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Register a new userSemiparametric efficiency in GMM models with auxiliary data
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Added by
Alessandro Tarozzi
Publication date
2008
fields
Mathematical Statistics
and research's language is
English
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We study semiparametric efficiency bounds and efficient estimation of parameters defined through general moment restrictions with missing data. Identification relies on auxiliary data containing information about the distribution of the missing variables conditional on proxy variables that are observed in both the primary and the auxiliary database, when such distribution is common to the two data sets. The auxiliary sample can be independent of the primary sample, or can be a subset of it. For both cases, we derive bounds when the probability of missing data given the proxy variables is unknown, or known, or belongs to a correctly specified parametric family. We find that the conditional probability is not ancillary when the two samples are independent. For all cases, we discuss efficient semiparametric estimators. An estimator based on a conditional expectation projection is shown to require milder regularity conditions than one based on inverse probability weighting.
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We consider X 1 ,. .. , X n a sample of data on the circle S 1 , whose distribution is a twocomponent mixture. Denoting R and Q two rotations on S 1 , the density of the X i s is assumed to be g(x) = pf (R --1 x) + (1 -- p)f (Q --1 x), where p $in$ (0, 1) and f is an unknown density on the circle. In this paper we estimate both the parametric part $theta$ = (p, R, Q) and the nonparametric part f. The specific problems of identifiability on the circle are studied. A consistent estimator of $theta$ is introduced and its asymptotic normality is proved. We propose a Fourier-based estimator of f with a penalized criterion to choose the resolution level. We show that our adaptive estimator is optimal from the oracle and minimax points of view when the density belongs to a Sobolev ball. Our method is illustrated by numerical simulations.
It is well known that the minimax rates of convergence of nonparametric density and regression function estimation of a random variable measured with error is much slower than the rate in the error free case. Surprisingly, we show that if one is willing to impose a relatively mild assumption in requiring that the error-prone variable has a compact support, then the results can be greatly improved. We describe new and constructive methods to take full advantage of the compact support assumption via spline-assisted semiparametric methods. We further prove that the new estimator achieves the usual nonparametric rate in estimating both the density and regression functions as if there were no measurement error. The proof involves linear and bilinear operator theories, semiparametric theory, asymptotic analysis regarding Bsplines, as well as integral equation treatments. The performance of the new methods is demonstrated through several simulations and a data example.
This paper develops theory for feasible estimators of finite-dimensional parameters identified by general conditional quantile restrictions, under much weaker assumptions than previously seen in the literature. This includes instrumental variables nonlinear quantile regression as a special case. More specifically, we consider a set of unconditional moments implied by the conditional quantile restrictions, providing conditions for local identification. Since estimators based on the sample moments are generally impossible to compute numerically in practice, we study feasible estimators based on smoothed sample moments. We propose a method of moments estimator for exactly identified models, as well as a generalized method of moments estimator for over-identified models. We establish consistency and asymptotic normality of both estimators under general conditions that allow for weakly dependent data and nonlinear structural models. Simulations illustrate the finite-sample properties of the methods. Our in-depth empirical application concerns the consumption Euler equation derived from quantile utility maximization. Advantages of the quantile Euler equation include robustness to fat tails, decoupling of risk attitude from the elasticity of intertemporal substitution, and log-linearization without any approximation error. For the four countries we examine, the quantile estimates of discount factor and elasticity of intertemporal substitution are economically reasonable for a range of quantiles above the median, even when two-stage least squares estimates are not reasonable.
In this paper we develop an online statistical inference approach for high-dimensional generalized linear models with streaming data for real-time estimation and inference. We propose an online debiased lasso (ODL) method to accommodate the special structure of streaming data. ODL differs from offline debiased lasso in two important aspects. First, in computing the estimate at the current stage, it only uses summary statistics of the historical data. Second, in addition to debiasing an online lasso estimator, ODL corrects an approximation error term arising from nonlinear online updating with streaming data. We show that the proposed online debiased estimators for the GLMs are consistent and asymptotically normal. This result provides a theoretical basis for carrying out real-time interim statistical inference with streaming data. Extensive numerical experiments are conducted to evaluate the performance of the proposed ODL method. These experiments demonstrate the effectiveness of our algorithm and support the theoretical results. A streaming dataset from the National Automotive Sampling System-Crashworthiness Data System is analyzed to illustrate the application of the proposed method.
Let $X_1,dots, X_n$ be i.i.d. random variables sampled from a normal distribution $N(mu,Sigma)$ in ${mathbb R}^d$ with unknown parameter $theta=(mu,Sigma)in Theta:={mathbb R}^dtimes {mathcal C}_+^d,$ where ${mathcal C}_+^d$ is the cone of positively definite covariance operators in ${mathbb R}^d.$ Given a smooth functional $f:Theta mapsto {mathbb R}^1,$ the goal is to estimate $f(theta)$ based on $X_1,dots, X_n.$ Let $$ Theta(a;d):={mathbb R}^dtimes Bigl{Sigmain {mathcal C}_+^d: sigma(Sigma)subset [1/a, a]Bigr}, ageq 1, $$ where $sigma(Sigma)$ is the spectrum of covariance $Sigma.$ Let $hat theta:=(hat mu, hat Sigma),$ where $hat mu$ is the sample mean and $hat Sigma$ is the sample covariance, based on the observations $X_1,dots, X_n.$ For an arbitrary functional $fin C^s(Theta),$ $s=k+1+rho, kgeq 0, rhoin (0,1],$ we define a functional $f_k:Theta mapsto {mathbb R}$ such that begin{align*} & sup_{thetain Theta(a;d)}|f_k(hat theta)-f(theta)|_{L_2({mathbb P}_{theta})} lesssim_{s, beta} |f|_{C^{s}(Theta)} biggr[biggl(frac{a}{sqrt{n}} bigvee a^{beta s}biggl(sqrt{frac{d}{n}}biggr)^{s} biggr)wedge 1biggr], end{align*} where $beta =1$ for $k=0$ and $beta>s-1$ is arbitrary for $kgeq 1.$ This error rate is minimax optimal and similar bounds hold for more general loss functions. If $d=d_nleq n^{alpha}$ for some $alphain (0,1)$ and $sgeq frac{1}{1-alpha},$ the rate becomes $O(n^{-1/2}).$ Moreover, for $s>frac{1}{1-alpha},$ the estimators $f_k(hat theta)$ is shown to be asymptotically efficient. The crucial part of the construction of estimator $f_k(hat theta)$ is a bias reduction method studied in the paper for more general statistical models than normal.
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