Right censoring in a discrete life model

Although there are many sophisticated models for estimation of failure rate based on censored data in continuous distributions, not much work has been done in the discrete case. We introduce a discrete model for life lengths and consider its properties. For this model, we derive the corresponding maximum likelihood estimators of the parameters under Type I and Type II right-censoring. Received May 2000     

Penalized Quantile Regression with Semiparametric Correlated Effects: An Application with Heterogeneous Preferences

This paper proposes new ?₁‐penalized quantile regression estimators for panel data, which explicitly allows for individual heterogeneity associated with covariates. Existing fixed‐effects estimators can potentially suffer from three limitations which are overcome by the proposed approach: (i) incidental parameters bias in nonlinear models with large N and small T ; (ii) lack of efficiency; and (iii) inability to estimate the effects of time‐invariant regressors. We conduct Monte Carlo simulations to assess the small‐sample performance of the new estimators and provide comparisons of new and existing penalized estimators in terms of quadratic loss. We apply the technique to an empirical example of the estimation of consumer preferences for nutrients from a demand model using a large transaction‐level dataset of household food purchases. We show that preferences for nutrients vary across the conditional distribution of expenditure and across genders, and emphasize the importance of fully capturing consumer heterogeneity in demand modeling. Copyright © 2016 John Wiley & Sons, Ltd.     

Exponentially tilted likelihood inference on growing dimensional unconditional moment models

Growing-dimensional data with likelihood function unavailable are often encountered in various fields. This paper presents a penalized exponentially tilted (PET) likelihood for variable selection and parameter estimation for growing dimensional unconditional moment models in the presence of correlation among variables and model misspecification. Under some regularity conditions, we investigate the consistent and oracle properties of the PET estimators of parameters, and show that the constrained PET likelihood ratio statistic for testing contrast hypothesis asymptotically follows the chi-squared distribution. Theoretical results reveal that the PET likelihood approach is robust to model misspecification. We study high-order asymptotic properties of the proposed PET estimators. Simulation studies are conducted to investigate the finite performance of the proposed methodologies. An example from the Boston Housing Study is illustrated.     

Smooth estimators for estimating order restricted scale parameters of two gamma distributions

We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation.     

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.     

Nonparametric estimation for a current status right‐censored data model

We consider the Case 1 interval censoring approach for right‐censored survival data. An important feature of the model is that right‐censored event times are not observed exactly, but at some inspection times. The model covers as particular cases right‐censored data, current status data, and life table survival data with a single inspection time. We discuss the nonparametric estimation approach and consider three nonparametric estimators for the survival function of failure time: maximum likelihood, pseudolikelihood, and the naïve estimator. We establish strong consistency of the estimators with the L₁ rate of convergence. Simulation results confirm consistency of the estimators.     

On solving bias‐corrected non‐linear estimation equations with an application to the dynamic linear model

In a seminal paper, Mak, Journal of the Royal Statistical Society B, 55, 1993, 945, derived an efficient algorithm for solving non‐linear unbiased estimation equations. In this paper, we show that when Mak's algorithm is applied to biased estimation equations, it results in the estimates that would come from solving a bias‐corrected estimation equation, making it a consistent estimator if regularity conditions hold. In addition, the properties that Mak established for his algorithm also apply in the case of biased estimation equations but for estimates from the bias‐corrected equations. The marginal likelihood estimator is obtained when the approach is applied to both maximum likelihood and least squares estimation of the covariance matrix parameters in the general linear regression model. The new approach results in two new estimators when applied to the profile and marginal likelihood functions for estimating the lagged dependent variable coefficient in the dynamic linear regression model. Monte Carlo simulation results show the new approach leads to a better estimator when applied to the standard profile likelihood. It is therefore recommended for situations in which standard estimators are known to be biased.     

Instrumental variable estimation based on grouped data

The paper considers the estimation of the coefficients of a single equation in the presence of dummy intruments. We derive pseudo ML and GMM estimators based on moment restrictions induced either by the structural form or by the reduced form of the model. The performance of the estimators is evaluated for the non-Gaussian case. We allow for heteroscedasticity. The asymptotic distributions are based on parameter sequences where the number of instruments increases at the same rate as the sample size. Relaxing the usual Gaussian assumption is shown to affect the normal asymptotic distributions. As a result also recently suggested new specification tests for the validity of instruments depend on Gaussianity. Monte Carlo simulations confirm the accuracy of the asymptotic approach.     

