica 73 (4), 1103–1123] and Otsu [2006. Generalized empirical likelihood inference for nonlinear and time series models under weak identification. Econometric Theory 22 (3), 513–527]. The main feature of these tests is that their empirical null rejection probabilities are not affected much by the strength or weakness of identification. More precisely, we show that the statistics are asymptotically distributed as chi-square under both classical asymptotic theory and weak instrument asymptotics of Stock and Wright [2000. GMM with weak identification. Econometrica 68 (5), 1055–1096]. We also introduce a modification to Otsu's (2006) statistic that is computationally more attractive. A Monte Carlo study reveals that the finite-sample performance of the suggested tests is very competitive.     

Small sample corrections for LTS and MCD

The least trimmed squares estimator and the minimum covariance determinant estimator [6] are frequently used robust estimators of regression and of location and scatter. Consistency factors can be computed for both methods to make the estimators consistent at the normal model. However, for small data sets these factors do not make the estimator unbiased. Based on simulation studies we therefore construct formulas which allow us to compute small sample correction factors for all sample sizes and dimensions without having to carry out any new simulations. We give some examples to illustrate the effect of the correction factor.     

Conditional empirical likelihood estimation and inference for quantile regression models

This paper considers two empirical likelihood-based estimation, inference, and specification testing methods for quantile regression models. First, we apply the method of conditional empirical likelihood (CEL) by Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 1667–1714] and Zhang and Gijbels [2003. Sieve empirical likelihood and extensions of the generalized least squares. Scandinavian Journal of Statistics 30, 1–24] to quantile regression models. Second, to avoid practical problems of the CEL method induced by the discontinuity in parameters of CEL, we propose a smoothed counterpart of CEL, called smoothed conditional empirical likelihood (SCEL). We derive asymptotic properties of the CEL and SCEL estimators, parameter hypothesis tests, and model specification tests. Important features are (i) the CEL and SCEL estimators are asymptotically efficient and do not require preliminary weight estimation; (ii) by inverting the CEL and SCEL ratio parameter hypothesis tests, asymptotically valid confidence intervals can be obtained without estimating the asymptotic variances of the estimators; and (iii) in contrast to CEL, the SCEL method can be implemented by some standard Newton-type optimization. Simulation results demonstrate that the SCEL method in particular compares favorably with existing alternatives.     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.