104,138 research outputs found
Targeted Learning of the Mean Outcome Under an Optimal Dynamic Treatment Rule
We consider estimation of and inference for the mean outcome under the optimal dynamic two time-point treatment rule defined as the rule that maximizes the mean outcome under the dynamic treatment, where the candidate rules are restricted to depend only on a user-supplied subset of the baseline and intermediate covariates. This estimation problem is addressed in a statistical model for the data distribution that is nonparametric beyond possible knowledge about the treatment and censoring mechanism. This contrasts from the current literature that relies on parametric assumptions. We establish that the mean of the counterfactual outcome under the optimal dynamic treatment is a pathwise differentiable parameter under conditions, and develop a targeted minimum loss-based estimator (TMLE) of this target parameter. We establish asymptotic linearity and statistical inference for this estimator under specified conditions. In a sequentially randomized trial the statistical inference relies upon a second order difference between the estimator of the optimal dynamic treatment and the optimal dynamic treatment to be asymptotically negligible, which may be a problematic condition when the rule is based on multivariate time-dependent covariates. To avoid this condition, we also develop targeted minimum loss based estimators and statistical inference for data adaptive target parameters that are defined in terms of the mean outcome under the estimate of the optimal dynamic treatment. In particular, we develop a novel cross-validated TMLE approach that provides asymptotic inference under minimal conditions, avoiding the need for any empirical process conditions. We offer simulation results to support our theoretical findings. This work expands upon that of an earlier technical report (van der Laan, 2013; van der Laan and Luedtke, 2014) with new results and simulations, and is accompanied by a work which explores the estimation of the optimal rule (Luedtke and van der Laan, 2014)
High-dimensional estimation with geometric constraints
Consider measuring an n-dimensional vector x through the inner product with
several measurement vectors, a_1, a_2, ..., a_m. It is common in both signal
processing and statistics to assume the linear response model y_i = +
e_i, where e_i is a noise term. However, in practice the precise relationship
between the signal x and the observations y_i may not follow the linear model,
and in some cases it may not even be known. To address this challenge, in this
paper we propose a general model where it is only assumed that each observation
y_i may depend on a_i only through . We do not assume that the
dependence is known. This is a form of the semiparametric single index model,
and it includes the linear model as well as many forms of the generalized
linear model as special cases. We further assume that the signal x has some
structure, and we formulate this as a general assumption that x belongs to some
known (but arbitrary) feasible set K. We carefully detail the benefit of using
the signal structure to improve estimation. The theory is based on the mean
width of K, a geometric parameter which can be used to understand its effective
dimension in estimation problems. We determine a simple, efficient two-step
procedure for estimating the signal based on this model -- a linear estimation
followed by metric projection onto K. We give general conditions under which
the estimator is minimax optimal up to a constant. This leads to the intriguing
conclusion that in the high noise regime, an unknown non-linearity in the
observations does not significantly reduce one's ability to determine the
signal, even when the non-linearity may be non-invertible. Our results may be
specialized to understand the effect of non-linearities in compressed sensing.Comment: This version incorporates minor revisions suggested by referee
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Robust misspecification tests for the Heckman’s two-step estimator
We construct and evaluate LM and Neyman’s C(α) tests based on bivariate Edgeworth expansions for the consistency of the Heckman’s two-step estimator in selection models, that is, for the marginal normality and linearity of the conditional expectation of the error terms. The proposed tests are robust to local misspecification in nuisance distributional parameters. Monte Carlo results show that instead of testing bivariate normality, testing marginal normality and linearity of the conditional expectations separately have a better size performance. Moreover, the robust variants of the tests have better size and similar power to non-robust tests, which determines that these tests can be successfully applied to detect specific departures from the null model of bivariate normality. We apply the tests procedures to women’s labor supply data
Asymptotically Optimal Tests when Parameters are Estimated
The main purpose of this paper is to provide an asymptotically optimal test.
The proposed statistic is of Neyman-Pearson-type when the parameters are
estimated with a particular kind of estimators. It is shown that the proposed
estimators enable us to achieve this end. Two particular cases, AR(1) and ARCH
models were studied and the asymptotic power function was derived
On Convex Least Squares Estimation when the Truth is Linear
We prove that the convex least squares estimator (LSE) attains a
pointwise rate of convergence in any region where the truth is linear. In
addition, the asymptotic distribution can be characterized by a modified
invelope process. Analogous results hold when one uses the derivative of the
convex LSE to perform derivative estimation. These asymptotic results
facilitate a new consistent testing procedure on the linearity against a convex
alternative. Moreover, we show that the convex LSE adapts to the optimal rate
at the boundary points of the region where the truth is linear, up to a log-log
factor. These conclusions are valid in the context of both density estimation
and regression function estimation.Comment: 35 pages, 5 figure
Efficient inference about the tail weight in multivariate Student distributions
We propose a new testing procedure about the tail weight parameter of
multivariate Student distributions by having recourse to the Le Cam
methodology. Our test is asymptotically as efficient as the classical
likelihood ratio test, but outperforms the latter by its flexibility and
simplicity: indeed, our approach allows to estimate the location and scatter
nuisance parameters by any root- consistent estimators, hereby avoiding
numerically complex maximum likelihood estimation. The finite-sample properties
of our test are analyzed in a Monte Carlo simulation study, and we apply our
method on a financial data set. We conclude the paper by indicating how to use
this framework for efficient point estimation.Comment: 23 page
Moment-based estimation of smooth transition regression models with endogenous variables
Nonlinear regression models have been widely used in practice for a variety of time series and cross-section datasets. For purposes of analyzing univariate and multivariate time series data, in particular, Smooth Transition Regression (STR) models have been shown to be very useful for representing and capturing asymmetric behavior. Most STR models have been applied to univariate processes, and have made a variety of assumptions, including stationary or cointegrated processes, uncorrelated, homoskedastic or conditionally heteroskedastic errors, and weakly exogenous regressors. Under the assumption of exogeneity, the standard method of estimation is nonlinear least squares. The primary purpose of this paper is to relax the assumption of weakly exogenous regressors and to discuss moment based methods for estimating STR models. The paper analyzes the properties of the STR model with endogenous variables by providing a diagnostic test of linearity of the underlying process under endogeneity, developing an estimation procedure and a misspecification test for the STR model, presenting the results of Monte Carlo simulations to show the usefulness of the model and estimation method, and providing an empirical application for inflation rate targeting in Brazil. We show that STR models with endogenous variables can be specified and estimated by a straightforward application of existing results in the literature.Smooth transition, nonlinear models, nonlinear instrumental variables, generalized method of moments, endogeneity, inflation targeting.
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