A smoothed maximum score estimator for the binary choice panel data model with an application to labour force participation

In a binary choice panel data model with individual effects and two time periods, Manski proposed the maximum score estimator based on a discontinuous objective function and proved its consistency under weak distributional assumptions. The rate of convergence is low ( N ^1/3) and its limit distribution cannot easily be used for statistical inference. In this paper we apply the idea of Horowitz to smooth Manski's objective function. The resulting smoothed maximum score estimator is consistent and asymptotically normal with a rate of convergence that can be made arbitrarily close to N ^1/2, depending on the strength of the smoothness assumptions imposed. The estimator can be applied to panels with more than two time periods and to unbalanced panels. We apply the estimator to analyze labour force participation of married Dutch females.     

Asymptotics of SIMEX-based variance estimation

In this paper, we study the asymptotic properties of simulation extrapolation (SIMEX) based variance estimation that was proposed by Wang et al. (J R Stat Soc Series B 71:425–445, 2009). We first investigate the asymptotic normality of the parameter estimator in general parametric variance function and the local linear estimator for nonparametric variance function when permutation SIMEX (PSIMEX) is used. The asymptotic optimal bandwidth selection with respect to approximate mean integrated squared error (AMISE) for nonparametric estimator is also studied. We finally discuss constructing confidence intervals/bands of the parameter/function of interest. Other than applying the asymptotic results so that normal approximation can be used, we recommend a nonparametric Monte Carlo algorithm to avoid estimating the asymptotic variance of estimator. Simulation studies are carried out for illustration.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

Sequential point estimation of the powers of a normal scale parameter

We consider the sequential point estimation problem of the powers of a normal scale parameter σ^r with r≠ 0 when the loss function is squared error plus linear cost. It is shown that the regret due to using our fully sequential procedure in ignorance of σ is asymptotically minimized for estimating σ⁻². We also propose a bias-corrected procedure to reduce the risk and show that the larger the distance between r and −2 is, the more effective our bias-corrected procedure is. Received August 2000     

Characterization of the asymptotic distribution of semiparametric M-estimators

This paper develops a concrete formula for the asymptotic distribution of two-step, possibly non-smooth semiparametric M-estimators under general misspecification. Our regularity conditions are relatively straightforward to verify and also weaker than those available in the literature. The first-stage nonparametric estimation may depend on finite dimensional parameters. We characterize: (1) conditions under which the first-stage estimation of nonparametric components do not affect the asymptotic distribution, (2) conditions under which the asymptotic distribution is affected by the derivatives of the first-stage nonparametric estimator with respect to the finite-dimensional parameters, and (3) conditions under which one can allow non-smooth objective functions. Our framework is illustrated by applying it to three examples: (1) profiled estimation of a single index quantile regression model, (2) semiparametric least squares estimation under model misspecification, and (3) a smoothed matching estimator.     

Empirical Bayesball Remixed: Empirical Bayes Methods for Longitudinal Data

Empirical Bayes methods for Gaussian and binomial compound decision problems involving longitudinal data are considered. A recent convex optimization reformulation of the nonparametric maximum likelihood estimator of Kiefer and Wolfowitz (Annals of Mathematical Statistics 1956; 27 : 887–906) is employed to construct nonparametric Bayes rules for compound decisions. The methods are illustrated with an application to predict baseball batting averages, and the age profile of batting performance. An important aspect of the empirical application is the general bivariate specification of the distribution of heterogeneous location and scale effects for players that exhibits a weak positive association between location and scale attributes. Prediction of players' batting averages for 2012 based on performance in the prior decade using the proposed methods shows substantially improved performance over more naive methods with more restrictive treatment of unobserved heterogeneity. Comparisons are also made with nonparametric Bayesian methods based on Dirichlet process priors, which can be viewed as a regularized, or smoothed, version of the Kiefer–Wolfowitz method. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonparametric model validations for hidden Markov models with applications in financial econometrics

We address the nonparametric model validation problem for hidden Markov models with partially observable variables and hidden states. We achieve this goal by constructing a nonparametric simultaneous confidence envelope for transition density function of the observable variables and checking whether the parametric density estimate is contained within such an envelope. Our specification test procedure is motivated by a functional connection between the transition density of the observable variables and the Markov transition kernel of the hidden states. Our approach is applicable for continuous-time diffusion models, stochastic volatility models, nonlinear time series models, and models with market microstructure noise.     

