Kairat T. Mynbaev

CENTRAL LIMIT THEOREMS FOR WEIGHTED SUMS OF LINEAR PROCESSES: LP -APPROXIMABILITY VERSUS BROWNIAN MOTION

Standardized slowly varying regressors are shown to be L -approximable. This fact allows us to provide alternative proofs of asymptotic expansions of nonstochastic quantities and central limit results due to P.C.B. Phillips, under a less stringent assumption on linear processes. The recourse to stochastic calculus related to Brownian motion can be completely dispensed with.

Regressions with asymptotically collinear regressors

We investigate the asymptotic behavior of the OLS estimator for regressions with two slowly varying regressors. It is shown that the asymptotic distribution is normal one-dimensional and may belong to one of four types depending on the relative rates of growth of the regressors. The analysis establishes, in particular, a new link between slow variation and

Reducing bias in nonparametric density estimation via bandwidth dependent kernels: L1 view

L_p

A class of nonparametric density derivative estimators based on global Lipschitz conditions

-approximability. A revised version of this paper has been published in Econometrics Journal (2011), volume 14, pp. 304--320.

Regressions with Asymptotically Collinear Regressor

We define a new bandwidth-dependent kernel density estimator that improves existing convergence rates for the bias, and preserves that of the variation, when the error is measured in L1. No additional assumptions are imposed to the extant literature.

Consistency and asymptotic normality for a nonparametric prediction under measurement errors

Abstract Estimators for derivatives associated with a density function can be useful in identifying its modes and inflection points. In addition, these estimators play an important role in plug-in methods associated with bandwidth selection in nonparametric kernel density estimation. In this paper, we extend the nonparametric class of density estimators proposed by Mynbaev and Martins-Filho (2010) to the estimation of m-order density derivatives. Contrary to some existing derivative estimators, the estimators in our proposed class have a full asymptotic characterization, including uniform consistency and asymptotic normality. An expression for the bandwidth that minimizes an asymptotic approximation for the estimators’ integrated squared error is provided. A Monte Carlo study sheds light on the finite sample performance of our estimators and contrasts it with that of density derivative estimators based on the classical Rosenblatt–Parzen approach.

Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model

We investigate the asymptotic behavior of the OLS estimator for regressions with two slowly varying regressors. It is shown that the asymptotic distribution is normal one-dimensional and may belong to one of four types depending on the relative rates of growth of the regressors. The analysis establishes, in particular, a new link between slow variation and

Asymptotic distribution of the OLS estimator for a mixed spatial model

L_p

Convergence Almost Everywhere

-approximability. A revised version of this paper has been published in Econometrics Journal (2011), volume 14, pp. 304--320.

Improving bias in kernel density estimation

Nonparametric prediction of a random variable Y conditional on the value of an explanatory variable X is a classical and important problem in Statistics. The problem is significantly complicated if there are heterogeneously distributed measurement errors on the observed values of X used in estimation and prediction. Carroll et?al. (2009) have recently proposed a kernel deconvolution estimator and obtained its consistency. In this paper we use the kernels proposed in Mynbaev and Martins-Filho (2010) to define a class of deconvolution estimators for prediction that contains their estimator as one of its elements. First, we obtain consistency of the estimators under much less restrictive conditions. Specifically, contrary to what is routinely assumed in the extant literature, the Fourier transform of the underlying kernels is not required to have compact support, higher-order restrictions on the kernel can be avoided and fractional smoothness of the involved densities is allowed. Second, we obtain asymptotic normality of the estimators under the assumption that there are two types of measurement errors on the observed values of X . It is apparent from our study that even in this simplified setting there are multiple cases exhibiting different asymptotic behavior. Our proof focuses on the case where measurement errors are super-smooth and we use it to discuss other possibilities. The results of a Monte Carlo simulation are provided to compare the performance of the estimator using traditional kernels and those proposed in Mynbaev and Martins-Filho (2010).