Estimating Nonlinear Models with Multiple Fixed Effects: A Computational Note*

In this paper we consider estimation of nonlinear panel data models that include multiple individual fixed effects. Estimation of these models is complicated both by the difficulty of estimating models with possibly thousands of coefficients and also by the incidental parameters problem; that is, noisy estimates of the fixed effects when the time dimension is short contaminate the estimates of the common parameters due to the nonlinearity of the problem. We propose a simple variation of existing bias‐corrected estimators, which can exploit the additivity of the effects for numerical optimization. We exhibit the performance of the estimators in simulations.     

An integrated maximum score estimator for a generalized censored quantile regression model

Quantile regression techniques have been widely used in empirical economics. In this paper, we consider the estimation of a generalized quantile regression model when data are subject to fixed or random censoring. Through a discretization technique, we transform the censored regression model into a sequence of binary choice models and further propose an integrated smoothed maximum score estimator by combining individual binary choice models, following the insights of Horowitz (1992) and Manski (1985). Unlike the estimators of Horowitz (1992) and Manski (1985), our estimators converge at the usual parametric rate through an integration process. In the case of fixed censoring, our approach overcomes a major drawback of existing approaches associated with the curse-of-dimensionality problem. Our approach for the fixed censored case can be extended readily to the case with random censoring for which other existing approaches are no longer applicable. Both of our estimators are consistent and asymptotically normal. A simulation study demonstrates that our estimators perform well in finite samples.     

One-step ahead adaptive D-optimal design on a finite design space is asymptotically optimal

We study the consistency of parameter estimators in adaptive designs generated by a one-step ahead D-optimal algorithm. We show that when the design space is finite, under mild conditions the least-squares estimator in a nonlinear regression model is strongly consistent and the information matrix evaluated at the current estimated value of the parameters strongly converges to the D-optimal matrix for the unknown true value of the parameters. A similar property is shown to hold for maximum-likelihood estimation in Bernoulli trials (dose–response experiments). Some examples are presented.     

A minimum chi-squared method for indirect parameter estimation from Poisson data

In this paper, we introduce a new algorithm for estimating non-negative parameters from Poisson observations of a linear transformation of the parameters. The proposed objective function fits both a weighted least squares (WLS) and a minimum χ² estimation framework, and results in a convex optimization problem. Unlike conventional WLS methods, the weights do not need to be estimated from the datas, but are incorporated in the objective function. The iterative algorithm is derived from an alternating projection procedure in which "distance" is determined by the chi-squared test statistic, which is interpreted as a measure of the discrepancy between two distributions. This may be viewed as an alternative to the Kullback-Leibler divergence which corresponds to the maximum likelihood (ML) estimation. The algorithm is similar in form to, and shares many properties with, the expectation maximization algorithm for ML estimation. In particular, we show that every limit point of the algorithm is an estimator, and the sequence of projected (by the linear transformation into the data space) means converge. Despite the similarities, we show that the new estimators are quite distinct from ML estimators, and obtain conditions under which they are identical.     

Fixed effects demeaning in the presence of interactive effects in treatment effects regressions and elsewhere

The present paper shows that cross-section demeaning with respect to time fixed effects is more useful than commonly appreciated, in that it enables consistent and asymptotically normal estimation of interactive effects models with heterogeneous slope coefficients when the number of time periods, T, is small and only the number of cross-sectional units, N, is large. This is important when using OLS but also when using more sophisticated estimators of interactive effects models whose validity does not require demeaning, a point that to the best of our knowledge has not been made before in the literature. As an illustration, we consider the problem of estimating the average treatment effect in the presence of unobserved time-varying heterogeneity. Gobillon and Magnac (2016) recently considered this problem. They employed a principal components-based approach designed to deal with general unobserved heterogeneity, which does not require fixed effects demeaning. The approach does, however, require that T is large, which is typically not the case in practice, and the results reported here confirm that the performance can be extremely poor in small-T samples. The exception is when the approach is applied to data that have been demeaned with respect to fixed effects.     