Efficient estimation of the semiparametric spatial autoregressive model

Efficient semiparametric and parametric estimates are developed for a spatial autoregressive model, containing non-stochastic explanatory variables and innovations suspected to be non-normal. The main stress is on the case of distribution of unknown, nonparametric, form, where series nonparametric estimates of the score function are employed in adaptive estimates of parameters of interest. These estimates are as efficient as the ones based on a correct form, in particular they are more efficient than pseudo-Gaussian maximum likelihood estimates at non-Gaussian distributions. Two different adaptive estimates are considered, relying on somewhat different regularity conditions. A Monte Carlo study of finite sample performance is included.     

Direct and indirect effects of continuous treatments based on generalized propensity score weighting

This paper proposes semi- and nonparametric methods for disentangling the total causal effect of a continuous treatment on an outcome variable into its natural direct effect and the indirect effect that operates through one or several intermediate variables called mediators jointly. Our approach is based on weighting observations by the inverse of two versions of the generalized propensity score (GPS), namely the conditional density of treatment either given observed covariates or given covariates and the mediator. Our effect estimators are shown to be asymptotically normal when the GPS is estimated by either a parametric or a nonparametric kernel-based method. We also provide a simulation study and an empirical illustration based on the Job Corps experimental study.     

Nonparametric stochastic frontiers: A local maximum likelihood approach

This paper proposes a new approach to handle nonparametric stochastic frontier (SF) models. It is based on local maximum likelihood techniques. The model is presented as encompassing some anchorage parametric model in a nonparametric way. First, we derive asymptotic properties of the estimator for the general case (local linear approximations). Then the results are tailored to a SF model where the convoluted error term (efficiency plus noise) is the sum of a half normal and a normal random variable. The parametric anchorage model is a linear production function with a homoscedastic error term. The local approximation is linear for both the production function and the parameters of the error terms. The performance of our estimator is then established in finite samples using simulated data sets as well as with a cross-sectional data on US commercial banks. The methods appear to be robust, numerically stable and particularly useful for investigating a production process and the derived efficiency scores.     

On the asymptotic behaviour of the Stringer bound1

The Stringer bound is a widely used nonparametric 100(1 -α)% upper confidence bound for the fraction of errors in an accounting population. This bound has been found in practice to be rather conservative, but no rigorous mathematical proof of the correctness of the Stringer bound as an upper confidence bound is known and also no counterexamples are available. In a pioneering paper Bickel (1992) has given some fixed sample support to the bound's conservatism together with an asymptotic expansion in probability of the Stringer bound, which has led to his claim of the asymptotic conservatism of the Stringer bound. In the present paper we obtain expansions of arbitrary order of the coefficients in the Stringer bound. As a consequence we obtain Bickel's asymptotic expansion with probability 1 and we show that the asymptotic conservatism holds for confidence levels 1 -α, with α∈ (0,1/2]. It means that in general also in a finite sampling situation the Stringer bound does not necessarily have the right confidence level. Based on our expansions we propose a modified Stringer bound which has asymptotically precisely the right nominal confidence level. Finally, we discuss other consequences of the expansions of the Stringer bound such as a central limit theorem, a law of the iterated logarithm and the functional versions of them.     

On estimating the reliability after sequentially estimating the mean: The exponential case

Sequential estimation problems for the mean parameter of an exponential distribution has received much attention over the years. Purely sequential and accelerated sequential estimators and their asymptotic second-order characteristics have been laid out in the existing literature, both for minimum risk point as well as bounded length confidence interval estimation of the mean parameter. Having obtained a data set from such sequentially designed experiments, the paper investigates estimation problems for the associatedreliability function. Second-order approximations are provided for the bias and mean squared error of the proposed estimator of the reliability function, first under a general setup. An ad hoc bias-corrected version is also introduced. Then, the proposed estimator is investigated further under some specific sequential sampling strategies, already available in the literature. In the end, simulation results are presented for comparing the proposed estimators of the reliability function for moderate sample sizes and various sequential sampling strategies.     

Uniform confidence bands for functions estimated nonparametrically with instrumental variables

This paper is concerned with developing uniform confidence bands for functions estimated nonparametrically with instrumental variables. We show that a sieve nonparametric instrumental variables estimator is pointwise asymptotically normally distributed. The asymptotic normality result holds in both mildly and severely ill-posed cases. We present methods to obtain a uniform confidence band and show that the bootstrap can be used to obtain the required critical values. Monte Carlo experiments illustrate the finite-sample performance of the uniform confidence band.     

Instrumental variables estimators of nonparametric models with discrete endogenous regressors

This paper discusses estimation of nonparametric models whose regressor vectors consist of a vector of exogenous variables and a univariate discrete endogenous regressor with finite support. Both identification and estimators are derived from a transform of the model that evaluates the nonparametric structural function via indicator functions in the support of the discrete regressor. A two-step estimator is proposed where the first step constitutes nonparametric estimation of the instrument and the second step is a nonparametric version of two-stage least squares. Linear functionals of the model are shown to be asymptotically normal, and a consistent estimator of the asymptotic covariance matrix is described. For the binary endogenous regressor case, it is shown that one functional of the model is a conditional (on covariates) local average treatment effect, that permits both unobservable and observable heterogeneity in treatments. Finite sample properties of the estimators from a Monte Carlo simulation study illustrate the practicability of the proposed estimators.     

Estimation in the generalized Poisson model via robust testing

An estimation method is presented which compromises robust efficiency with computational feasibility in the case of the generalized Poisson model. The formal setup is built on flexible nonparametric extensions of the underlying model. The estimation efficiency is expressed via minimax properties of tests resulting from expansions of estimators. The nonparametric neighborhoods related to the proposed score function are exemplified and a real data case is analysed. The resulting method balances several qualitative features of statistical inference: strong differentiability (asymptotic derivations are more accurate), efficiency and natural model extension (quality of formal basic assumptions).     

Quantile-based nonparametric inference for first-price auctions

We propose a quantile-based nonparametric approach to inference on the probability density function (PDF) of the private values in first-price sealed-bid auctions with independent private values. Our method of inference is based on a fully nonparametric kernel-based estimator of the quantiles and PDF of observable bids. Our estimator attains the optimal rate of Guerre et al. (2000), and is also asymptotically normal with an appropriate choice of the bandwidth.     

Nonparametric estimation of distribution functions

Summary Using Silverman and Young’s (1987) idea of rescaling a rescaled smoothed empirical distribution function is defined and investigated when the smoothing parameter depends on the data. The rescaled smoothed estimator is shown to be often better than the commonly used ordinary smoothed estimator.     

Effect of factors with time-dependent covariates on the survival status of the elderly in Taiwan

The elderly population has been increasing rapidly in recent years because of improvements in medical care and the progress in economics in Taiwan. Now, the population of old people age above 65 has approached 10%, and problems of the elderly have become a major concern for public health. In this study, we use five waves of Survey of Health and Living Status of the Elderly in Taiwan held from 1989 to 2003 to explore 15 variables related to demographic characteristic, health status, health behavior and home condition etc. affect the survival status of the elderly by employing Cox proportional hazard model. The results show that there are nine variables, e.g. age, gender, ethnic group, ADL, self-rated health, physical function, smoking, chewing betel nuts and marital status, were strongly related to the survival status of the elderly. In addition, Aalen’s nonparametric additive model is not only used as an alternative to a proportional hazards to describe the effects of covariates on survival time but assist in detecting and describing the nature of time-dependent effects of covariates used for Cox proportional model. We find there are some time-dependent covariates (e.g. self-rate health, ADL function, and physical function), and Aalen’s nonparametric additive model provides a flexible and nonparametric method for investigating the time-dependent variables through plots of the estimated cumulative regression coefficients, with confidence intervals.     

Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data

In this paper, we propose a new approach to the empirical likelihood inference for the parameters in heteroscedastic partially linear single-index models. In the growing dimensional setting, it is proved that estimators based on semiparametric efficient score have the asymptotic consistency, and the limit distribution of the empirical log-likelihood ratio statistic for parameters \((\beta ^{\top },\theta ^{\top })^{\top }\) is a normal distribution. Furthermore, we show that the empirical log-likelihood ratio based on the subvector of \(\beta \) is an asymptotic chi-square random variable, which can be used to construct the confidence interval or region for the subvector of \(\beta \). The proposed method can naturally be applied to deal with pure single-index models and partially linear models with high-dimensional data. The performance of the proposed method is illustrated via a real data application and numerical simulations.     

Nonparametric estimation in case of endogenous selection

This paper addresses the problem of estimation of a nonparametric regression function from selectively observed data when selection is endogenous. Our approach relies on independence between covariates and selection conditionally on potential outcomes. Endogeneity of regressors is also allowed for. In the exogenous and endogenous case, consistent two-step estimation procedures are proposed and their rates of convergence are derived. Pointwise asymptotic distribution of the estimators is established. In addition, bootstrap uniform confidence bands are obtained. Finite sample properties are illustrated in a Monte Carlo simulation study and an empirical illustration.