A new bounded log-linear regression model

In this paper we introduce a new regression model in which the response variable is bounded by two unknown parameters. A special case is a bounded alternative to the four parameter logistic model. The four parameter model which has unbounded responses is widely used, for instance, in bioassays, nutrition, genetics, calibration and agriculture. In reality, the responses are often bounded although the bounds may be unknown, and in that situation, our model reflects the data-generating mechanism better. Complications arise for the new model, however, because the likelihood function is unbounded, and the global maximizers are not consistent estimators of unknown parameters. Although the two sample extremes, the smallest and the largest observations, are consistent estimators for the two unknown boundaries, they have a slow convergence rate and are asymptotically biased. Improved estimators are developed by correcting for the asymptotic biases of the two sample extremes in the one sample case; but even these consistent estimators do not obtain the optimal convergence rate. To obtain efficient estimation, we suggest using the local maximizers of the likelihood function, i.e., the solution to the likelihood equations. We prove that, with probability approaching one as the sample size goes to infinity, there exists a solution to the likelihood equation that is consistent at the rate of the square root of the sample size and it is asymptotically normally distributed.     

On the existence of locally mvu estimators

It is proved that there exists an unbiased estimator for some real parameter of a class of distributions, which has minimal variance for some fixed distribution among all corresponding unbiased estimators, if and. only if the corresponding minimal variances for all related unbiased estimation problems concerning finite subsets of the underlying family of distributions are bounded. As an application it is shown that there does not exist some unbiased estimator for θ^k+c(ε≥0) with minimal variance for θ =0 among all corresponding unbiased estimators on the base of k i.i.d. random variables with a Cauchy-distribution, where θ denotes some location parameter.     

Common correlated effect cross‐sectional dependence corrections for nonlinear conditional mean panel models

This paper provides an approach to estimation and inference for nonlinear conditional mean panel data models, in the presence of cross‐sectional dependence. We modify Pesaran's (Econometrica, 2006, 74(4), 967–1012) common correlated effects correction to filter out the interactive unobserved multifactor structure. The estimation can be carried out using nonlinear least squares, by augmenting the set of explanatory variables with cross‐sectional averages of both linear and nonlinear terms. We propose pooled and mean group estimators, derive their asymptotic distributions, and show the consistency and asymptotic normality of the coefficients of the model. The features of the proposed estimators are investigated through extensive Monte Carlo experiments. We also present two empirical exercises. The first explores the nonlinear relationship between banks' capital ratios and riskiness. The second estimates the nonlinear effect of national savings on national investment in OECD countries depending on countries' openness.     

On parameter estimation of partly observed bilinear discrete-time stochastic systems

The parameter estimation problem of a partly observed nonlinear discrete-time stochastic system is considered. The unobserved component of the system is a q-dimensional stable autoregressive process of the pth order with random parameters, observed in the presence of multiplicative and additive noises. The distributions of all the noises of the system are supposed to be unknown. The problem is to estimate the mean of the drifting parameters of the object and variances of the additive noises of the system. Asymptotic correlation estimators of all these parameters are investigated and sequential estimators with given mean square accuracy of the mean of the drifting autoregressive parameters are obtained.     

Estimating and Forecasting with a Dynamic Spatial Panel Data Model*

This study focuses on the estimation and predictive performance of several estimators for the dynamic and autoregressive spatial lag panel data model with spatially correlated disturbances. In the spirit of Arellano and Bond (1991) and Mutl (2006) , a dynamic spatial generalized method of moments (GMM) estimator is proposed based on Kapoor, Kelejian and Prucha (2007) for the spatial autoregressive (SAR) error model. The main idea is to mix non‐spatial and spatial instruments to obtain consistent estimates of the parameters. Then, a linear predictor of this spatial dynamic model is derived. Using Monte Carlo simulations, we compare the performance of the GMM spatial estimator to that of spatial and non‐spatial estimators and illustrate our approach with an application to new economic geography.     

Estimation of copula-based semiparametric time series models

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric marginal distributions and parametric copula functions, while the copulas capture all the scale-free temporal dependence of the processes. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. These results are used to obtain root-n consistent and asymptotically normal estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles. The semiparametric conditional quantile estimators are automatically monotonic across quantiles, which is attractive for portfolio conditional value-at-risk calculations.     

THE BIVARIATE EXPONENTIAL DISTRIBUTION; A REVIEW AND SOME NEW RESULTS

Various models have been proposed as bivariate forms of the exponential distribution. A brief but comprehensive review is presented which classifies, interrelates and contrasts the different models and outlines what is known about distributional properties, applicability and estimation and testing of parameters (particularly the association parameter). Some new results are presented for one particular model. Maximum likelihood, and moment–type, estimators of the association parameter are examined. Asymptotic variances are derived and attention is given to the relative efficiency of the estimators and to problems of their evaluation.